ISSN sub w ww divided by hline to the power of B to the power of hline from hline to Banach a n sub : sub period 1 e 735 m i s hyphen\ begin 8787{ a l i periodg n ∗} sub d e slash j open parenthesis o electronic u r n al s slash B closing parenthesis sub J M A slash to the power ofISSN a c h ˆ J{ subB J ˆ{\ periodr u l Math e {3em o}{ u r0.4 n apt period} ˆ{\ subr u l l Anale {3em o}{ f period0.4 pt 5}} M a{ tBanach open parenthesis}} a 20 n h e} 1{ mw 1 a\ closingf r a c { parenthesisww }{\ r u l e {3em}{0.4 pt }}}ˆ{ a tc sub h i to the J power{ J } of comma. Math sub c no{ ao l period u 1 r A comma n a sub} n. 1 sub{ al 0} l 1Anal y endash{ subo s i f 1} s 35. Capital 5 Sigma{ M hyphen a } t ( 20 { h e } 1 { m } 1 { a } ) { t }ˆ{ , } { , i } { c } no { a l } . 1 { A } CONVERGENCE B Banach ana chJJ.Matho urna.l Analof .5Mat(20he1m1a)ticnoal. 1A,n1a0l1y −−si1s35 ISSNw ww , GABRIEL{ n } 1 NGUETSENG{ a } 0 to{ thel power} :.1e1735 ofmis{ 1−y AND8787}.de NILS −−/j(oelectronic SVANSTEDT{ surnals i /B})J to1MA the{ powers /} of35 2 *} { : { . } 1 { e } 735 { m iCommunicated s } − 8787 by C{ period. } Badea{ d e / j } ( { o } e l e c t r oΣ n− i cCONVERGENCE{ u r n a l s / B } ) Abstract{ J } periodMA/ .. We discuss}\\\ twoSigma new concepts− CONVERGENCE of convergence in L to the power of p hyphen spaces comma the so hyphen 1 2∗ \endcalled{ a l i weak g n ∗} Capital Sigma hyphenGABRIEL convergence NGUETSENG and strongAND Capital NILS Sigma SVANSTEDT hyphen convergence comma which are intermediate between classical weak convergence and strongCommunicated convergence by period C . Badea We also introduce \ centerlinethe concept{ ofGABRIEL CapitalAbstract Sigma $ . NGUETSENG hyphenWe discuss convergence ˆ{ two1 new}$ for concepts ANDRadon NILS measures of convergence $ SVANSTEDT period in OurLp− basic ˆspaces{ 2 tool , isthe∗ the } so$ classi - } hyphen cal Gelfand representationcalled weak theory Σ− convergence period Apart and from strong being Σ− a naturalconvergence generalization , which are of intermediate \ centerlinewell hyphen{ knownCommunicatedbetween two hyphen classical by scale C weak convergence . Badea convergence} theory and comma strong theconvergence present study . We lays also the introduce foundation of the mathematicalthe frameworkconcept of Σ that− convergence is needed to for undertake Radon measures a systematic . Our study basic tool is the classi - \ centerlineof deterministic{ Abstract homogenizationcal Gelfand . \quad representation problemsWe discuss beyond theory the two . Apart usual new fromperiodic concepts being setting a natural of period convergence generalization in of $ L ˆ{ p } − $ spaces , the so − } A few homogenizationwell - known problems two are - scale worked convergence out by way theory of illustration , the present period study lays the foundation \ centerline1 period .. Introduction{ calledof the weak mathematical $ \Sigma framework− $ that convergence is needed to undertake and strong a systematic $ \Sigma study − $ convergence , which are intermediate } To systematically passof deterministic to the l imit homogenization in a product of problemstwo weakly beyond convergent the usual s e hyphen periodic setting . \ centerlinequences one{ classicallybetweenA requires classical few homogenization that open weak parenthesis convergence problems at are least worked and closing outstrong parenthesis by way convergence of illustration one of the two . We s equences also introduce converges } strongly period .... More precisely comma ....1 .let CapitalIntroduction Omega be an open set in the N hyphen dimensional numerical \ centerlinespace R to{ thetheTo power systematically concept of N open of parenthesis $ pass\Sigma to the N greater l− imit$ equal in convergence a product 1 closing ofparenthesis for two Radon weakly comma measures convergent let open . parenthesis Our s e - basic u sub tool epsilon is the classi − } closing parenthesisquences sub one epsilon classically greater requires0 be a sequence that ( in at L least to the ) power one of of the p open two parenthesis s equences Capital converges Omega closing parenthesis open\ centerline parenthesisstrongly{ cal Capital . Gelfand OmegaMore precisely representationprovided with , let LebesgueΩ be theory an open . Apart set in from the N being− dimensional a natural numerical generalization of } N p measurespace closingR parenthesis(N ≥ 1), andlet let(uε open)ε>0 parenthesisbe a sequence v sub in epsilonL (Ω)(Ω closingprovided parenthesis with sub Lebesgue epsilon greater 0 be a s equence in L \ centerline { w e l l − known two − scale convergencep0 theory , the present1 study laysp the foundation } to the powermeasure of p to the ) and power let of( primevε)ε>0 openbe a parenthesis s equence Capital in L (Ω) Omega, where closing1 < parenthesis p < ∞ and commap 0 =1 where− 1 less1. p less infinity and p p0 p 1 dividedIt by i prime s a classical = 1 minus fact hline that from if pu toε → 1 subu in periodL (Ω) ( strong ) and vε → v in L (Ω)− weak \ centerline { of the mathematical framework1 that is needed to undertake a systematic study } It i s a classicalas ε → fact0, then that ifu uεv subε → epsilonu0v0 in rightL arrow(Ω)− weak u in L to. the power of p open parenthesis Capital Omega closing parenthesis open parenthesisHowever strong closing , in parenthesis a great number and v sub of s epsilon ituations right aris arrow ing v in in mathematicalL to the power of analysis p to the it power i of prime open parenthesis\ centerlines Capital often{ of Omegacrucial deterministic closing to investigate parenthesis homogenization the hyphen l imiting weak behaviorsproblems of beyond products the of usual the preceding periodic form setting . } as epsilonin right spite arrow of the 0 factcomma that then u none sub epsilon of the vtwo sub sequences epsilon right i s arrow allowed u sub to0 strongly v sub 0 in L to the power of 1 open parenthesis\ centerline Capital{A few Omega homogenization closing parenthesis problems hyphen weak are period worked out by way of illustration . } However comma .. in a great number of s ituations aris ing in mathematical analysis it \ centerlinei s often crucial{1 .to\ investigatequad Introduction the l imiting behaviors} of products of the preceding form in spite of the fact that .. none of the two sequences i s allowed to strongly \ hspacehline ∗{\ f i l l }To systematicallyDate : Received pass : 4 Mayto the 20 1l 0 ; imit Accepted in :a 9 product July 20 1 0 of . two weakly convergent s e − Date : Received : 4 May 20 1 0 semicolon Accepted∗ Corresponding : 9 July 20 author 1 0 period . \noindent* Corresponding201quences 0 Mathematics author one period classically Subject Classification requires . thatPrimary ( at46 J least 1 0 ; Secondary ) one of 35 B the 40 , two 28 A s 33 equences . converges 201 0 MathematicsKey words Subject and phrases Classification . Homogenization period .. Primary , homogenization 46 J 1 0 semicolon algebras Secondary, Σ− convergence 35 B 40 comma , Gelfand 28 A 33 period \noindentKey wordstransformations and t r o n phrases g l y . . period\ h f i l.. l HomogenizationMore precisely comma , \ homogenizationh f i l l l e t $ algebras\Omega comma$ be Capital an open Sigma set hyphen in convergence the $N −comma$ dimensional Gelfand numerical 1 1 transformation period \noindent1 1 space $ R ˆ{ N } (N \geq 1 ) , $ l e t $ ( u {\ varepsilon } ) {\ varepsilon > 0 }$ be a sequence in $L ˆ{ p } ( \Omega )( \Omega $ provided with Lebesgue

\noindent measure ) and let $ ( v {\ varepsilon } ) {\ varepsilon > 0 }$ be a s equence in $ L ˆ{ p ˆ{\prime }} ( \Omega ) , $ where $ 1 < p < \ infty $ and $ p \ f r a c { 1 }{\prime } = 1 − \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } { . }$

\noindent It i s a classical fact that if $ u {\ varepsilon }\rightarrow u $ in $ L ˆ{ p } ( \Omega ) ($ strong )and $v {\ varepsilon }\rightarrow v $ in $ L ˆ{ p ˆ{\prime }} ( \Omega ) − $ weak

\noindent as $ \ varepsilon \rightarrow 0 , $ then $ u {\ varepsilon } v {\ varepsilon } \rightarrow u { 0 } v { 0 }$ in $ L ˆ{ 1 } ( \Omega ) − $ weak .

However , \quad in a great number of s ituations aris ing in mathematical analysis it i s often crucial to investigate the l imiting behaviors of products of the preceding form in spite of the fact that \quad none of the two sequences i s allowed to strongly

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline {Date : Received : 4 May 20 1 0 ; Accepted : 9 July 20 1 0 . }

\ centerline { $ ∗ $ Corresponding author . }

\ centerline {201 0 Mathematics Subject Classification . \quad Primary 46 J 1 0 ; Secondary 35 B 40 , 28 A 33 . }

Key words and phrases . \quad Homogenization , homogenization algebras $ , \Sigma − $ convergence , Gelfand transformation .

\ centerline {1 1 } 1 2 .. G period NGUETSENG comma N period SVANSTEDT \noindentconverge period1 2 \ ..quad ForG example . NGUETSENG in homogenization , N . SVANSTEDT theory open square bracket 3 6 comma .. 1 4 comma .. 3 7 comma .. 3 commaconverge .. 3 5 comma. \quad .. 3For 8 comma example .. 2 comma in homogenization 2 .. 1 closing square theory bracket [ it 3 is 6 , \quad 1 4 , \quad 3 7 , \quad 3 , \quad 3 5 , \quad 3 8 , \quad 2 , 2 \quad 1 ] i t i s 1 2 G . NGUETSENG , N . SVANSTEDT converge . For example in homogenization frequent to have to compute l imits such as \noindenttheoryfrequent [ 3 6, to have 1 4 , to 3 compute 7 , 3 , l imits 3 5 , such 3 8 , as 2 , 2 1 ] it is Equation:frequent open parenthesis to have1 to period compute 1 closing l imits parenthesis such as .. limint less 0 epsilon right arrow 0 integral sub Capital Omega u sub epsilon open parenthesis x closing parenthesis psi open parenthesis x comma x divided by epsilon to the power of closing parenthesis dx\ begin comma{ a l i g n ∗} \limwhere{ ....< u sub{ 0 epsilon}\ rightvarepsilon arrow u .... in\rightarrow .... L toZ the power0 of}\ px ) openint parenthesis{\Omega Capital} u Omega{\ closingvarepsilon parenthesis} hyphen( x ) \ psi ( x , \ f r a c { x }{\ varepsilonlim uε(x)}ψˆ({x, ) dx,} dx , \ tag ∗{$ ( 1(1 ..1) 1 ) $} weak as .... epsilon right arrow 0 comma .... and< psi0ε→ in0 LΩ to the powerε of p to the power of prime open parenthesis Capital Omega semicolon\end{ a l i Cg n sub∗} per open parenthesis Y closing parenthesis closing parenthesis .... with Y = p p0 parenleftbigwhere minusuε 1→ dividedu in byL 2(Ω) sub− commaweak as1 dividedε → by0 2, and parenrightbigψ ∈ toL the(Ω; C powerper(Y )) ofwith N subY comma= C sub per open parenthesis\noindent( Y−where closing1 1 )N parenthesis\Ch f( iY l l) $ .... u being{\being thevarepsilon space the space of those}\ of continuous thoserightarrow continuous complexu functions complex $ \ h f f functionsoni l l in \ hf f ion l l $ L ˆ{ p } ( \Omega ) 2 , 2−, $per weak as \ h f i l l $ \ varepsilon \rightarrow 0 , $ \ h f i l l and $ \ psi \ in R to the powerN of N .... that .... are .... Y hyphen periodic comma .... i period e period comma ....N that .... satisfy .... f open L ˆ{ p ˆ{\R primethat are}} Y −( periodic\Omega , i .;C e . , that{ per satisfy} (f(y + Yk) )= f )(y $) for\ hy f i l∈ l withR and $ Y = $ parenthesis y plusN k closing parenthesis = f open parenthesis y closing parenthesis .... for .... y in R to the power of N .... and k ∈ Z (Z denotes the integers ), Cper(Y ) provided with the supremum norm . It i k in Z tos the of power interest of N to open recall parenthesis here that Z denotes the integers closing parenthesis comma C sub per open parenthesis Y closing \noindent $ ( − \ f r a c { 1 }{ 2 } { , }\ f r a c { 1 }{ 2 } 0) ˆ{ N } { , } C { per } ( Y ) $ parenthesis provided with the supremum norm period .. It ε p \ h f i l l being the space of those continuous complexψ → functionsψe in L (Ω)− $weak f $ as on ε → 0, (1.2) i s of interest to recallε here that x R where ψ (x) = ψ(x, ε ) and ψe(x) = Y ψ(x, y)dy for x ∈ Ω( see , e . g . , [ 2 psi to the6 ] power ) . of epsilon right arrow psi-tildewide in L to the power of p to the power of prime open parenthesis Capital Omega\noindent closing$ parenthesis R ˆ{ N hyphen}$ \ h weak f i l l asthat epsilon\ h right f i l l arroware 0\ commah f i l l open$ Y parenthesis− $ 1 p periode r i o d i 2 c closing , \ h parenthesis f i l l i . e . , \ h f i l l that \ h f i l l s a t i s f y \ h f i l l Furthermore , unless ψ is constant with respect to the periodicity variable y = $f(y+k)=f(y)$where psi to the power of epsilon open parenthesis x closing\ h parenthesis f i l l f o r =\ h psi f i l parenleftbig l $ y \ xin commaR x ˆ{ dividedN }$ by\ epsilonh f i l l and (y1, ..., yN)( this is a quite trivial o ccurrence ) , it is hopeless to try to get strong parenrightbig .... and psi-tildewide open parenthesis x closing parenthesis = integral sub Y psi open parenthesis x comma y closing \noindentconvergence$ k \ in in ( 1Z . ˆ 2{ )N } ( see( , e Z$ . g . ,denotes [ 3 ] ) the . integersThus , it i s $) beyond , the C classical{ per } ( Y ) $ parenthesisresources dy .... for x of in mathematicalCapital Omega open analysis parenthesis to compute see comma the .... l imit e period in ( g 1 period . 1 ) . comma .... open square bracket 2 .... 6provided closing square with bracket the closingsupremum parenthesis norm period. \quad I t i s of interestIt was to precisely recall to here overcome that such difficulties that the first author introduced in Furthermore1 989 comma basic .. ideas unless on psi two .. is -.. scale constant convergence with respect ( to see the periodicity [ 2 7 ] ) variable . Shortly y = after , the open parenthesisdirection y 1pointed comma out period by period further period pioneering comma papersy N closing ( parenthesis s ee [ 2 open 8 parenthesis , 1 ] ) this on is a quite trivial o ccurrence\ hspace ∗{\ closingf i l parenthesis l } $ \ psi commaˆ{\ varepsilon it is hopeless to}\ try torightarrow get strong \ widetilde {\ psi } $ in $ L ˆ{ p ˆ{\prime }} ( \Omegatwo -) s cale− convergence$ weak as init $ \ iatedvarepsilon a great activity\rightarrow that increased0 ,in interest ( 1 over . the 2 ) $ convergenceyears in . open See parenthesis , e . g 1. period , [ 2 2 3closing ] and parenthesis the references .. open therein parenthesis . see comma e period g period comma .. open square bracket 3 closing square bracket closing parenthesis period .. Thus comma it i s beyond the classical resources of \noindent whereWithout $ going\ psi toˆ{\ deeplyvarepsilon into details} ( , xlet ) us recall = \ thepsi main( ideas x underlying , \ f r a c { x }{\ varepsilon } mathematicaltwo - analysis scale convergence to compute the theory l imit in. open To parenthesis begin , for 1 period the benefit 1 closing of parenthesis the reader period it should ) $It was\ h preciselyf i l l and to overcome $ \ widetilde such difficulties{\ psi } that( the first x author ) = introduced\ int in{ Y }\ psi ( x , y ) dy $ be reminded that a sequence (u ) in Lp(Ω) (1 ≤ p < ∞) is said to weakly two - \ h1 f i 989 l l basicf o r ideas $ x on two\ in hyphen\Omega scale convergence( $ε seeε> open0 , parenthesis\ h f i l l e see . .. g open . , square\ h f i lbracket l [ 2 2\ 7h closing f i l l 6 square ] ) bracket . closing scale converge in Lp(Ω) to some u ∈ Lp(Ω; Lp (Y )) (Lp (Y ) stands for the parenthesis period .. Shortly after comma the direction 0 per per \noindent Furthermore , \quad u n l e s s $ \ psi $ p \quadN ) i s \quad constant with respect to the periodicity variable pointedspace out by furtherof Y − pioneeringperiodic paperscomplex .. open functions parenthesis in L sloc ee(R ..y open) if square as ε → bracket0, 2 .. 8 comma .. 1 closing square bracket $ y = $ closing parenthesis .. on two hyphen sZ cale convergence ZZ $( y 1 , . . . ,ε y N ) ($ this isaquite trivial occurrence) , it is hopelesstotrytogetstrong init iated a great activity that increaseduε(x in)ψ interest(x)dx over→ the yearsu0( periodx, y)ψ(x, .. Seey)dxdy comma e period g period comma .. open square bracketconvergence 2 3 closing in square ( 1 bracket . 2 ) \quadΩ ( see , e . gΩ× .Y , \quad [ 3 ] ) . \quad Thus , it i s beyond the classical resources of mathematical analysisp0 to compute the l imit in ( 1 . 1q ) . and thefor references all ψ therein∈ L period(Ω; Cper(Y )). A sequence (vε)ε>0 in L (Ω) (1 ≤ q < ∞) i s said to q q q Withoutstrongly going to deeply two - into scale details converge comma in ..L let(Ω) .. usto recall some thev main0 ∈ L ideas(Ω; L underlyingper(Y )) i f for all η > 0 and \ hspacetwo hyphen∗{\ f i scale lq l } It convergence was precisely theory period to ..overcome To begin comma such difficultiesforη the benefit of thatthe reader the it first should authorbe introduced in f ∈ L (Ω; Cper(Y )) satisfying k v0 − f kLq(Ω×Y )≤ 2 , one can find some α > 0 such that reminded that aε sequence .. open parenthesis u sub epsilon closing parenthesis sub epsilon greater 0 in L to the power of p open k vε − f k ≤ η provided 0 < ε ≤ α. parenthesis\noindent Capital1 989 Omega basicLq(Ω) closing ideas parenthesis on two open− scale parenthesis convergence 1 less or equal ( see p less\ infinityquad [ closing 2 7 parenthesis ] ) . \quad .. is saidShortly to weakly after , the direction If (u ) and (v ) are as above ( with the respective assigned two - scale con - twopointed hyphen out byε furtherε>0 pioneeringε ε>0 papers \quad ( s ee \quad [ 2 \quad 8 , \quad 1 ] ) \quad on two − s cale convergence vergence properties ) , it can be shown that when ε → 0, the sequence initscale iated converge a in great L to the activity power of p that open parenthesis increased Capital in interest Omega closing over parenthesis the years .... to . some\quad u subSee 0 in , L e to . the g power . , \quad [ 2 3 ] (u ) u Lp(Ω) ( u (x) = R u (x, y)dy, x ∈ Ω) and the referencesε ε>0 weakly therein converges . to e0 in with e0 Y 0 whereas of p parenleftbig Capital Omega semicolon L sub per to the power of p open parenthesisq Y closing parenthesis parenrightbig open (vε)ε>0 weakly converges to ve0 ( defined as ue0) in L (Ω) and further , there is parenthesisno L sub reason per to for the ourpower assuming of p open that parenthesis one of Y those closing two parenthesis sequences .... stands is strongly for the convergent . \ hspacespace of∗{\ Yf hyphen i l l }Without periodic complex going functions to deeply in L into sub loc details to the power , \ ofquad p parenleftbigl e t \quad R subus y to recall the power the of main N to the ideas power underlying Nevertheless , lett ing 1 = p + 1 and assuming that r ≥ 1, it can be shown that of parenrightbig closing parenthesis if asr epsilon right1 q arrow 0 comma r \noindentintegralwhen subtwo Capital−ε →scale0 Omega, the convergence s u equence sub epsilon( open theoryuεvε) parenthesisε>0 weakly . \quad x converges closingTo begin parenthesis in ,L for(Ω) psi the toto the thebenefit power function of of epsilon the open reader parenthesis it should be xreminded closing parenthesis that a dx sequence right arrow\quad integral$ integral ( u sub{\ Capitalvarepsilon Omega times} ) Y u{\ subvarepsilon 0 open parenthesis> x0 comma}$ in y closing $ L ˆ{ p } parenthesis( \Omega psi open) parenthesis ( 1 x\ commaleq yp closing< parenthesis\ infty dxdy) $ \quad is said to weakly two − (x ∈ Ω). z(x) = R u (x, y)v (x, y)dy for all psi in L to the power of p to the power of prime open parenthesis Capital Omega semicolonY 0 C0 sub per open parenthesis Y \noindent scale converge in $ L ˆ{ p } ( \Omega ) $ \ h f i l l to some $ u { 0 }\ in L ˆ{ p } closing parenthesisAs closing might parenthesis be expected period .. A , sequence strong open two parenthesis- scale convergence v sub epsilon closing implies parenthesis weak sub epsilon greater 0( in L\Omega to the power; of q L open ˆ{ p parenthesis} { per Capital} ( Omega Y closing ) ) parenthesis ( L open ˆ{ p parenthesis} { per 1} less( or equal Y q ) less $ infinity\ h f i closingl l stands for the two - scale convergence . The function u0 ( resp . v0) above is unique parenthesisand i s said is referredto to as the weak ( resp . strong ) two - scale limit of the \noindentstrongly twospace hyphen o f scale $ Yconverge− $ in L periodic to the power complex of q open functions parenthesis Capital in $ OmegaL ˆ{ p closing} { parenthesisl o c } ( to some R ˆ{ vN sub} 0{ y }ˆ{ ) } sequence (uε)( resp . (vε)). One in) L$ to i the f as power $ of\ varepsilon q parenleftbig Capital\rightarrow Omega semicolon0 L , sub $ per to the power of q open parenthesis Y closing parenthesis parenrightbig i f for all eta greater 0 \ [ and\ int f in L{\ toOmega the power} ofu q open{\ parenthesisvarepsilon Capital} ( Omega x semicolon ) \ psi C subˆ{\ per openvarepsilon parenthesis} Y( closing x parenthesis ) dx closing\rightarrow parenthesis\ int \ int satisfying{\Omega .. bar v sub\times 0 minus f barY } subu L q{ open0 } parenthesis( x Capital , yOmega ) times\ psi Y closing( parenthesis x , less y or ) equal dxdy \ ] eta divided by 2 sub comma one can find some alpha greater 0 such that .. bar v sub epsilon minus f to the power of epsilon bar sub L q open parenthesis Capital Omega closing parenthesis less\noindent or equal etaf o r provided a l l $0 less\ psi epsilon\ in less orL equal ˆ{ alphap ˆ{\ periodprime }} ( \Omega ;C { per } (Y) )If open . $ parenthesis\quad Asequence u sub epsilon closing $ ( parenthesis v {\ varepsilon sub epsilon greater} ) 0{\ andvarepsilon open parenthesis v> sub0 epsilon}$ closing in $ parenthesis L ˆ{ q } sub( epsilon\Omega greater) 0 are ( as above 1 \ openleq parenthesisq < with\ infty the respective) $ assigned i s s two a i d hyphen to scale con hyphen stronglyvergence properties two − scale closing converge parenthesis in comma $ L .. ˆ it{ ..q can} be( ..\ shownOmega that) .. $ when to epsilon some right $ v arrow{ 0 0}\ commain .. theL ˆ..{ q } sequence( \Omega .. open parenthesis; L ˆ{ uq sub} epsilon{ per closing} ( parenthesis Y ) sub )$ epsilon ifforall greater 0 $ \eta > 0 $ andweakly $ converges f \ in to u-tildewideL ˆ{ q } sub( 0 in L\Omega to the power;C of p open{ parenthesisper } ( Capital Y Omega ) )$closing satisfying parenthesis open\quad parenthesis$ \ parallel withv { tildewide-u0 } − subf 0 open\ parallel parenthesis x{ closingL qparenthesis ( \ =Omega integral sub\times Y u sub 0Y) open parenthesis}\ leq x comma\ f r a c y{\ closingeta parenthesis}{ 2 } { , }$ dyone comma can xfind in Capital some Omega $ \alpha closing parenthesis> 0 $ whereas suchopen that parenthesis\quad v sub$ epsilon\ parallel closing parenthesisv {\ varepsilon sub epsilon greater} − 0 weaklyf ˆ{\ convergesvarepsilon to v-tildewide}\ parallel sub 0 open{ parenthesisL q defined( \Omega as tildewide-u) }\ subleq 0 closing\eta parenthesis$ provided in L to the $ power 0 of< q open\ varepsilon parenthesis Capital\ leq Omega\alpha closing parenthesis. $ .. and further comma there is no I freason $ ( for u our{\ assumingvarepsilon that one} of those) {\ twovarepsilon sequences is strongly> convergent0 }$ and period $ ( v {\ varepsilon } ) {\ varepsilon > Nevertheless0 }$ are comma as above lett ing ( 1 divided with the by r respective= hline from p assigned to 1 plus 1 divided two − bys c q a land e con assuming− that r greater equal 1 comma itvergence can be shown properties that ) , \quad i t \quad can be \quad shown that \quad when $ \ varepsilon \rightarrow 0when , $ epsilon\quad rightthe arrow\quad 0 commasequence the s equence\quad .. open$ ( parenthesis u {\ uvarepsilon sub epsilon v} sub) epsilon{\ closingvarepsilon parenthesis> sub0 epsilon}$ greaterweakly 0 weakly converges converges to in L$ to\ widetilde the power of{u r} open{ parenthesis0 }$ in Capital $ L Omegaˆ{ p } closing( parenthesis\Omega ..) to the ( function $ with $ \ widetilde {u} { 0 } (Equation: x ) z open = parenthesis\ int { Y x closing} u parenthesis{ 0 } ( = integral x , sub Yy u sub ) 0 open dy parenthesis , x x\ commain \ yOmega closing parenthesis) $ whereas v sub$ 0 ( open v parenthesis{\ varepsilon x comma y} closing) {\ parenthesisvarepsilon dy .. open> parenthesis0 }$ x weakly in Capital converges Omega closing to parenthesis $ \ widetilde period{v} { 0 } ( $As .. defined might .. beas .. expected$ \ widetilde comma{ ..u} strong{ 0 two} hyphen) $ scale in .. $ convergence L ˆ{ q } .. implies( \Omega .. weak .. two) $ hyphen\quad scaleand further , there is no reasonconvergence for period our assuming .. The .. function that one u sub of 0 open those parenthesis two sequences resp period is v sub strongly 0 closing convergent parenthesis .. .above .. is .. unique andNevertheless is referred to .. , as lett ing $\ f r a c { 1 }{ r } = \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } + \ f r a c { 1 }{ q }$ andthe assuming weak .. open that parenthesis $ r resp\geq period1 .. strong , $ itclosing can parenthesis be shown .. that two hyphen scale limit .. of the sequence .. open parenthesiswhen $ u\ subvarepsilon epsilon closing\ parenthesisrightarrow open parenthesis0 , $ resp the period s equence open parenthesis\quad v$ sub ( epsilon u {\ closingvarepsilon parenthesis} closingv {\ varepsilon } parenthesis) {\ varepsilon period .. One > 0 }$ weakly converges in $ L ˆ{ r } ( \Omega ) $ \quad to the function \ begin { a l i g n ∗} \ tag ∗{$ z ( x ) = \ int { Y } u { 0 } ( x , y ) v { 0 } ( x , y ) dy $} ( x \ in \Omega ). \end{ a l i g n ∗}

As \quad might \quad be \quad expected , \quad strong two − s c a l e \quad convergence \quad i m p l i e s \quad weak \quad two − s c a l e convergence . \quad The \quad f u n c t i o n $ u { 0 } ( $ resp $ . v { 0 } ) $ \quad above \quad i s \quad unique and is referred to \quad as the weak \quad ( resp . \quad strong ) \quad two − s c a l e l i m i t \quad of the sequence \quad $ ( u {\ varepsilon } ) ($ resp $. ( v {\ varepsilon } ) ) . $ \quad One Capital Sigma hyphen CONVERGENCE .. 1 3 \ hspaceof the∗{\ .. majorf i l l ..} results$ \Sigma .. in .. two− $ hyphen CONVERGENCE scale .. convergence\quad ..1 theory 3 i s .. the .. so hyphen called two hyphen scale compactness theorem open parenthesis open square bracket 2 .. 7 closing square bracket comma open square bracket 2 3 comma Σ− CONVERGENCE 1 3 Theorem\noindent 7 closingo f the square\quad bracketmajor comma\quad open squarer e s u l bracket t s \quad 2 6 commain \quad Theoremtwo 1− closings c a l squaree \quad bracketconvergence closing parenthesis\quad :theory .. i s \quad the \quad so − c a l l e d two − s c a l e compactnessof the theorem major ( [ results 2 \quad in7 ] , two [ 23 - scale , Theorem7 convergence ] , [ theory26 , Theorem1i s the ] so ) : \quad from any bounded from any bounded- called two - scale compactness theorem ( [ 2 7 ] , [ 2 3 , Theorem 7 ] , [ 2 6 , sequencesequence open $ ( parenthesis u {\ uvarepsilon sub epsilon sub{ nn closing}} parenthesis) { n sub\ in n inN N in}$ L to in the $ power L ˆ{ ofp p open} ( parenthesis\Omega Capital) Theorem 1 ] ) : from any bounded sequence (u ) in Lp(Ω) (1 < p < ∞), where Omega( 1 closing< parenthesisp < open\ infty parenthesis) 1 , less $ p where less infinity $ 0 closing<εn parenthesisn\∈Nvarepsilon comma where{ n }\ 0 lessrightarrow epsilon sub n right0 arrow$ as 0 < ε → 0 as n → ∞, one can 0$ as n n right\rightarrow arrown infinity comma\ infty one can, $ one can extract a subsequence that weakly two - scale converges in Lp(Ω). The two - scale extract a subsequence that weakly two hyphen scale converges in L to the power of p open parenthesis Capital Omega closing \noindentcompactnessextract a theorem subsequence i s the that corner weakly stone two of a− byscale now converges well - known in homogenization $ L ˆ{ p } ( \Omega ) parenthesisapproach period .. The , the two so hyphen - called scale two - scale convergence method ( see , e . g . , [ 1 . $compactness\quad The theorem two i− s thes c a corner l e stone of a by now well hyphen known homogenization compactness1, theorem 12, 13,2 i s the 5,39, corner stone 23]). of a by now well − known homogenization approach commaIn fact the , so weak hyphen two called - s cale two convergence hyphen scale convergence is intended method to supply .. open the parenthesis deficiency see of comma usual .. e period g period commaapproach .. open , square the so bracket− c a 1 l ..l e 1 d comma two − ..scale 1 2 comma convergence .. 1 3 comma method 2 .. 5 comma\quad ( see , \quad e . g . , \quad [ 1 \quad 1 , \quad 1 2 , \quad 1 3 , 2 \quad 5 , 3 9 , \quadweak2 3 convergence ] ) . ( observe that the former implies the latter ) 3 9 commawhereas .. 2 3 closing strong square two bracket - scale closing convergence parenthesis i period s fitted to t emper the st iffness of usual In fact comma weak two hyphen s cale convergence is intended to supply the deficiency of usual In fact ,strong weak convergence two − s cale ( indeed convergence , the latter is intended implies the to former supply ) . the For deficiency further results of usual weak .. convergenceconcerning .. two open - scale parenthesis convergence observe .. we that refer .. the to .. [ former 9 , 4.. 0implies , 2 .. 3 the ] and .. latter the references closing parenthesis .. whereas ..weak strong\quad convergence \quad ( observe \quad that \quad the \quad former \quad i m p l i e s \quad the \quad l a t t e r ) \quad whereas \quad strong two − scaletherein convergence . i s fitted to t emper the st iffness of usual strong convergence two hyphenThe scale present convergence study i s fitted i s intended to t emper to the generalize st iffness theof usual two strong - scale convergence convergence theory to (open indeed parenthesis , the indeed latter comma implies the latter the implies former the ) former . \quad closingFor parenthesis further period results .. For concerning further results two concerning− s c a l two e convergencenonperiodic we refer s ett to ings [ 9 , so , \ truequad i s4 it 0 that , 2 two\quad - scale3 ] convergence and the references i s strict ly therein relevant . hyphen scaleto periodic structures . It goes without saying that such an undertaking requires convergence we refer to open square bracket 9 comma .. 4 0 comma 2 .. 3 closing square bracket and the references therein The presentappropriate study i materials s intended , the to usual generalize material the for two two− - scalescale convergence convergence theory theory being period obsolete in the forthcoming general framework . In this connection a fundamental toThe nonperiodic present study si s ett intended ings to ,generalize so true the i two s hyphen it that scale two convergence− scale theory convergence i s strict ly relevant to periodicrole will structures be played . by\quad so - calledIt goes homogenization without saying algebras that . such One an of undertaking our main tools requires to nonperiodicwill be s ettthe ings classical comma Gelfand so true i srepresentation it that two hyphen theory scale convergence ( see , ie s strict . g . ly , relevant [ 2 2 , 1 to periodic5 ] structures ) . Most period of .. the It goes main without results saying proved that here such anare undertaking stated ( without requires proofs ) in some \noindentappropriateappropriate materials comma materials the usual material , the usualfor two hyphen material scale for convergence two − theoryscale being convergence theory being obsoletearticles in the by forthcoming the first author general with framework reference to . \quad an unpublishedIn this connection paper [ 2 a 9 fundamental ] as obsoleteregards in the forthcoming the proofs general . Algebras framework with period mean.. In this values connection was a fundamental first introduced in rolerole will will be beplayed played by so hyphen by so called− called homogenization homogenization algebras period algebras .. One of. our\quad mainOne tools of our main tools will be[ the 4 1 ] classical but a Gelfand complete representation theory adapted theory for e .\quad g .( homogenization see , \quad e theory . g . in , \quad [ 2 2 , \quad 1 5 ] ) . \quad Most o f will be thethe classical present Gelfand form was representation first introduced theory .. in open [ 2 parenthesis 9 ] . see comma .. e period g period comma .. open square bracketthe main 2 2 comma results .. 1 5 proved closing square here bracket are stated closing parenthesis ( without period proofs .. Most ) in of some articles by the f i r s t \quadTheauthor rest of with the paper reference is organized to \quad as followsan unpublished . Section paper 2 deals\quad with[ homogeniza 2 9 ] \quad as regards the proofs . the main- results t ion proved algebras here introduced are stated open earlier parenthesis in [ 3 without 0 ] proofs . closing Several parenthesis concrete in examples some articles of by the Algebrasfirst .. author with with\quad referencemean to values .. an unpublished\quad was paper\quad .. openf i r square s t \quad bracketintroduced 2 9 closing square\quad bracketin \quad .. as regards[ 4 1 the ] \quad but \quad a complete \quad theory adaptedhomogeniza for e . g -. \ tquad ion algebrashomogenization are considered theory . in The the special present case form of almost was first periodic introduced proofs periodhomogenization algebras is discussed . In Section 3 we discuss weak inAlgebras [ 2 9 with ] . .. mean values .. was .. first .. introduced .. in .. open square bracket 4 1 closing square bracket .. but .. a p complete ..Σ theory− convergence and strong Σ− convergence in L It i s of great interest to stress The rest of the paper is organized as follows . \quad .Section 2 deals with homogeniza − adaptedhere for e periodthat all g period the main .. homogenization re - sults theory achieved in the present in two form - s was cale first introduced convergence theory tin ion open algebras square bracket introduced 2 9 closing earlier square bracket in [ period 3 \quad 0 ] . \quad Several concrete examples of homogeniza − carry over mutatis mutandis to Σ− convergence theory . Thus , it i s no wonder tThe ion rest algebras of the paper are is consideredorganized as follows . \quad periodThe .. Section special 2 deals case with of homogeniza almost periodic hyphen homogenization that the Σ− convergence method i s a mere adaptation of the two - scale convergence a lt g ion e b r algebras a s \quad introducedi s \quad earlierd i ins cu open s s e d square . \quad bracketIn 3\ ..quad 0 closingSec squaretion 3 bracket\quad periodwe \ ..quad Severald i s concrete c u s s \quad examplesweak of \quad $ \Sigma method− $ . convergence In S ection\quad 4 weand introduce strong the concept of Σ− convergence of homogenizameasures hyphen . Finally , in Section 5 we show how $t ion\Sigma algebras− are$ considered convergence period .. in The $special\ l e f t case.L\ ofbegin almost{ array periodic}{ c homogenization} p \\ . \end{ array } I t \ right .$ i s of great interest to stress here that all the main re − s u l t s \quadΣ− convergenceachieved \ theoryquad in is applied two − tos studyc a l e \ homogenizationquad convergence problems theory beyond carry the over usual\quad mutatis \quad mutandis to algebrasperiodic .. is .. discussed sett ing period . .. In .. Section 3 .. we .. discuss .. weak .. Capital Sigma hyphen convergence .. and strong $Capital\Sigma Sigma− hyphen$ convergence convergence in theory Row 1 p Row . \quad 2 periodThus . i s , of it great i interests no wonder to stress that here that the all $the\Sigma main re hyphen− $ convergence method i s a mere adaptationExcept of where the otherwise two − scale stated convergence , vector spaces method are . considered\quad In over S ectionC( the 4 com we introduce - sults .. achievedplex numbers .. in two ) hyphen and scalar s cale .. functions convergence are theory assumed carry over to take .. mutatis complex .. mutandis values to . We theCapital\quad Sigmaconcept hyphen\ convergencequad o f theory $ \Sigma period ..− Thus$ commaconvergence it i s no ofwonder measures that the . Capital\quad SigmaF i n a hyphen l l y , convergence\quad in \quad Sec tion 5 we \quad show how will mostly fo llow the standard notation . For example if X and F denote a method i s a lo cally compact space and a Banach space , respectively , we write C(X; F ) for the \noindentmere adaptation$ \Sigma of the two− hyphen$ convergence scale convergence theory method is period applied .. In S to ection study 4 we homogenization introduce problems beyond the space of continuous mappings of X into F, B(X; F ) for the space of bounded contin usualthe .. concept periodic .. of sett Capital ing Sigma . hyphen convergence of measures period .. Finally comma .. in .. Section 5 we .. show how - uous functions of X into F, and K(X; F ) for the space of compactly supported Capital Sigma hyphen convergence theory is applied to study homogenization problems beyond the continuous functions of X into F. The norm in B(X; F ) will be the supremum Exceptusual periodic where otherwisesett ing period stated , vector spaces are considered over $ C ( $ the com − norm k u k ∞ = sup k u(x) k, where k . k stands for the norm in plexExcept numbers where otherwise ) and scalarstated comma functionsx∈ vectorX spaces are assumedare considered to overtake C complex open parenthesis values the . com\quad hyphenWe w i l l F. K(X; F ) i s provided with the usual inductive l imit topology . For shortness we mostlyplex numbers fo llow closing the parenthesis standard and notation scalar functions . \quad are assumedFor example to take complex if $ valuesX $ period\quad ..and We will $ F $ denote a lo cally compactwill space write andC( aX) Banachfor spaceC(X; C ,), respectivelyB(X) for B ,(X we; C write) and $CK(X) (for XK ;(X; FC). ) $ for the space mostly foLikewise llow the standard we notation period .. For example if X .. and F denote a lo cally ofcontinuousmappingsofcompact space and a Banach space $X$ comma respectively into $F comma , we B write ( C open X parenthesis ; F X )$ semicolon\quad F closingfor the parenthesis space of bounded contin − foruous the space functions of $X$ into $F , $ and $K ( X ; F ) $ for the space of compactly supported continuousof continuous functions mappings of ofX into $X$ F comma into B open $ Fparenthesis . $ X\quad semicolonThenormin F closing parenthesis $B ..( for X the space ; Fof bounded )$ continwill hyphenbe the supremum normuous\ functionsquad $ of X\ parallel into F commau and K\ openparallel parenthesis\ infty X semicolon= F closing\sup parenthesis{ x \ forin theX space}\ of compactlyparallel supportedu ( xcontinuous ) \ parallel functions of, X $ into\quad F periodwhere .. The\quad norm in$ B\ parallel open parenthesis. X\ semicolonparallel F$ closing\quad parenthesisstands will\quad be thef o r the \quad norm \quad in \quad supremum$ F . K ( X ; F ) $ inorm s provided .. bar u bar with infinity the = usual supremum inductive sub x in l X imit bar uopen topology parenthesis . \quad x closingFor parenthesis shortness bar we comma will .. write where .. bar period$ C bar ..( stands X .. ) for $ the\quad .. normf o.. r in\ ..quad F period$C(X;C) K open parenthesis X semicolon F,B(X)$ closing parenthesis \quad f o r \quad $ Bi s provided ( X with ; the C usual ) inductive $ \quad l imitand topology\quad period$ K .. For ( shortness X) we $ will\ writequad f o r \quad $ K ( X ; CC ) open . parenthesis $ \quad XLikewise closing parenthesis\quad we .. for .. C open parenthesis X semicolon C closing parenthesis comma B open parenthesis X closing parenthesis .. for .. B open parenthesis X semicolon C closing parenthesis .. and .. K open parenthesis X closing parenthesis .. for .. K open parenthesis X semicolon C closing parenthesis period .. Likewise .. we 1 4 .. G period NGUETSENG comma N period SVANSTEDT \noindentwill put L1 to 4 the\quad powerG of . p NGUETSENG open parenthesis , N X . closing SVANSTEDT parenthesis for L to the power of p open parenthesis X semicolon C closingw i l l parenthesis put $ L comma ˆ{ p and} ( L sub X loc to ) the $ power f o r of $ p L open ˆ{ parenthesisp } ( X X closing ; parenthesis C ) for ,$and$Lˆ L sub loc to the power{ p } of{ p l o c } 1 4 G . NGUETSENG , N . SVANSTEDT will put Lp(X) for Lp(X; ), and Lp (X) for open( Xparenthesis ) $ X f o semicolon r $ L ˆC{ closingp } { parenthesisl o c } period( X .. We ; generally C ) refer . to $ \quadC We generallyloc refer to Lp (X; ). We generally refer to open squareloc bracketC 4 comma .. 5 comma .. 1 8 closing square bracket for integration theory period \noindent2 period[ .. 4[ Homogenization , 4 , 5\ ,quad 1 85 ]algebras for, \quad integration1 8 ] theory for integration . theory . 2 period 1 period .. Preliminaries period2 . ..Homogenization Let N be a positive integer algebras period .. For any real epsilon greater 0 comma we set N ε > 0, \ centerlineEquation:2 . open{ 12 . . parenthesis\Preliminariesquad Homogenization 2 period . 1 closingLet algebras parenthesisbe a positive} .. H sub integer epsilon . open For parenthesis any real x closingwe parenthesis = open parenthesisset x sub 1 divided by epsilon to the power of alpha sub 1 sub comma period period period comma x sub N divided by epsilon\noindent to the2 power . 1 of . alpha\quad subPreliminaries N to the power of closing . \quad parenthesisLet $N$ comma x be = aopen positive parenthesis integer x sub 1 comma . \quad periodFor period any r e a l $ \ varepsilon > 0 , $ we s e t ) period comma x sub N closing parenthesis in R tox the1 powerxN of N comma N Hε(x) = ( ..., , x = (x1, ..., xN ) ∈ R , (2.1) where .. open parenthesis alpha sub i closing parenthesisεα1 , εαN sub 1 less or equal i less or equal N is a given family of positive integers period\ begin ..{ Thisa l i g givesn ∗} a family H = H {\ varepsilon } ( x ) = ( \ f r a c { x { 1 }}{\ varepsilon ˆ{\alpha { 1 }}} { , } open parenthesiswhere H( subαi)1 epsilon≤i≤N is closing a given parenthesis family subof positive epsilon greater integers 0 of . mappings This of gives R to the a family power ofH N= into R to the power ..., \ f r a c { x { N }}{\N varepsilonN ˆ{\alpha { N }}}ˆ{ ) } , x = ( x { 1 } of N with the(H followingε)ε>0 of mappings properties : of R into R with the following properties : , . . . , x { N } ) \ in R ˆ{ N } N , \ tag ∗{$ ( 2 . 1 ) $} open parenthesis(H H closing)1 limε parenthesis→0 | Hε(x) 1|= limint +∞ subfor epsilonany x right∈ R arrowwith 0 barx 6= Hω, subwhere epsilon| . open| and parenthesisω denote x closing parenthesis bar\end ={ plusa l i g infinity n ∗} for any x in R to the power of N with x equal-negationslash omega comma where bar period bar and omega denote 1 N \noindent where \quad Euclidean$ ( \alpha { i } )the { 1 in\ leq R i. \ leq N }$ is a given family of positive integers . \quad This gives a family the open parenthesis subthe( H sub closinglim parenthesisnorm|H (and 2 to the0origin power of Euclidean∈, N respectively sub l im. sub epsilon right arrow 0 norm H )2 ε→0 ε x)|= forallx R vextendsingle-vextendsingle-vextendsingle$ H = $ H 1 divided by epsilon open parenthesis and x closing parenthesis vextendsingle-vextendsingle- $ ( H {\ varepsilon } ) {\ varepsilon > 0 }$ of mappings of $Rˆ{ N }$ i n t o $ R ˆ{ N }$ vextendsingle = to the power1 ofN the) 0N origin for all x to the powerN of in in comma from R to the power of N period to R to the with the followingFor u ∈ L propertiesloc(Ry (Ry :denotes the space R of variables y = (y1, ..., yN)), we will power of Nput respectively for s implicity period For u in L sub loc to the power of 1 parenleftbig R sub y to the power of N to the power of parenrightbig open parenthesis R \ hspace ∗{\ f i l l }( H $ ) 1 \lim {\ varepsilon \rightarrow 0 }\mid H {\ varepsilon } sub y to the power of N denotes the space R toε the power of N of variablesN y = open parenthesis y 1 comma period period period u (x) = u(Hε(x)) (x ∈ R ). comma( x y N ) closing\mid parenthesis= + closing\ infty parenthesis$ fcomma o r any we $ x \ in R ˆ{ N }$ with $ x \ne \omega , $ where $ \mid . \mid $ and $ \omega $ denote N will put forNow s implicity , the family H = (Hε)ε>0 generates a mean value on R as fo llows . ∞ ∞ N N u to theLet powerΠ of epsilon= Π open(R parenthesisy ; H) be x closing the parenthesis space = of u those open parenthesis functions H subu epsilon∈ open B(Ry parenthesis) x closing \ begin { a l i g n ∗} ε ∞ N parenthesisfor closing which parenthesis a parenleftbig complex number x in R to theue exists power such of N parenrightbig that u → ue periodin L (Rx )− weak ∗ as the { ( } { H }ˆ{ Euclidean } { ) 2 } { l im {\N )varepsilon \rightarrow 0 ∞}} norm{\arrowvert } Now commaε → 0 ... theThis family yields H = a open linear parenthesis operator HM subfrom epsilonB(R closingy to parenthesisC whose domain sub epsilon is D greater(M) = 0Π .. generates a mean H \ f r a c { 1 }{\ varepsilon } ( and { x ) \arrowvert = }ˆ{ the } 0 o r i g i n { f o r a l l value on R toand the whose power of value N .. asat ..u fo∈ llowsD(M period) is M ..(u Let) = u( the above l imit ) . x }ˆ{ in }\ in , ˆ{ R ˆ{ N } . } { R ˆ{ eN }} respectively . Capital Pi toIt theis not power hard of toinfinity check = that CapitalΠ∞ Piis to a the closed power vector of infinity subspace parenleftbig of B( RN sub) containing y to the power the of N semicolon H \end{ a l i g n ∗} R parenrightbigconstants .. be .. the . .. Furthermore space .. of those , the .. functions fo llowing .. u properties in B parenleftbig are trivial R sub y: toM the(u) power≥ of0 Nfor parenrightbig .. for .. which .. a u ∈ Π∞ with u ≥ 0,M(1) = 1. Finally , Π∞ i s translation invariant , i . e . , we have For $ u \ in ∞ L ˆ{ 1 } { l o∞ c } ( RN ˆ{ N } { y }ˆ{ ) } ( R ˆ{ NN } { y }$ denotes the space complexτa numberu ∈ Π u-tildewidewhenever existsu ∈ Π suchand thata u∈ toRy the( where powerτ ofau epsilon(y) = u( righty − a arrow) for y tildewide-u∈ R ), and in further L to the power of infinity parenleftbig$ R ˆ{ N } R$ofvariables$y sub x to the power of N parenrightbig = ( hyphen y 1weak *, as epsilon . . right . arrow , 0 period y N .. This ) ) ,$we will putM( forτau) s = M implicity(u). This follows immediately by a s imple adaptation of the proof of [ 3 yields a linear operator M from B parenleftbig R sub y to the power of NN to the power of parenrightbig to C whose domain is 1 , Theorem 4 . 1 ] . Thus ,M i s a mean value on R ( see Definition 2 . 1 of [ 3 D open parenthesis M closing parenthesis = Capital Pi to the power of infinityN and 1 ] ) . Specifically ,M i s the mean valu e on R for H. \ [whose u ˆ{\ valuevarepsilon at u in D open} ( parenthesis x ) M closing = u parenthesis ( H is{\ M openvarepsilon parenthesis} u closing( x parenthesis ) ) = u-tildewide( x \ openin R ˆ{ N } 2 .). 2 . Definition\ ] and basic properties of a homogenization algebra parenthesis. the above l imit closing parenthesis period Let It is notthe hard basic to check notation that Capital be as Pi above to the . power of infinity is a closed vector subspace of B parenleftbig R to the power of N parenrightbig containing Now , \quadDefinitionthefamily 2 . 1 $H . We = t erm ( a Hhomogenization{\ varepsilon algebra} ) ({\ or anvarepsilon H - algebra >) on0 }$ \quad generates a mean value on the constantsN period .. Furthermore comma the fo llowingN properties are trivial : M open parenthesis u closing parenthesis R ( for H), any closed subalgebra A of B(Ry ) with the following properties : greater$ R ˆ{ equalN }$ 0 for\quad as \quad f o l l o w s . \quad Let $ \Pi ˆ{\ infty } = ( HA\Pi )1ˆA{\withinfty the} supremum( R ˆ norm{ N } i{ s separabley } ; . H ) $ \quad be \quad the \quad space \quad o f those \quad f u n c t i o n s \quad u in Capital Pi to the power of infinity( HA with)2 uA greatercontains equal the 0 comma constants M open . parenthesis 1 closing parenthesis = 1 period .. Finally$ u comma\ in CapitalB ( Pi to R the ˆ{ powerN } { ofy infinity} ) is $ translation\quad f invariant o r \quad commawhich i period\quad e perioda comma we have ( HA ) 3 If u ∈ A, then u ∈ A( u the complex conj ugate of u). complextau sub a number u in Capital $ Pi\ widetilde to the power{u of} infinity$ exists whenever such u in that Capital $ Pi u to ˆ the{\ powervarepsilon of infinity}\ and arightarrow in R sub y to the\ powerwidetilde {u} $ in $ L ˆ{\ infty } ( R ˆ{ N } { x } ) − $ weak $ ∗ $ as $ \ varepsilon \rightarrow of N open parenthesis where tau sub a u open parenthesis y closing parenthesis∞ = u open parenthesis y minus a closing parenthesis for0 y in . R$ to\ thequad powerThis of N closing parenthesis(HA)4 commaA ⊂ D(M) = Π . yieldsand further a linear M open parenthesisoperator tau $M$ sub a u fromclosing parenthesis $B ( = RˆM open{ N parenthesis} { y }ˆ u{ closing) }$ parenthesis to $C$ period whose .. This domain follows is immediately$ D ( by MIn a s the ) imple s = equel adaptation\Pi theˆ{\ H -infty algebra}$A andi s assumed to be equipped with the supremum whoseof the valueproofnorm of at open. $ square Thus u bracket\,Aini s a3D 1 commutative comma (M)$is$M( TheoremC∗ 4− periodalgebra 1 closing with squareidentity u bracket ) . = period We\ denotewidetilde .. Thus the comma{u} M i( sa $ mean the above l imit ) . value on R tospec the - power trum of of N openA by parenthesis∆(A)( the see Definition set of all nonzero multiplicative linear forms on A), It2 is period notthe 1 hard of latteropen to square beingcheck bracket endowed that 3 1 $ closing with\Pi ˆ the square{\ Gelfandinfty bracket}$ topology closing is a parenthesis closed, i . e . periodvector , the .. relative subspaceSpecifically weak comma of∗ $B M i s ( the mean Rˆ{ N } valu) $ e on containing Rtopology to the power on ofA N0 for( topological H period dual of A). As is classical ( see , e . g . , [ 2 2 the2 period constants, 2 period p . .71 ....\ ]quad Definition , [Furthermore 1 .... 5 and , .... p basic . , 304 the .... ] propertiesfo ) , llowing∆(A) ....i s of a properties acompact .... homogenization space are . trivial .... The algebra Gelfand $ period : M .... Let ( u ) \geqthe basic0transformation $ notation f o r be as above on periodA will be denoted by G. For the benefit of the reader we recall Definitionthat 2 periodG is defined1 period toWe be t erm the a mapping homogenization of algebra open parenthesis or an H hyphen algebra closing parenthesis on\noindent R to the power$ u of N\ openin parenthesis\Pi ˆ{\ forinfty }$ with $ u \geq 0 , M ( 1 ) = 1 . $ \quadH closingF i n a parenthesis l l y $ , comma\Pi anyˆ{\ closedinfty subalgebra}$ i sA oftranslation B parenleftbig invariant R sub y to the , i power . e of . N , parenrightbig we have .. with the following$ \tau properties{ a } : u \ in \Pi ˆ{\ infty }$ whenever $ u \ in \Pi ˆ{\ infty }$ and $ a \ inopenR parenthesis ˆ{ N } { HAy closing} ( parenthesis $ where 1 A $ with\tau the{ supremuma } u norm ( i s separabley ) period = u ( y − a ) $ f o r $ yopen\ parenthesisin R ˆ{ HAN closing} ) parenthesis , $ 2 A contains the constants period andopen further parenthesis $M HA closing ( parenthesis\tau { a 3} If uu in A )comma = then M overbar ( u in A ) open . parenthesis $ \quad toThis the power follows of hline immediately u the by a s imple adaptation complexof the conj proof ugate of of u [ closing 3 1 ,parenthesis Theorem period 4 . 1 ] . \quad Thus $ , M$ i s ameanvalue on $Rˆ{ N } ( $open see parenthesis Definition HA closing parenthesis 4 A subset D open parenthesis M closing parenthesis = Case 1 infinity Case 2 period 2.1of[31]).In the s equel the H hyphen\quad algebraSpecifically A i s assumed to be$ equipped, M$ with i sthe the supremum mean valu e on $Rˆ{ N }$ f o r $ H . $norm period .. Thus comma A i s a commutative C to the power of * hyphen algebra with identity period .. We denote the spec hyphen \noindenttrum of A2 by . Capital 2 . \ Deltah f i l lopenD e parenthesis f i n i t i o n A\ h closing f i l l and parenthesis\ h f i l open l b a parenthesis s i c \ h f i l the l p set r o of p e all r t nonzeroi e s \ h multiplicativef i l l o f a \ linearh f i l l homogenization \ h f i l l algebra . \ h f i l l Let forms on A closing parenthesis comma the \noindentlatter beingthe endowed basic with notation the Gelfand be topology as above comma . i period e period comma the relative weak * topology on A to the power of prime open parenthesis topological dual of A closing parenthesis period .. As is classical .. open parenthesis see\noindent comma ..Definition e period g period 2 . comma 1 . We .. open t erm square a homogenization bracket 2 2 comma algebra .. p period ( 71 or closing an H square− algebra bracket ) comma on $R .. open ˆ{ N } square( $ bracketf o r 1 .. 5 comma .. p period 304 closing square bracket closing parenthesis comma $HCapital ) Delta ,$ open anyclosedparenthesis A closing subalgebra parenthesis $A$ i s a compact of $B space ( period Rˆ ..{ TheN Gelfand} { y } transformation) $ \quad onwith A will the be following properties : denoted by \ centerlineG period ..{ For( HA the benefit $ ) of 1 the reader A$ we with recall the that supremum G is defined norm to be i the s mapping separable of . } \ centerline {(HA $ ) 2 A$ contains the constants . }

\ centerline {( HA ) 3 I f $ u \ in A , $ then $\ overline {\}{ u }\ in A ( ˆ{\ r u l e {3em}{0.4 pt }} u$ the complex conj ugate of $u ) . $ }

\ [ \ l e f t . ( HA ) 4 A \subset D ( M ) = \Pi\ begin { a l i g n e d } & \ infty \\ &. \end{ a l i g n e d }\ right . \ ]

In the s equel the H − algebra $ A $ i s assumed to be equipped with the supremum norm . \quad Thus $ , A$ i s a commutative $Cˆ{ ∗ } − $ algebra with identity . \quad We denote the spec − trum of $A$ by $ \Delta ( A ) ( $ the set of all nonzero multiplicative linear forms on $ A ) , $ the latter being endowed with the Gelfand topology , i . e . , the relative weak $ ∗ $ topology on $ A ˆ{\prime } ($ topological dual of $A ) . $ \quad As is classical \quad ( see , \quad e . g . , \quad [ 2 2 , \quad p . 71 ] , \quad [ 1 \quad 5 , \quad p . 304 ] ) , $ \Delta ( A )$ i sacompact space . \quad The Gelfand transformation on $ A $ will be denoted by $ G . $ \quad For the benefit of the reader we recall that $ G $ is defined to be the mapping of Capital Sigma hyphen CONVERGENCE .. 1 5 \ hspaceA into∗{\ C openf i l lparenthesis} $ \Sigma Capital− Delta$ CONVERGENCE open parenthesis\ Aquad closing1 parenthesis 5 closing parenthesis such that G open parenthesis u closing parenthesis open parenthesis s closing parenthesis = angbracketleft s comma u right angbracket for s in Capital Delta Σ− CONVERGENCE 1 5 open\noindent parenthesis$ A closing $ i n parenthesist o $ C and ( u in\Delta A comma where( A the ) )$suchthat$G ( u ) ( s A into C(∆(A)) such that G(u)(s) = hs, ui for s ∈ ∆(A) and u ∈ A, where the brackets )brackets = \ standlangle for thes duality , pairing u \ betweenrangle A$ to the f o r power $ sof prime\ in and\ ADelta period .. One( classical A ) result $ and on $ u \ in A , $stand where for the the duality pairing between A0 and A. One classical result on which wewhich will greatly we will lean greatly i s the so lean hyphen i s called the so commutative - called commutative Gelfand endash Gelfand Naimark – theo Naimark hyphen theo bracketsrem open square stand bracket for the 2 2 .. duality comma p period pairing 277 closing between square $ bracket A ˆ{\ commaprime which}$ says and that $ G A is an i. sometric $ \quad is omorphismOne classical result on - rem [ 2 2 , p . 277 ] , which says that G is an i sometric is omorphism of the of the C to the power of * hyphen algebra C∗− algebra A onto the C∗− algebra C(∆(A)). It results from this that the space \noindentA onto thewhich C to the we power will of *greatly hyphen algebra lean C i open s the parenthesis so − called Capital commutative Delta open parenthesis Gelfand A closing−− Naimark parenthesis theo closing− C(∆(A)) i s separable , thanks to ( HA ) 1 . We deduce using a classical result parenthesisrem [ 2 period 2 \quad .. It results, p . from 277 this ] , that which .. the says space thatC open parenthesis$G$ is Capital an i Delta sometric open parenthesis is omorphism A closing of parenthesis the $C ˆ{ ∗ } ( see , e . g . , [ 6 , TGX . 24 ] ) that the compact space ∆(A) i s metrizable . −closing$ algebra parenthesis Except where otherwise stated , ∆(A) is provided with the so - called M− measure $A$i s separable onto comma the .. $Cˆ thanks{ to ∗ .. } open − $ parenthesis algebra HA $C closing parenthesis( \Delta 1 period( .. A We deduce ) ) using . a $ classical\quad resultIt ..results from this that \quad the space for A, denoted below by β. It is worth reminding that β i s the positive Radon open$ C parenthesis ( \Delta see comma( .. eA period ) g period ) $ comma .. open square bracket 6 comma measure on ∆(A), of total mass 1 , such that iTGX s separable period 24 closing , \quad squarethanks bracket to closing\quad parenthesis( HA ) 1 that . the\quad compactWe deduce space Capital using Delta a classical open parenthesis result A closing\quad ( see , \quad e . g . , \quad [ 6 , parenthesisTGX . 24 i s ] metrizable ) that periodthe compact spaceZ $ \Delta ( A )$ i s metrizable . Except where otherwise stated commaM Capital(u) = DeltaG open(u)(s) parenthesisdβ(s)(u ∈ AA closing). parenthesis is provided with the so hyphen ∆(A) calledExcept M hyphen where measure otherwise stated $ , \Delta ( A ) $ is provided with the so − c a l l e d $ M − $for Ameasure commaWe refer .. denoted to below [ 3 by 0 beta] for period more ..detail It is worth about remindingβ. that beta i s the positive Radon f omeasure r $ AThe on Capital ,next $ proposition\ Deltaquad opendenoted parenthesis includes below Aa fewclosing by other $ parenthesis\beta useful propertiescomma. $ \ ofquad total of H massIt -algebras is 1 comma worth . such remindingPropo- that that $ \beta $ i s the positive Radon M open parenthesissition u 2 closing . 2 parenthesis. Let = integralp ∈ subR, Capital p > Delta0. openFor parenthesisu ∈ A closingA, we parenthesis have G open parenthesis umeasure closing parenthesis| onu |p $∈ \ openDeltaA parenthesiswith ( A s closing ) parenthesis ,$ of d total beta open mass1 parenthesis , such s closing that parenthesis open parenthesis u in A closing parenthesis period \ [ M ( u ) = \ int {\Delta (A) } G ( u ) ( s ) d \beta ( We refer to .. open square bracket 3 .. 0 closingZ square bracket for more detail about beta period sThe ) next ( proposition u \ in includesA).andM a few(| otheru |p\)] =useful properties| G(u)(s) of|p Hdβ hyphen(s). algebras period G(| u |p) =| G(u) |p Proposition .. 2 period 2 period .. Let p in R∆( commaA) p greater 0 period .. For u in A comma .. we .. have .. bar u bar to the power of p in A .. with Proof . For p and u as stated , it i s clear that | G(u) |p lies in C(∆(A)). \noindentEquation:We G open r e f e parenthesis r to \quad bar u[ bar 3 \ toquad the power0 ] for of p moreclosing detail parenthesis about = bar $ G\ openbeta parenthesis. $ u closing parenthesis Therefore , we may consider v ∈ A such that G(v) = | G(u) |p . bar to the power of p .. and MN open parenthesis bar u bar to the power of p closing parenthesisp =p integral sub Capital Delta open \noindentForThe nexty ∈ propositionR , it follows includesv(y) a few= G(v other)(δy) =useful| G(u)( propertiesδy) | = | u(y of) | , Hwhere− a l g eδy b r a s . parenthesis A closing parenthesis bar G open parenthesisN u closing parenthesis open parenthesis s closing parenthesis bar to the Propositiondenotes\quad the Dirac2 . 2 measure . \quad onLetR $at p y. \Hencein R the proposition , p > fo0 llows . readily $ \quad. For $ u \ in power of p d beta open parenthesisWe turn s closing now our parenthesis attention period to a concept of degeneracy . AProof , $ period\quad .. Forwe p\ andquad u ashave stated\quad comma$ it i\mid s clear thatu ..\ barmid Gˆ open{ p parenthesis}\ in uA closing $ \quad parenthesiswith bar to the power Definition 2 . 3 . The H - algebra A i s said to be nondegenerate if the only of p lies in C open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis period .. Therefore comma function u ∈ A verifying u ≥ 0 and M(u) = 0 i s the zero function in B( N ). Otherwise \ beginwe ..{ maya l i g .. n ∗} consider .. v in A .. such .. that .. G open parenthesis v closing parenthesisR = bar G open parenthesis u closing \ tag ∗{$ GA is (t ermed\mid degenerateu \mid . ˆ{ p } ) = \mid G ( u ) \mid ˆ{ p }$} and M ( parenthesisProposition bar to the power 2 of . p 4 period . ..The For .. fol y lowingin R to the two power ass ertions of N comma are equivalent.. it .. follows . \midv openu parenthesis\mid ˆ{ yp closing} ) parenthesis = \ int = G{\ openDelta parenthesis(A) v closing parenthesis}\mid openG parenthesis ( u delta ) (sub y s closing ) ( i )A is nondegenerate . parenthesis\mid ˆ{ p =} bard G open\beta parenthesis( u s closing ) parenthesis . open parenthesis delta sub y closing parenthesis bar to the power of p =\end bar{ ua openl i g n ∗} parenthesis y closing parenthesis bar to the power of p comma where delta sub y denotes the Dirac measure (ii)Suppβ = ∆(A). on R to the power of N at y period .. Hence the proposition fo llows readily period square \noindentWe turnProof nowProof our . attention . \Supposequad toFor a concept ( $p$i ) of holds degeneracy and . $u$ We period claim as that stated ( ii , ) it i i s strue clear . Otherwisethat \quad $ \mid G (Definition u )let 2 periodr\midbe 3ˆ{ someperiodp }$ .. point The l i e H s hyphen in in $∆( algebra CA) (lying A i s\ saidDelta off to be nondegenerateSuppβ( A( the ) if )the support only . $ function\quad of β)Therefore. , weu in\quad A verifyingBymay Urysohn\ uquad greaterc o equal ’ n s s i d 0 e lemma r and\quad M open we$ parenthesis may v consider\ in u closingA $ some parenthesis\quad ϕsuch =∈ 0 i\quad sC(∆( theA zero))that functionsuch\quad that in B$ parenleftbig G ( Rv to the) power = of\ϕmid N parenrightbig≥ G0, ϕ ((r) period u = ..1 ) Otherwiseand\mid ˆϕ{ p=} 0 .on $ \quad For \quad $ y \ in R ˆ{ N } , $ \quad i t \quad f o l l o w s $v(y)=G(v)(R \ delta { y } ) = \mid G−1 ( u ) ( \ delta { y } A is t ermedSuppβ. degenerateClearly periodβ(ϕ) ≡ ∆(A) ϕ(s)dβ(s) = 0. Therefore , lett ing u = G (ϕ), ) Proposition\mid ˆ{ 2p period} = 4 period\mid .. Theu fol lowing( y two ) ass ertions\mid areˆ{ equivalentp } , $ period where $ \ delta { y }$ denotes the Dirac measure on $ Rit ˆ{ followsN }$M at(u) $ = y 0. .S $ ince\quadu ≥ Hence0 ( indeed the proposition , it i s a classical fo llows fact readily that G and $ . \ square $ open parenthesisG−1 are i closing order parenthesis preserving A is ) nondegenerate , we s eeperiod by ( i ) that u = 0. Hence open parenthesis ii closing parenthesis Supp beta = Capital Delta open parenthesis A closing parenthesis period \ centerlineϕ(s){We = turnG(u)( nows) our = 0attentionfor any s to∈ a∆(A concept), a contradiction of degeneracy and so . } ( ii ) i s true . Proof periodReciprocally .. Suppose , .. open assume parenthesis ( i i closing ) and parenthesis let u ∈ A ..with holdsu period≥ 0 and .. WeM claim(u) that = 0 ... openThen parenthesis ii closing parenthesis .. i s true period .. Otherwise let r be .. some \noindentϕ =DefinitionG(u) ≥ 0 and 2 .β 3(ϕ .) =\quad 0. ConsequentlyThe H − algebraϕ = $ 0 A $on iSuppβ s said( tosee be , nondegenerate e . g . , if the only function point ..[ in 4 .. , Capital p . Delta 69 open ] ) ; parenthesis hence Aϕ closing(s) = parenthesis 0 for .. lying all s ..∈ off∆( ..A Supp), according beta open to parenthesis ( i i ) . the .. support .. of beta$ u closing\ in parenthesisA$ verifying period .. By .. $ Urysohn u \geq quoteright0$ s .. and lemma $M we ( u ) = 0$ isthezerofunctionin Therefore u = 0 and so ( i ) follows .  $ Bmay (consider R ˆ ..{ someN } ..) phi in . C $ open\quad parenthesisOtherwise Capital Delta open parenthesis A closing parenthesis closing parenthesis .. $ A $2 is . 3 t . ermedAlmost degenerate periodic . H - algebras . Our purpose is to present typical such that ..examples phi greater equal 0 comma phi open parenthesis r closing parenthesis = 1 .. and .. phi = 0 .. on Supp beta period .... Clearly beta open parenthesis phi closing parenthesis equiv integral sub Capital Delta open parenthesis A \noindentofProposition H - algebras . 2 First . 4 . of\ allquad , weThe recall fol that lowing by an two almost ass periodic ertions continuous are equivalent func - . closing parenthesis phi openN parenthesis s closing parenthesisN d beta open parenthesis s closingN parenthesis = 0 period .... Therefore t ion on R i s meant any u ∈ B(R ) whose translates {τau : a ∈ R }( recall that comma .... lett ing u = G to the power of minusN 1 open parenthesis phi closing parenthesis comma N τau(y) = u(y − a) for y ∈ R ) form a relatively compact set in B(R ). The \ centerlineit follows M{ open( i parenthesis $ ) A$ u closing is nondegenerate parenthesis = 0 period . } .. S ince u greater equal 0 open parenthesis indeed comma it i s a classical fact that G and G to the power of minus 1 \ [are ( .. iorder i preserving ) Supp closing\beta parenthesis= comma\Delta .. we(A). .. s ee by .. open\ parenthesis] i closing parenthesis .. that .. u = 0 period .. Hence .. phi open parenthesis s closing parenthesis = G open parenthesis u closing parenthesis open parenthesis s closing parenthesis = 0 \noindentfor any s inProof Capital . Delta\quad openSuppose parenthesis\quad A closing( i ) parenthesis\quad holds comma . a\ contradictionquad We claim and so that .. open\quad parenthesis( i i ii ) closing\quad i s true . \quad Otherwise let parenthesis$ r $ be ..\ iquad s truesome period .. Reciprocally comma .. assume .. open parenthesis i i closing parenthesis pointand let\quad u in A within \ uquad greater$ equal\Delta 0 and M( open A parenthesis ) $ \ uquad closingl yparenthesis i n g \quad = 0o period f f \quad .. Then$ phi Supp = G open\beta parenthesis( $ uthe closing\quad parenthesissupport greater\quad equalo f 0 and $ \ betabeta open parenthesis) . $ phi\quad closingBy parenthesis\quad Urysohn = 0 period ’ s \quad lemma we mayConsequently consider ..\ phiquad = 0some .. on\ ..quad Supp beta$ \ openvarphi parenthesis\ in seeC( comma ..\ eDelta period g period( A comma ) .. open) $ square\quad bracketsuch 4 that \quad comma$ \varphi .. p period\geq .. 69 closing0 , square\varphi bracket closing( parenthesis r ) = semicolon 1 $ ..\ hencequad ..and phi open\quad parenthesis$ \varphi s closing= parenthesis 0 $ =\quad 0 .. foron .. all s in Capital Delta open parenthesis A closing parenthesis comma according to open parenthesis i i closing parenthesis period .. Therefore\noindent u = 0$ and Supp so open\beta parenthesis. $ i closing\ h f i parenthesis l l C l e a r l follows y $ period\beta square( \varphi ) \equiv \ int {\Delta (A)2 period 3}\ periodvarphi .... Almost( periodic s H ) hyphen d algebras\beta period( .... s Our ) purpose = is 0 to present . $ typical\ h f i l examplesl Therefore , \ h f i l l l e t t ing $ uof H = hyphen G algebrasˆ{ − period1 } ....( First\varphi of all comma) we , recall $ that by an almost periodic continuous func hyphen t ion on R to the power of N i s meant any u in B parenleftbig R to the power of N parenrightbig whose translates braceleftbig tau\noindent sub a u : aitfollows in R to the power $M of N ( bracerightbig u ) open = parenthesis 0 .$ recall\quad thatS i n c e $ u \geq 0 ( $ indeed , it i s a classical fact that $ Gtau $ sub and a u open $ G parenthesisˆ{ − 1 y}$ closing parenthesis = u open parenthesis y minus a closing parenthesis .... for y in R to the powerare of\quad N closingorder parenthesis preserving .... form ) a , relatively\quad compactwe \quad .... sets ee .... byin B\ parenleftbigquad ( i R) to\quad the powerthat of\ Nquad parenrightbig$ u period = 0 ..... $ The\quad Hence \quad $ \varphi (s)=G(u)(s)=0$ f o r any $ s \ in \Delta ( A ) , $ a contradiction and so \quad ( i i ) \quad i s true . \quad Reciprocally , \quad assume \quad ( i i ) and l e t $ u \ in A $ with $ u \geq 0$and$M ( u ) = 0 .$ \quad Then $ \varphi = G ( u ) \geq 0 $ and $ \beta ( \varphi ) = 0 . $ Consequently \quad $ \varphi = 0 $ \quad on \quad $ Supp \beta ( $ see , \quad e . g . , \quad [ 4 , \quad p . \quad 69 ] ) ; \quad hence \quad $ \varphi ( s ) = 0 $ \quad f o r \quad a l l $ s \ in \Delta ( A ) ,$ accordingto(ii). \quad Therefore $u = 0 $ and so ( i ) follows $ . \ square $

\noindent 2 . 3 . \ h f i l l Almost periodic H − a l g e b r a s . \ h f i l l Our purpose is to present typical examples

\noindent o f H − a l g e b r a s . \ h f i l l First of all , we recall that by an almost periodic continuous func −

\noindent t ion on $ R ˆ{ N }$ i s meantany $u \ in B ( R ˆ{ N } ) $ whose translates $ \{\tau { a } u : a \ in R ˆ{ N }\} ( $ recall that

\noindent $ \tau { a } u ( y ) = u ( y − a ) $ \ h f i l l f o r $ y \ in R ˆ{ N } ) $ \ h f i l l form a relatively compact \ h f i l l s e t \ h f i l l in $ B ( R ˆ{ N } ) . $ \ h f i l l The 1 6 .. G period NGUETSENG comma N period SVANSTEDT \noindentspace of such1 6 functions\quad G is commonly . NGUETSENG denoted , N by . AP SVANSTEDT parenleftbig R to the power of N parenrightbig comma and is a Banach space 1 6 G . NGUETSENG , N . SVANSTEDT \noindentunder the supremumspace of norm such period functions .. Specifically is commonly comma AP denoted parenleftbig by R to $ the AP power ( of RN parenrightbig ˆ{ N } ) with , the $ supremum and is a Banach space space of such functions is commonly denoted by AP ( N ), and is a Banach space under normunder and the supremum norm . \quad Specifically $ , APR ( Rˆ{ N } ) $ with the supremum norm and the supremum norm . Specifically , AP ( N ) with the supremum norm and the usual thethe usual usual algebra algebra operations operations in B parenleftbig in $ B R to the( power RR ˆ{ ofN N} parenrightbig) $ i s i s a a commutative commutative C to $Cˆ the power{ ∗ of } * −hyphen$ algebra with identity . algebra operations in B( N ) i s a commutative C∗− algebra with identity . On the algebraOn the with other identity hand period , \quad givenR $ u \ in AP ( R ˆ{ N } ) , $ \quad i t \quad can be shown that the closed con − other hand , given u ∈ AP ( N ), it can be shown that the closed con - vex vexOn hthe u l otherl o f hand $ \{\ commatau .. given{ ua in} APu parenleftbigR : a R\ toin the powerR ˆ{ ofN N}\} parenrightbig$ \quad commain .. $it ..B can ( be shown R ˆ{ thatN } hull of {τ u : a ∈ N } in B( N ) contains one and only one constant m(u) called the the) $ closed\quad con hyphencontainsa oneR and onlyR one constant $m ( u ) $ mean of u( see [ 2 0 , p . 94 ] and [ 3 1 ] ) . This yields a mapping u → m(u) of calledvex hull themeanof of braceleftbig tau $u sub a u ($ : a in R see to the [ 20 power ,p of N . bracerightbig 94 ] and .. [ in 31 B parenleftbig ] ) . \quad R to theThis power yields of N parenrightbig a mapping $ u \rightarrowAP (RN ) into mC, which ( iu s l inear ) $ , o positivef , translation invariant , and which attains .. contains onethe and value only 1 one on constant the constant m open function parenthesis 1 . u closing Therefore parenthesis , this determines a mean value $called AP the ( mean R of ˆ{ uN open} parenthesis) $ into see open $ C square , $ bracket which 2 0 i comma s l inear p period , 94 positive closing square , translation bracket and open invariant square , and which attains m on N with D(m) = AP ( N ), called the mean value ( on N ) for AP ( N ). Interesting bracketthe value 3 1 closing 1 onR square the bracket constant closing functionR parenthesis 1 period . \quad .. ThisTherefore yields a mappingR , this u rightdeterminesR arrow m opena mean parenthesis value u closing $ m $enough on $ R,M ˆ{ N (}the$with$D mean value on (RN mfor )H) =is AP an extension ( Rˆ of{ Nm,} as) , $ called the mean value ( on parenthesisshown of below . $ RAP ˆ{ parenleftbigN } )$ R to for the power $AP of N ( parenrightbig Rˆ{ N into} ) C comma . $ which i s l inear comma positive comma translation invariant Proposition 2 . 5 . We have AP ( N ) ⊂ Π∞ and m(u) = M(u) for al commaInteresting and which enough attains $ , M ( $ the meanR value on $Rˆ{ N }$ \quad f o r $ H ) $ \quad is an extension of l u ∈ $ mthe value , $ 1 on\quad the constantas function 1 period .. Therefore comma this determines a mean value m on R to the power of N with D open parenthesis m closing parenthesis = AP parenleftbig R to the power of N parenrightbig \noindent shown below . comma called the mean value open parenthesis on R to the powerN of N closing parenthesis for AP parenleftbig R to the power of N AP (R ). parenrightbig period \noindent Proposition \ h f i l l 2 . 5 . \ h f i l l We \ h f i l l have \ h f i l l $ AP ( R ˆ{ N } ) \subset Interesting enough comma M open parenthesis the mean value on R to the powerN of N .. for H closing parenthesis .. is an Proof . To begin , let Γ be the algebra of all functions u : R → C of the form extension\Pi ˆ{\ ofinfty m comma}$ ..\ ash f i l l and$m ( u ) =M ( u )$ \ h f i l l f o r \ h f i l l a l l $ u \ inshown$ below period X N u(y) = ck exp(2iπk · y)(y ∈ R ), Proposition .... 2 period 5 period .... We .... have .... AP parenleftbig R to the power of N parenrightbig subset Capital Pi to the\ begin power{ a of l i infinityg n ∗} .... and m open parenthesis u closing parenthesis = M openk parenthesis u closing parenthesis .... for .... al l u AP ( R ˆ{ N } ). in N \end{ a l i gwhere n ∗} k ranges over a finite subset of R ( depending on u), and the dot denotes AP parenleftbig R to the power of N parenrightbig periodN the usual Euclidean inner product in R . Each such u i s called a trigonometric Proof period .. To begin commaN let Capital Gamma be theN algebra of all functions u : R to the powerN of N right arrow C of the polynomial on R . We have Γ ⊂ AP (R ) and further Γ is dense in AP (R ) ( see form\noindent Proof . \quad To begin , let $ \Gamma $ be the algebra of all functions $u : Rˆ{ N } \rightarrow, e .C$ g . , of [ the 2 form 0 , chap . 5 ] , [ 2 2 , chap . 1 0 ] ) . Thus , the Line 1 uproposition open parenthesis is y closing proved parenthesis if we = sum can c sub check k exponent that open parenthesis for each 2 i piu k times∈ yΓ, closing parenthesis parenleftbig y in R to the powerε of N parenrightbig comma∞ LineN 2 k \ [ \ begin {wea l i g n ehave d } uu (→ ym )(u) = in \sumL (R c)− {weakk }\∗exp as (ε → 20. iClearly\ pi it ik s \cdot y where k ranges over a finite subset of R to the power ofN N open parenthesis depending on u closing parenthesisN comma and the enough to verify this for u = γk (k ∈ R ), where γk(y) = exp (2iπk · y) for y ∈ R . dot) denotes ( y \ in R ˆ{ N } ), \\ k \end{Ina l iother g n e d }\ words] , the whole problem reduces to showing that , given any arbitrary the usual Euclidean1 N inner product in R to the power of N period .. Each such u i s called a trigonometric f ∈ L (Rx )(f independent of ε), we have as ε → 0, polynomial on R to the power of N period .. We have Capital Gamma subset AP parenleftbig R to the power of N parenrightbig .. and further Capital Gamma is dense in AP parenleftbigZ R to theZ power of N parenrightbig γεfdx → m(γk) fdx \noindentopen parenthesiswhere see $ comma k $ .. ranges e period over g period a finitek comma .. subset open square of bracket$ R ˆ{ 2N ..} 0 comma( $ .. depending chap period on 5 closing $u square ) bracket, $ and comma the .. dot open denotes square bracket 2 .. 2 comma .. chap period 1 0 closing square bracket closing parenthesis period .. Thus for all k ∈ N . This i s trivial if k = ω ( the origin in N ), because m(1) = 1. commathe usual .. the .. Euclidean propositionR .. inner is .. proved product .. if we in $ R ˆ{ N } . $ \quadR Each such $ u $ i s called a trigonometric So assume that k 6= ω. Recalling that m(γk) = 0 in this case , we see that polynomialcan .. check .. on that $ .. R for ˆ{ ..N each} ... u $ in Capital\quad GammaWe have comma $ \ ..Gamma we .. have\subset .. u to the powerAP of ( epsilon R ˆ right{ N arrow} ) m $ open\quad and f u r t h e r the proposition i s proved once we have verified that lim Ff(−H (k)) = 0, where Ff parenthesis$ \Gamma u$ closing isdensein parenthesis .. $AP in .. L to ( the powerRˆ{ ofN infinity} ) $parenleftbigε R→0 to the powerε of N parenrightbig sub hyphen weak stands for the Fourier transform of f. But this fo llows immediately by the ..( * .. see as , \quad e . g . , \quad [ 2 \quad 0 , \quad chap . 5 ] , \quad [ 2 \quad 2 , \quad chap . 1 0 ] ) . \quad Thus , \quad the \quad proposition \quad i s \quad proved \quad i f we Riemann – Lebesgue lemma . canepsilon\quad rightcheck arrow\ 0quad periodthat .. Clearly\quad it if o s r enough\quad toeach verify\ thisquad for u$ = u gamma\ in k parenleftbig\Gamma k, in $ R to\quad the powerwe \ ofquad N have \quad Thus , AP ( N ) i s a closed subalgebra of B( N ) verifying properties ( HA parenrightbig$ u ˆ{\ varepsilon comma where}\ gammaRrightarrow k open parenthesism ( y closing u parenthesis ) $ R\quad = in \quad $ L ˆ{\ infty } ( R ˆ{ N } ) 2 - ) exp{ − open }$ parenthesis weak \quad 2 i pi k$ times∗ $ y closing\quad parenthesisas for y in R to the power of N period .. In other words comma the whole ( HA ) 4 . Unfortunately AP ( N ) fails to carry out ( HA ) 1 and hence we are led problem$ \ varepsilon reduces to showing\rightarrow 0R . $ \quad Clearly it i s enough to verify this for $ u = \gamma tok ( k \ in R ˆ{ N } ) , $ where $ \gamma k ( y ) = $ that commarestrict given ourselves any arbitrary to some f in Lspecific to the power subalgebras of 1 parenleftbig . R sub x to the power of N parenrightbig open parenthesis f independentexp $ ( of 2 epsilon i closing\ pi parenthesisk \cdot comma wey have ) $as epsilon f o r right $ y arrow\ in 0 commaR ˆ{ N } . $ \quad In other words , the whole problem reduces to showing Let R be a countable subgroup of the additive group N . We define thatintegral , given gamma any sub k arbitrary to the power of $ epsilon f \ in fdx rightL arrowˆ{ 1 m} open( parenthesis R ˆ{ NR}y gamma{ x } k closing) parenthesis( f $ integral independent fdx of $ \forvarepsilon all k in R to the) power ,$ of N wehaveas period .. This i$ s trivial\ varepsilon if k = omega\ openrightarrow parenthesis the0 origin , $ in R to the power of N closing AP ( N ) = {u ∈ AP ( N ): Sp(u) ⊂ R} parenthesis comma .. because m open parenthesisR R 1 closing parenthesisR = 1 period \ [ \ int \gamma ˆ{\ varepsilonN } { k } fdx \rightarrow m ( \gamma k ) \ int fdx \ ] So assumewith thatSp ..( ku) negationslash-equal = {k ∈ R : M( omegaγ periodk) 6= 0} .. Recalling( spectrum that of m openu). parenthesisNote that gamma the spectrum k closing parenthesis = 0 in N N this case commaof any .. we function see that in the AP (R ) i s a countable set , and so the definition of APR(R ) P propositionmakes i s proved sense once . we Now have verified , let thatΓR limintbe the sub epsilons et of right all functionsarrow 0 F f ofopen the parenthesis form minusk ckγk H sub epsilon open \noindent f o r a l l $ k \ in R ˆ{ N } . $ \quad ThisN i s trivial if $k = \omega ( $ parenthesiswith k closingck parenthesis∈ C and closingγk parenthesis(y) = exp = 0 comma(2iπk · wherey)(y ∈ R ), where k ranges over the origin in $Rˆ{ N } ) , $ \quad because $m ( 1 ) = 1 .$ F f .. standsan arbitrary for the Fourier finite transform subset of f period of R ... ButThe this fo llows set immediatelyΓR is by a the subalgebra of So assume thatN \quad $ k N\not= \omega . $ \quad Recalling that $m ( \gamma k RiemannAP endash(R ), Lebesgueand lemmaAPR(R period) square )Thus = comma 0$ AP inthiscase parenleftbig R , to\ thequad powerwe of see N parenrightbig that the .. i s a closed subalgebra of B parenleftbig R to the power of Nproposition parenrightbig .. i verifying s proved properties once .. we open have parenthesis verified HA closing that parenthesis $ \lim {\ 2 hyphenvarepsilon \rightarrow 0 } F fopen ( parenthesis− H {\ HAvarepsilon closing parenthesis} ( 4 period k .... ) Unfortunately ) = 0AP parenleftbig ,$where R to the power of N parenrightbig .... fails to carry$ F out f open $ \ parenthesisquad stands HA closing for the parenthesis Fourier 1 and transform hence we are of led $ to f . $ \quad But this fo llows immediately by the restrict ourselves to some specific subalgebras period \noindentLet R be aRiemann countable−− subgroupLebesgue of the lemma additive group $ . R\ subsquare y to the $ power of N period .. We define AP sub R parenleftbig R to the power of N parenrightbig = braceleftbig u in AP parenleftbig R to the power of N parenrightbig :\ hspace Sp open∗{\ parenthesisf i l l }Thus u closing $, parenthesis AP subset ( Rˆ R bracerightbig{ N } ) $ \quad i s a closed subalgebra of $B ( Rˆ{ N } ) $with\quad Sp openverifying parenthesis u properties closing parenthesis\quad = braceleftbig( HA ) 2 − k in R to the power of N : M open parenthesis to the power of hline gamma sub k closing parenthesis equal-negationslash 0 bracerightbig open parenthesis spectrum of u closing parenthesis period .. Note\noindent that the( spectrum HA ) 4 . \ h f i l l Unfortunately $AP ( Rˆ{ N } ) $ \ h f i l l fails to carry out ( HA ) 1 and hence we are led to of any function in AP parenleftbig R to the power of N parenrightbig i s a countable set comma and so the definition of AP sub R\noindent parenleftbigrestrict R to the power ourselves of N parenrightbig to some specific subalgebras . makes sense period .. Now comma .. let Capital Gamma sub R be the s et of all functions of the form sum sub k c sub k gamma k\ centerline with { Let $ R $ be a countable subgroup of the additive group $ R ˆ{ N } { y } . $ \quad We d e f i n e } c sub k in C .. and gamma k open parenthesis y closing parenthesis = .. exp open parenthesis 2 i pi k times y closing parenthesis parenleftbig\ [ AP { R y in} R( to the R power ˆ{ N of} N parenrightbig) = \{ commau ..\ wherein ..AP k ranges ( .. R over ˆ{ ..N an} arbitrary) : Sp ( u ) \subset R finite\}\ ..] subset .. of R period .. The .. set .. Capital Gamma sub R .. is .. a .. subalgebra .. of AP parenleftbig R to the power of N parenrightbig comma .. and .. AP sub R parenleftbig R to the power of N parenrightbig \noindent with $Sp ( u ) = \{ k \ in R ˆ{ N } : M ( ˆ{\ r u l e {3em}{0.4 pt }} \gamma { k } ) \ne 0 \} ($ spectrumof $u ) .$ \quad Note that the spectrum of any function in $AP ( Rˆ{ N } ) $ i s a countable set , and so the definition of $AP { R } ( R ˆ{ N } ) $ makes sense . \quad Now , \quad l e t $ \Gamma { R }$ be the s et of all functions of the form $ \sum { k } c { k }\gamma k $ with $ c { k }\ in C $ \quad and $ \gamma k ( y ) = $ \quad exp $ ( 2 i \ pi k \cdot y ) ( y \ in R ˆ{ N } ) , $ \quad where \quad $ k $ ranges \quad over \quad an arbitrary f i n i t e \quad subset \quad o f $ R . $ \quad The \quad s e t \quad $ \Gamma { R }$ \quad i s \quad a \quad subalgebra \quad o f $ AP ( R ˆ{ N } ) , $ \quad and \quad $ AP { R } ( R ˆ{ N } ) $ Capital Sigma hyphen CONVERGENCE .. 1 7 \ hspacecoincides∗{\ withf i l l the} $ closure\Sigma of Capital− $ Gamma CONVERGENCE sub R in B\quad parenleftbig1 7 R to the power of N parenrightbig open parenthesis see comma e period g period comma .... open square bracket 2 .... 0 comma p period 93 comma Proposition 5 period 4 closing square Σ− CONVERGENCE 1 7 bracket\noindent closingcoincides parenthesis periodwith the closure of $ \Gamma { R }$ in $ B ( R ˆ{ N } ) ($ see,e.g., \ h f i l l [ 2 \ h f i l l 0 , p . 93 , Proposition 5 . 4 ] ) . coincides with the closure of Γ in B( N )( see , e . g . , [ 2 0 , p . 93 , Proposition Hence comma recalling Proposition 2 periodR 5 commaR it becomes an elementary exercise to verify that \noindent5 .Hence 4 ] ) . , recalling Proposition 2 . 5 , it becomes an elementary exercise to verify that AP subHence R parenleftbig , recalling R to theProposition power of N 2 parenrightbig . 5 , it becomes is a homogenization an elementary algebra exercise on R to to theverify power that of N open parenthesis for H closing parenthesis period .... We will refer to AP sub R parenleftbig R to the power of N parenrightbig AP ( N ) is a homogenization algebra on N ( for H). We will refer to AP ( N ) \noindentas the almostR$R periodicAP { R H hyphen} ( algebra R ˆ{ attachedN } ) to $ R periodisR a homogenization algebra on $R RR ˆ{ N } ( $ f o r as the almost periodic H - algebra attached to R. $ HBefore ) going . $any\ furtherh f i l l commaWe will let us refer recall a to classical $ AP notion{ R we} will( need R period ˆ{ N ..} If G) is $ Before going any further , let us recall a classical notion we will need . If G is a lo cally compact Abelian group comma .. we denote it s dual by G-hatwide sub comma .. i period e period comma hatwide-G G ., G is\noindent the groupa loas cally the compactalmost periodic Abelian group H − algebra , we denote attached it s to dual $R by b, . $i . e b is the G = {ξ ∈ : | ξ |= 1}. of all continuousgroup of homomorphisms all continuous of homomorphisms G into the unit circle of U =into open the brace unit xi circle in C : barU xi barC = 1 closing brace period \ hspaceWith the∗{\With topologyf i l l the} Before of topology compact going of convergence compact any further on convergence G comma , let G-hatwide on usG, recallGb ii s s a a loa lo cally classical cally compact compact notion Abelian Abelian we will need . \quad I f $ Ggroup $ periodi sgroup .. .Points Points in G-hatwide in Gb are are the the so hyphen - called called continuous continuous characters characters of ofG. G periodIf γ ..∈ If gammaGb and in hatwide-G and y in G commay ∈ G, itit is is customary customary to denote to denote gammaγ( openy) by parenthesishγ, yi or y closinghy, γi. parenthesis by angbracketleft gamma comma y right angbracket\noindent ora .. angbracketleftloHaving cally made compact y comma this pointAbelian gamma , right group let angbracket us keep , \quad in period mindwe denote that the it countable s dual bysubgroup $ \widehatR {G} { , }$ N \quadHavingi made.of eR this $ .introduced point , comma\widehat above .. let us is{G keep naturally} $ in mind is the provided that group the countable with the subgroup discrete R topology . Con - ofof allR to continuousthesequently power of , N it homomorphisms .... s dual introduced group aboveRb ofis iscompact naturally$ G $ providedinto ( see the , ewith . unit g the . , discrete circle [ 2 2 topology , p $ . U 1 22 period = ] ) . ....\{\ Con We hyphenxi \ in C : \mid \ xi \mid = 1 \} . $ sequentlywill comma also it need s dual the group ( group R-hatwide ) homomorphism is compact .. openϕ : parenthesisRN → Rb defined see comma at each e periody ∈ gR periodN by comma .. open square bracket 2 2 comma p period 1 22 closing square bracket closing parenthesis period .. We will also need \noindentthe open parenthesisWith the group topology closing parenthesisofhϕ( compacty), ki = homomorphismγk convergence(y) = exp(2iπk phi· on :y)( R to $k the∈ G R power). , of\ Nwidehat right arrow{G} R-hatwide$ i s a defined lo cally at each compact Abelian y in R to the power of N by N N \noindentangbracketleftThegroup function phi . open\quadϕ parenthesismapsPoints continuously y closingin $ parenthesis\widehatR into{ commaG} Rb$and k are right , the angbracket on so the− othercalled = gamma hand continuous k, open ϕ(R parenthesis) characters y closing of parenthesis$ G . $i = s exponent dense\quad inI open fRb( $ parenthesisthis\gamma is a classical 2 i\ piin k times result\widehat y closing; use{ ,G parenthesis e} .$ g . and , open [ 1 parenthesis6 , p . 98 ,k in R( 22 closing . 1 1parenthesis . period $The y function\5in ) ] if phiG need maps , be $ continuously ) . it is customary R to the power to denote of N .. into $ \ R-hatwidegamma and( comma y ) .. $ on by the other$ \ langle hand comma\gamma phi , y \rangle $ or \quad $ \ langle y , \gamma \rangle . $ parenleftbigFinally R to the , power the canonical of N parenrightbig isomorphism of R onto Rb( dual group of Rb) will be denoted i s dense in R-hatwide open parenthesis this is a classical result semicolon use comma e period g period comma .. open square \ hspace ∗{\byf iψ. l l }HavingIt is good made to recall this that pointψ ,i s\quad given byleth usψ(k) keep, γi = h inγ, k mindi for k that∈ R, theγ ∈ Rb countable. subgroup $ R $ bracket 1 6We comma are p now period in 98 a positioncomma .. opento prove parenthesis the fo 22 llowing period result1 1 period . 5 closing parenthesis closing square bracket if need be closing parenthesis period N \noindentPropositiono f $ R ˆ{ N 2} .$ 6\ . h f iLet l l introducedA = APR(R above). Then is , naturally the compact provided space with∆(A) thecan discrete topology . \ h f i l l Con − Finally commabe the canonical isomorphism of R onto R-hatwide-hatwide open parenthesis dual group of hatwide-R closing parenthesis will be denoted \noindentprovidedsequently with a, group it s dual operation group under $ \widehat which i{ tR is} $ an i Abelians compact group\quad and( further see , e . g . , \quad [22,p.122]). \quad We will also need by psi periodthe Haar .. It is measure good to recall on that∆(A psi) iis s given precis by ely angbracketleft the M - measure psi open parenthesisβ. k closing parenthesis comma gamma rightthe angbracket ( group = ) angbracketleft homomorphism gamma $ comma\varphi k right: angbracket R ˆ{PN for}\ k inrightarrow R comma gamma\ inwidehat R-hatwide{R} sub$ period defined at each Proof . For each function of the form u = k ckγk (ck ∈ C), where k ranges over $ yWe are\ in now inR a ˆ position{ N }$ to by prove the fo llowing result period P a finite subset of R depending so lely on u, let T (u) = k ckψ(k). This defines a Proposition .... 2 period 6 period .... Let A = AP sub R parenleftbig R to the power of N parenrightbig period .... Then comma linear mapping T :Γ → C(Rb) such that k T (u) k ∞ =k u k and ....\ [ the\ langle .... compact\varphi space Capital( DeltaR y open ) parenthesis , k A\rangle closing parenthesis= \gamma∞ .... can be k ( y ) = \exp ( 2provided i \ pi with ak group\cdot .... operationy under ) which( k i t is\ ....in an AbelianR). group\ ....] and further the Haar measure on Capital Delta open parenthesis A closing parenthesisN .. is precis ely the M hyphen measure beta period T (u)(ϕ(y)) = u(y)(y ∈ R ) (2.2) Proof period .... For each function of the form u = sum sub k c sub k gamma k open parenthesis c sub k in C closing parenthesis \noindent The function $ \varphi $ maps continuously $ R ˆ{ N }$ \quad i n t o $ \widehat{R} $ comma wherefor k all rangesu over∈ Γ . Thanks to the fact that Γ is dense in A, we see that we can and , \quad on the otherR hand $ , \varphi (R R ˆ{ N } ) $ a finite subsetextend of RT dependingby continuity so lely to on u comma a continuous let T open parenthesisl inear umapping closing parenthesis , st il = l denotedsum sub k c sub k psi open i s dense in $ \widehat{R} ( $ this is a classical result ; use , e . g . , \quad [ 1 6 , p . 98 , \quad (22 . 11 . 5) ] if needbe) . parenthesisby k closingT, parenthesisof A into period ..C( ThisRb). definesMoreover , the latter is an i sometric a linear mapping T : Capital Gamma sub R right∗ arrow C open parenthesis to the power of R-hatwide closing parenthesis such that\noindent .. bar Thomomorphism openFinally parenthesis , the u closingcanonical of the parenthesisC isomorphismalgebra bar infinityA of =into bar $ uR bar $ the sub ontoC∗ infinity− algebra $ \ andwidehatC(Rb), {\andwidehat ( {R}} ( $ dual group of − $ \Equation:widehat{ openR} parenthesis) $ will 2 period be denoted 2 closing parenthesis .. T open parenthesis u closing parenthesis open parenthesis phi open parenthesisby $ \ psi2 y .closing 2 ). parenthesis $ holds\quad forclosingIt is all parenthesis goodu ∈ to =A. recall u openBy parenthesis using that $ \ ypsi closing$ parenthesis i s givenby parenleftbig $ \ ylangle in R to the\ powerpsi of( k ) , \gamma \rangle = \ langle \gamma , k \rangle $ f o r $ k \ in R, N parenrightbigthe classical property that Rb i s total in C(Rb), it can be shown without difficulty \gammafor all uthat in\ in CapitalT \iswidehat Gamma surj ective{ subR} R and{ period. therefore}$ .. Thanks an to isometric the fact that isomorphism Capital Gamma of the sub RC∗− .. isalgebra dense inA A comma .. we see that we canonto the C∗− algebra C(Rb). This being so , let L be the mapping of C(∆(A)) into \noindent We are now in a position to prove the fo llowing result . extend TC( byRb) continuitydefined to by .. aL continuous(f) = T (G ..− l1 inear(f)) ..for mappingf ∈ comma C(∆(A)) .., stwhere il l denotedG is the by T Gelfand comma .. of A .. intotransformation .. C open parenthesis on A. to theThis power mapping of R-hatwide is clearly closing an parenthesis i sometric period i somorphism .. Moreover of comma the C∗ ..− the .. latter .. is .. an\noindent .. i sometricProposition .. homomorphism\ h f .. i l of l the2 . .. 6Case . \ 1h * f Case i l l 2Let hyphen $ A = AP { R } ( R ˆ{ N } ) . $ \ h f i l l Then , \ h f i l l the \ h f i l l compact space algebra C(∆(A)) onto the C∗− algebra C(Rb). Consequently , according to [ 2 2 , $ \algebraDelta A ..( into A .. the ) C $ to the\ h f power i l l can of * behyphen algebra C open parenthesis to the power of R-hatwide closing parenthesis comma .. andp . .. 90 open , Theorem parenthesis 4 2 . period 1 . 4 2 ] closing , there parenthesis exists a homeomorphism .. holds .. for .. all uh inof AR periodb onto ..∆( ByA using) such \noindent provided with a group \ h f i l l operation under which i t is \ h f i l l an Abelian group \ h f i l l and f u r t h e r the classicalthat propertyL(f)(t) = thatf(h R-hatwide-hatwide(t)) (t ∈ Rb) for i any s totalf ∈ in C(∆( C openA)). parenthesisNow , to for thes1 power, s2, of s hatwide-R∈ ∆(A), closing parenthesis −1 comma it canput bes shown1 + s2 without= h(t1t2 difficulty) and −s = h(t )( observe that Rb i s a multiplicative group ) , \noindent the Haar measure−1 on $ \Delta (− A1 ) $ \quad is precis ely the M − measure $ \beta that T iswhere surj ectiveti = andh therefore(si)(i an= isometric1, 2) isomorphismand t = h of(s) the. C toThis the definespower of a * binaryhyphen algebra relation A + . $onto the C to the power of * hyphen algebra C open parenthesis to the power of R-hatwide closing parenthesis period .. This being so comma let L be the mapping of C open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis into\noindent Proof . \ h f i l l For each function of the form $ u = \sum { k } c { k }\gamma kC ( open c parenthesis{ k }\ toin the powerC of ) R-hatwide ,$ where closing parenthesis $k$ ranges .. defined over by L open parenthesis f closing parenthesis = T open parenthesis G to the power of minus 1 open parenthesis f closing parenthesis closing parenthesis .. for f in C open parenthesis Capital\noindent Deltaa open finite parenthesis subset A closing of $R$ parenthesis dependingso closing parenthesis lely comma on where$u G ,$ is the Gelfandlet $T ( u ) = \sumtransformation{ k } c on{ A periodk }\ ..psi This mapping( k is clearly ) . an $ i sometric\quad iThis somorphism defines of the aC linear to the power mapping of * hyphen $T algebra : C\ openGamma parenthesis{ R }\ Capitalrightarrow Delta open parenthesisC ( ˆ A{\ closingwidehat parenthesis{R}} closing) $ parenthesis such that .. \quad onto$ \ parallel the C to the powerT of ( * hyphen u algebra) \ parallel C open parenthesis\ infty to the power= \ ofparallel R-hatwide closingu parenthesis\ parallel period{\ ..infty Consequently}$ and comma according to .. open square bracket 2 2 comma \ beginp period{ a l i 90 g n comma∗} Theorem 4 period 1 period 4 closing square bracket comma there exists a homeomorphism h of R-hatwide ontoT Capital ( u Delta ) open ( parenthesis\varphi A closing(y))=u(y)(y parenthesis such that \ in R ˆ{ N } ) \ tag ∗{$ ( 2L open . parenthesis2 ) $} f closing parenthesis open parenthesis t closing parenthesis = f open parenthesis h open parenthesis t closing parenthesis\end{ a l i g closing n ∗} parenthesis open parenthesis t in R-hatwide closing parenthesis .. for any f in C open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis period .. Now comma .. for s sub 1 comma s sub 2 comma s in Capital Delta\noindent open parenthesisf o r a l l A closing$ u parenthesis\ in \Gamma comma { R } . $ \quad Thanks to the fact that $ \Gamma { R }$ \quadput sis sub dense 1 plus sin sub $A 2 = h open ,$ parenthesis\quad we t sub see 1 thatt sub 2 we closing can parenthesis and minus s = h open parenthesis t to the powerextend of minus $ T 1closing $ by parenthesis continuity open to parenthesis\quad a observe continuous that R-hatwide\quad i sl a i multiplicative n e a r \quad groupmapping closing , parenthesis\quad st comma il l denoted by $ Twhere , t $ sub\ iquad = h too the f power of minus 1 open parenthesis s sub i closing parenthesis open parenthesis i = 1 comma 2 closing parenthesis$ A $ ....\quad and ti n= th o to\quad the power$ C of minus ( ˆ 1{\ openwidehat parenthesis{R}} s closing) parenthesis . $ \quad periodMoreover .... This ,defines\quad a binarythe \ relationquad l a t t e r \quad i s \quad an \quad i sometric \quad homomorphism \quad o f the \quad plus$\ l e f t .C\ begin { a l i g n e d } & ∗ \\ & − \end{ a l i g n e d }\ right . $ algebra $ A $ \quad i n t o \quad the $ C ˆ{ ∗ } − $ algebra $C (ˆ{\widehat{R}} ) , $ \quad and \quad ( 2 . 2 ) \quad holds \quad f o r \quad a l l $ u \ in A . $ \quad By using

\noindent the classical property that $ \widehat{\widehat{R}} $ istotalin $C (ˆ{\widehat{R}} ) , $ it can be shown without difficulty

\noindent that $ T $ is surj ective and therefore an isometric isomorphism of the $ C ˆ{ ∗ } − $ algebra $A$ onto the $ C ˆ{ ∗ } − $ algebra $C (ˆ{\widehat{R}} ) . $ \quad This being so , let $L$ be the mapping of $C ( \Delta ( A ) ) $ i n t o $ C ( ˆ{\widehat{R}} ) $ \quad definedby $L ( f ) = T ( Gˆ{ − 1 } ( f ) ) $ \quad f o r $ f \ in C( \Delta ( A ) ) ,$ where $G$ istheGelfand

\noindent transformation on $ A . $ \quad This mapping is clearly an i sometric i somorphism of the $ C ˆ{ ∗ } − $ algebra $C ( \Delta ( A ) ) $ \quad onto the $ C ˆ{ ∗ } − $ algebra $C ( ˆ{\widehat{R}} ) . $ \quad Consequently , according to \quad [ 2 2 , p . 90 , Theorem 4 . 1 . 4 ] , there exists a homeomorphism $ h $ of $ \widehat{R} $ onto $ \Delta ( A )$ suchthat $L(f)(t)=f(h(t))(t \ in \widehat{R} ) $ \quad f o r any $ f \ in C( \Delta ( A ) ) . $ \quad Now , \quad f o r $ s { 1 } , s { 2 } , s \ in \Delta ( A ) , $ put $ s { 1 } + s { 2 } = h ( t { 1 } t { 2 } ) $ and $ − s = h ( t ˆ{ − 1 } ) ( $ observe that $ \widehat{R} $ i s a multiplicative group ) ,

\noindent where $ t { i } = h ˆ{ − 1 } ( s { i } ) ( i = 1 , 2 ) $ \ h f i l l and $ t = h ˆ{ − 1 } ( s ) . $ \ h f i l l This defines a binary relation $ + $ 1 8 .. G period NGUETSENG comma N period SVANSTEDT \noindentunder which1 8Capital\quad DeltaG . open NGUETSENG parenthesis , A N closing . SVANSTEDT parenthesis i s an Abelian topological group open parenthesis with the Gelfand topology closing parenthesis 1 8 G . NGUETSENG , N . SVANSTEDT \noindentand .. h ..under is .. a group which .. homomorphism $ \Delta ..( of R-hatwide A ) $ .. onto i s .. an Capital Abelian Delta open topological parenthesis group A closing ( parenthesis with the period Gelfand topology ) under which ∆(A) i s an Abelian topological group ( with the Gelfand topology ) .. It .. remains .. to .. verify that \noindentthe Haarand measureand \hquad onis Capital$ a h group Delta $ \ openquad homomorphism parenthesisi s \quad Aa closing group of parenthesisRb \quadonto homomorphism coincides∆(A). withIt beta\ remainsquad openo parenthesis f to$ \widehat the M hyphen{R} $ measure\quad onto forverify A closing\quad that parenthesis$ the\Delta Haar period measure( .. C A learly on )∆( itA) .coincides $ \quad withI t β\quad( the Mremains− measure\quad for toA).\quadC v e r i f y that theamounts Haarlearly to measure verifying it amounts onthat beta $ \ toDelta is verifying translation( that invariant Aβ )$is period translation coincides .. For this invariant purpose withcomma. $ \beta For introduce this( purpose $ the , $ M − $ measure for $ A )introduce . $ \quad C l e a r l y i t the mapping j : R to the powerN of N right arrow Capital Delta open parenthesis A closing parenthesisN .. defined by j open parenthesisamountsthe y to closing mapping verifying parenthesisj that: R = delta→ $ ∆(\ subbetaA) y opendefined$ is parenthesis translation by j(y) Dirac = δy measure invariant( Dirac at measure y in . R\quad to at theyFor power∈ R this) of. We N purpose closing parenthesis , introduce period need to show that j i s a group homomorphism . It suffices to \noindentcheckthemapping that $ j : Rˆ{ N }\rightarrow \Delta ( A ) $ \quad d e f i n e d by We .. need to .. show .. that .. j .. i s .. a group .. homomorphism periodN .. It .. suffices .. to .. check that $ jj = h ( circj =y phih ◦ openϕ )( parenthesis =usual\ compositiondelta usual{ compositiony )} .( $ closing Fix Dirac freely parenthesis measurey ∈ R period. at ....Letting $ Fix y freely\ubin= yG in(uR) R, ˆwe to{ theN have} power) of .N $period .... LettingWe \quad u-hatwideneed = to G open\quad parenthesisshow \quad u closingthat parenthesis\quad comma$ j $ we have\quad i s \quad a group \quad homomorphism . \quad I t \quad s u f f i c e s \quad to \quad check that u-hatwide open parenthesis h open parenthesis phi open parenthesis y closing parenthesis closing parenthesis closing parenthesis u(h(ϕ(y))) = L(u)ϕ(y) = T (u)(ϕ(y)) = u(y) = u(j(y)) for any u ∈ A. =\noindent L open parenthesisb $ j =hatwide-u h closingb\ circ parenthesis\varphi phi open( $ parenthesis usual composition y closingb parenthesis ) . \ h = f iT l l openFix parenthesis freely u $ closing y \ in R ˆ{ N } . $ \ h f i l l Letting $ \widehat{u} = G ( u ) ,$wehave parenthesisHence open parenthesisj(y) = h phi(ϕ(y open)) and parenthesis so j i s y aclosing group parenthesis homomorphism closing parenthesis , as claimed = u open . parenthesis With this y closing parenthesis = u-hatwide open parenthesis j open parenthesisN y closing parenthesis closing parenthesis for any u in A period in mind , let u ∈ A and a ∈ R . Then , clearly (τj(a)G(u))(j(y)) = G(τau)(j(y)) for all \ [ Hence\widehat j open{u parenthesis} ( y h closing ( parenthesis\varphi = h open( parenthesis y ) ) phi open ) parenthesis = L y ( closing\widehat parenthesis{u} closing) parenthesis\varphi y ∈ N . By the density of j( N ) in ∆(A)( this is a classical result ) , it follows and( so y j i s ) a groupR = homomorphism T ( u comma ) (as claimedR \varphi period ..( With y this ) ) = u ( y ) = \widehat{u} G(τ u) = τ G(u) for all a ∈ N and all u ∈ A. Therefore , using the fact (in j mind ( commaa y let ) u inj(a A) and for a in R toany the power uR of\ in N periodA. .. Then\ ] comma clearly parenleftbig tau sub j open parenthesis a that M i s translation invariant , we deduce β(τ f) = β(f) for all s ∈ j( N ) where f i s closing parenthesis G open parenthesis u closing parenthesis parenrightbigs open parenthesis j openR parenthesis y closing parenthesis freely fixed in C(∆(A)). Hence , the translation invariance of β( i . e . , β(τ f) = β(f) closing parenthesis = G open parenthesis tau sub a u closing parenthesis open parenthesis j open parenthesiss y closing parenthesis for f ∈ C(∆(A)), s ∈ ∆(A)) fo llows from the facts that j( N ) i s dense in ∆(A) and the closing\noindent parenthesisHence$j ( y ) = h ( \varphi R( y ) ) $ and so $ j $ i s a group homomorphism , as claimed . \quad With t h i s mapping s → β(τ f) sends continuously ∆(A) into . infor mind all y in , R let to the $ power u \ ofsin N periodA $ .. By and the $ density a of\ in j parenleftbigRC ˆ{ N R} to the. $ power\quad of NThen parenrightbig , clearly .. in Capital $ ( Delta\tau { j ( a )This} G(u))(j(y))=G( completes the proof .  \tau { a } u ) ( j open parenthesis AAs closing a direct parenthesis consequence open parenthesis of the above this is proposition a classical result , there closing is the parenthesis following comma corol it - (follows y G ) open ) parenthesis $ tau sub a u closing parenthesis = tau sub j open parenthesis a closing parenthesis G open parenthesis f o r a l llary $ y . \ in R ˆ{ N } . $ \quad Bythe density of $ j ( Rˆ{ N } ) $ \quad in u closing parenthesis .. for all a in R to the power of N and all u in A period ..N Therefore comma .. using the $ \Delta Corollary( A ) 2 ($ . 7 . thisThe is a H classical - algebra A result= APR )(R ,) it is nondegenerate ( s fact thatee M Definition i s translation 2 . invariant 3 ) . comma we deduce beta open parenthesis tau sub s f closing parenthesis = beta open parenthesisf o l l o w s f closing $ G parenthesis ( \tau for{ alla s in} j parenleftbigu ) = R to\ thetau power{ j of N ( parenrightbig a ) } G ( u ) $ \quad f o r a l l $ a \ inProofR . ˆ{ ConsideringN }$ and a that l l $ the u support\ in A of a Haar. $ \ measurequad Therefore on a lo cally , \quad compactusing the where f igroup s freely is fixed j ust in the C open said parenthesis group ( thisCapital i s Delta a classical open parenthesis result ) , A we closing s ee that parenthesis the corollary closing parenthesis period .. Hencefact comma that the $M$ translation i s invariance translation of beta invariant open parenthesis , we i period deduce e period $ \ commabeta ( \tau { s } f ) = follows immediately by Proposition 2 . 6 and use of Proposition 2.4. \betabeta open( parenthesis f )$ tau forall sub s f closing $s parenthesis\ in =j beta ( open R parenthesis ˆ{ N } ) f closing $ parenthesis for f in C open parenthesis Capitalwhere Delta $ f open $ parenthesis i s freely A closing fixed parenthesis in $C closing ( parenthesis\Delta comma( A s in ) Capital ) Delta . $ open\quad parenthesisHence A, theclosing translation invariance of $ \beta ( $ i . e . , parenthesis closing parenthesis fo. llowsInthe from the facts that j parenleftbigofProposition R to the power of N parenrightbigfound N that Remark∆(A) = 2.8Rb(uptoa courseofthetopologicalproof ),2.6whereweAhave=AP (R ). $i s\ densebeta in Capital( \tau Delta open{ s parenthesis} f ) A closing = \ parenthesisbeta (groupisomorphism and f the mapping ) $ f o s r right $ arrow f \ betain openR C( parenthesis\Delta tau sub s( f closing A parenthesis ) ) ,sends s continuously\ in \ CapitalDelta Delta( open A parenthesis ) )$ A closing fo llowsfromthe parenthesis into C facts period that $j ( R ˆ{ N } )The $ basic case of periodic H - algebras . Let A = This completes the proof period square 1 1 Cper(Y )( see Section 1 ) with Y = (− )N. It is an easy exercise to iAs s a dense directconsequence in $ \Delta of the above( Aproposition )$ comma andthemapping there is the2 , 2 following $s corol\rightarrow hyphen \beta ( \tau { s } check that A is an H - algebra . We have here M(u) = R u(y)dy for flary ) period $ sends continuously $ \Delta ( A )$ into $C .$ Y u ∈ A. Now , we observe that this H - algebra i s only a particular Corollary .. 2 period 7 period .. The H hyphen algebra A = AP sub R parenleftbig R to the power ofN N parenrightbig .. is C (Y ) = AP N ( ), \noindentalmostThis periodic completes H - the algebra proof . $ More . \ preciselysquare $ , we have per R=Z R nondegenerateas .. open is parenthesis easily verified s ee Definition . Hence , according to Remark 2 period 3 closing parenthesis period 2 .8, ∆(A) = N ( N− A = C (Y ), R = N ≡ \ hspaceProof period∗{\ f i l.... l } ConsideringAs aT direct ....the that consequence ....torus the .... ) support with of the .... above ofper a .... proposition Haarof course .... measure ; , ....there indeed on .... is ab ....theT lo following cally .... compact corol − ( / )N R = N ( group is jR ustZ thefor said groupZ .... opens ee parenthesis , e . g . this , [ i 2 s a classical 0 ] ) . result Letclosing us parenthesis stress that comma the weabove s ee that the corollary ∆(A) N \noindentfollows immediatelyequalityl a r y . between by Propositionand .... 2T periodactually 6 .... and proceeds use of Proposition from an identification .... 2 period 4 period by means square of a Remark( Capital topological Delta open ) parenthesis A closing parenthesis = 2 period 8 R-hatwide sub open parenthesis up to the power of π period\noindent to thegroup powerC o r o ofl l a Inisomorphism r y sub\quad to to the2 power. 7 . of In\ thequad this subThe a course connection H − ofalgebra the topological , let $A proof =be group the AP i somorphism{ isometricR } ( sub is closing R ˆ{ N parenthesis} ) $ C (Y ) C( N ) π(u)(p(y)) = u(y)(y ∈ N ) comma\quad tois theomorphism nondegenerate power of of of Propositionper\quad 2(onto period s ee 6 Definition whereT wesuch A have that = AP sub R to the power of foundR subfor parenleftbig R to the u ∈ C (Y ), p N power2 . of 3 N ) parenrightbig . per periodwhere to the powerdenotes of that the canonical homomorphism of R onto N . The .. basicT ..Then case .. , of periodic for any .. H hyphen algebras period .. Let .. A = C sub per open parenthesis Y closing parenthesis open\noindent parenthesisProof see .. . Section\ h f i l l Considering \ h f i l l that \ h f i l l the \ h f i l l support \ h f i l l o f a \ h f i l l Haar \ h f i l l measure \ h f i l l on \ h f i l l a \ h f i l l l o c a l l y \ h f i l l compact 1 closing parenthesis .. with Y = parenleftbig minus 1 divided by 2 sub comma 1 divided by 2 parenrightbig N sub period .. It \noindent group is j ust the said group \ h f i l lu ∈(A this= Cper i(Y s), wehave a classical result ) , we s ee that the corollary .. is an easy exercise to check that ..Z A is an H hyphenZ algebra period Z We .. have .. here .. M open parenthesisu(y)dy u= closing parenthesisG(u)(s)dβ(s =) =integralπ( subu)(z Y)dz, u open parenthesis y closing parenthesis dy .. \noindent follows immediately by Proposition \ h f i l l 2N . 6 \ h f i l l and use of Proposition \ h f i l l for .. u in A period .. Now comma .. weY .. observe∆( ..A that) .. this .. H hyphenT $ 2algebra . i s 4 only a. particular\ square almost $ periodic H hyphen algebra period .. More precisely comma we have N C sub perwhere opendz parenthesisdenotes HaarY closing measure parenthesis on the = compact AP sub R group = Z toT the. power of N parenleftbig R to the power of N parenrightbig\ [ Remark{\ commaDelta ..} as ..( is .. A easily ) .. verified = 2 period . .. Hence 8 {\ commawidehat ..{ accordingR}}ˆ{ ... to} ..{ Remark( up }ˆ{ In } { to }ˆ{ the } { a } course2 period 8of comma the Capital{ topological Delta open parenthesis} proof A closing{ group parenthesis i = somorphism T to the power}ˆ of{ Nof open Proposition parenthesis the N} hyphen{ ) torus, } closing2 . parenthesis 6 { where with A} =we C sub{ A per} openhave parenthesis{ = Y AP closing{ R parenthesis}}ˆ{ found comma} of{ course( } semicolonR ˆ{ N ..} indeed) R-hatwide . ˆ{ that }\ ] = T to the power of N equiv open parenthesis R slash Z closing parenthesis to the power of N for R = Z to the power of N open parenthesis s ee comma e periodThe \quad g periodb a comma s i c \quad open squarecase bracket\quad 2o .. f 0 p closing e r i o d i square c \quad bracketH − closinga l g e b parenthesis r a s . \quad periodLet .. Let\quad us stress$ that A the = above C { per } equality( Y between ) ( $ see \quad Sec tion 1Capital ) \quad Deltawith open parenthesis $ Y = A closing( − parenthesis \ f r a c { 1 and}{ T to2 the} { power, }\ off N r a actually c { 1 }{ proceeds2 } from)N an identification{ . }$ \ byquad meansI t \quad is an easy exercise to check that \quad of$ Aa open $ iparenthesis s an H − topologicalalgebra closing . parenthesis Wegroup\quad .. isomorphismhave \quad periodhere .. In\quad this .. connection$ M ( comma u .. ) let .. = pi be\ int the ..{ isometricY } u is omorphism ( y of ) dy $ \quad f o r \quad $ uC sub\ in per openA parenthesis . $ \quad Y closingNow parenthesis , \quad ..we onto\quad C parenleftbigobserve T\quad to the powerthat of\quad N parenrightbigt h i s \quad .. suchH − that pi open parenthesisalgebra u i closing s only parenthesis a particular open parenthesis almost p open periodic parenthesis H − yalgebra closing parenthesis . \quad closingMore parenthesis precisely = u , open we parenthesishave y closing$ C parenthesis{ per } parenleftbig( Y ) y in R = to the AP power{ R of N = parenrightbig Z ˆ{ N ..}} for u( in C R sub ˆ{ perN open} ) parenthesis , $ Y\quad closingas parenthesis\quad i s \quad e a s i l y \quad v e r i f i e d . \quad Hence , \quad according \quad to \quad Remark comma \noindentwhere p denotes$ 2 the . .. canonical 8 , ..\ homomorphismDelta ( .. A of R ) to the = power T ˆ of{ N ..} onto( $ .. T the to the $ power N − of$ N period torus .. ) Then with comma$ A ..= for C.. any{ per } ( Y ) ,$ ofcourse; \quad indeed $ \widehat{R} = T ˆ{ N }\equiv $ $u in ( A = R C sub / per Z open )parenthesis ˆ{ N }$ Y closing f o r parenthesis $ R = comma Z ˆ{ weN have} integral($ see,e.g.,[2 sub Y u open parenthesis y closing\quad parenthesis0 ] ) . \quad Let us stress that the above equality between dy$ =\ integralDelta sub Capital( A Delta ) $ open and parenthesis $ T ˆ{ AN closing}$ parenthesis actually G proceeds open parenthesis from u an closing identification parenthesis open by parenthesis means of s a ( topological ) closing parenthesis d beta open parenthesis s closing parenthesis = integral sub T to the power of N pi open parenthesis u closing parenthesis\noindent opengroup parenthesis\quad zisomorphism closing parenthesis . \ dzquad commaIn t h i s \quad connection , \quad l e t \quad $ \ pi $ be the \quad isometric is omorphism of $where C { dzper denotes} Haar( measure Y ) $on the\quad compactonto group $ TC to the ( power T ˆ{ ofN N period} ) $ \quad such that $ \ pi ( u )(p(y))=u(y)(y \ in R ˆ{ N } ) $ \quad f o r $ u \ in C { per } ( Y ) , $ where $ p $ denotes the \quad c a n o n i c a l \quad homomorphism \quad o f $ R ˆ{ N }$ \quad onto \quad $ T ˆ{ N } . $ \quad Then , \quad f o r \quad any

\ begin { a l i g n ∗} u \ in A = C { per } ( Y ) , we have \\\ int { Y } u ( y ) dy = \ int {\Delta (A) } G ( u ) ( s ) d \beta ( s ) = \ int { T ˆ{ N }} \ pi ( u ) ( z ) dz , \end{ a l i g n ∗}

\noindent where $ dz $ denotes Haar measure on the compact group $ T ˆ{ N } . $ Capital Sigma hyphen CONVERGENCE .. 1 9 \ hspaceRemark∗{\ ..f 2 i period l l } $ 9\ periodSigma .. More− $ .. generally CONVERGENCE comma ..\quad let .. open1 9 brace b sub 1 comma period period period comma b sub N closing brace .. be .. a .. open parenthesis non hyphen necessarily .. orthogonal closing parenthesis Σ− CONVERGENCE 1 9 \noindentbasis of RRemark to the power\quad of N2 open . 9parenthesis . \quad viewedMore as\quad an N hypheng e n e r dimensional a l l y , \quad vectorl spacee t \quad over R closing$ \{ parenthesisb { 1 period} , Remark 2 . 9 . More generally , let {b , ..., b } be a ( non - ... Let . S be . the set , b { N }\} $ \quad be \quad a \quad ( non1 −N necessarily \quad orthogonal ) necessarily orthogonal ) basis of N ( viewed as an N− dimensional vector space b aof s iall s k o fin R $ to R the ˆ{ powerN } of( N $ of the viewed form k as = sumanR sub $N i = 1− to$ the dimensional power of N t sub vector i b sub space i open parenthesisover $R t sub ) i in Z. $ \quad Let $S$ be the set N PN closing parenthesisover R comma). Let andS letbe the set of all k ∈ R of the form k = i=1 tibi(ti ∈ Z), and let o f a l l $ k \ in R ˆ{ N }$ of theform $k = \sum ˆ{ N } { i = 1 } t { i } b { i } Y = open brace y in R to the power of N : y = sum fromi=1 i = 1 to N r sub i b sub i comma minus 1 divided by 2 less or equal r ( t { i }\ in Z ) , $ and l e t X 1 1 sub i less or equal 1 divided by 2 closingY brace= {y ∈ periodN : y = r b , − ≤ r ≤ }. R i i 2 i 2 A continuous complex function u on R to the power of NN i s said to be Y hyphen periodic i f u open parenthesis y plus k closing parenthesis\ [ Y = = \{ y \ in R ˆ{ N } : y = \sum ˆ{ i = 1 } { N } r { i } b { i } , − \ f r a c {A1 continuous}{ 2 }\ complexleq r function{ i }\u leqon RN\ fi r a s c said{ 1 to}{ be2 }\}Y − periodic. \ ] i f u(y + k) = u open parenthesis y closing parenthesisN for all .. y in R to the power of NN and all k in S period .. We define P sub Y parenleftbig u(y) for all y ∈ R and all k ∈ S. We define PY (R ) to be the space of all such R to the power of N parenrightbig to be the space of all such N N functionsfunctions period .. There . There is no .. is serious no difficulty serious in difficulty showing that in showing P sub Y that parenleftbigPY (R R) to= AP theR power=S∗(R of) N parenrightbig = \noindentwhereA continuousS∗ = {l ∈ R complexN : l · k ∈ Z functionfor all k ∈ $S u}( $the on dot denotes$ R ˆ{ N the}$ usual i s Euclidean said tobe inner $Y − $ periodic i f AP sub R = S * parenleftbig RN to the power of N parenrightbigN N $ uwhere ( Sproduct to y the power+ ink of *R =)). braceleftbig =Thus $ ,P l inY R(R to) thei power s an ofHN - algebra : l times on k inR Z for( allfor k inH) S. bracerightbigIt can open parenthesis the$u dot denotes (be the y shown usual )$ Euclidean that forall the\quad above$ y development\ in R ˆ{ N regarding}$ andCper a l l(Y ) $carries k \ in overS . $ \quad We d e f i n e $ Pinner{ productY mutatis} ( .. in R R ˆ to{ N the} power) $ of N to closing be the parenthesis space period of all .. Thussuch comma P sub Y parenleftbig R to the power of N parenrightbigf u n c t i o nmutandis s .. . i s\quad an H to hyphenThere the present algebra i s no general on\quad R to thesettingserious power . of difficulty N open parenthesis in showing for H closing that parenthesis $ P { periodY } ..( It .. R can ˆ{ N } )be = .. shown AP2.4. ..{F thatR− u ..rther = the .. S above examples∗ .. } development( R of ˆ{ homogenization..N regarding} ) $ C sub per open algebras parenthesis . YThe closing space parenthesisA in .. carries .. over where $each S ˆ{ ∗ } = \{ l \ in R ˆ{ N } : l \cdot k \ in Z $ f o r a l l $ k \ in .. mutatis N S mutandis\} of( to the $ the following the present dot general examplesdenotes setting the has period proved usual Euclideanto be an H - algebra on R for H( see , e . g . , inner2 period product 4 period\ F-uquad rtherin .... $ examples R ˆ{ N ....} of) homogenization . $ \quad .... algebrasThus $ period , .... P The{ Y space} A( in each R ˆ{ N } ) $ \quad i s an H − algebra on $ Rof ˆ the{ N following} ( $ examples f o r has $ H proved ) to .be $ an H\quad hyphenI t algebra\quad oncan R to the power of N for H open parenthesis see comma e [30]). periodbe \quad g periodshown comma\quad that \quad the \quad above \quad development \quad regarding $ C { per } ( Y ) $ \quad c a r r i e s \quad over \quad mutatis open square bracket 3 0 closing square bracket closingN parenthesis) periodN ) Example 2 . 1 0 . Put A = B∞(Ry , where B∞(Ry denotes the space of those Example .... 2 period 1 0 period .... Put A = B sub infinity parenleftbig R sub y to the power of N to the power of parenrightbig continuous complex functions on N that converge ( to a finite number ) at infinity . comma\noindent wheremutandis B sub infinity to parenleftbig the present R sub general y toR they power setting of N to . the power of parenrightbig .... denotes the space of those continuousWe complex have functions here onM R(u sub) y = to thel im power|y|→∞ ofu N(y) thatfor convergeu ∈ openA, parenthesisand it to a i finite s evident number closing parenthesis \noindentthat$ 2A .is 4 a degenerate . F−u $ H r - t algebrah e r \ h f . i l l examples \ h f i l l of homogenization \ h f i l l a l g e b r a s . \ h f i l l The space at infinity period N ) $ AWe $ .. inhaveExample each here .. M open 2 parenthesis. 1 1 . u closingLet parenthesisA = =B ..∞,per l im(Y sub) barbe y the bar right closure arrow in infinityB(Ry u open parenthesis y P u closing parenthesisof the space.. for u of in A functions comma .. and of it the .. i s .. form evident ..u that= .. A ..ϕi isi .. awith a summation \noindentdegenerateofof finitely H hyphen the following algebra many period termsexamples , has proved to be an H − algebra on $ R ˆ{ N }$ f o r $ H ($ see,e.g.,   Example .. 2 period 1 1 periodN ) .. Let .. A = B sub infinity comma−1 per1 open parenthesis Y closing parenthesis .. be the .. closure where ϕi ∈ B∞(Ry , ui ∈ Cper(Y ) with Y = N. This is an H - algebra . in B parenleftbig R sub y to the power of N to the power of parenrightbig2, 2 .. of the space of \ beginfunctions{ a lExample i ..g n of∗} the .. form 2 .. . u 1 = 2 sum . phiMore i to the generally power of u , i .. let withR ..be a .. a summation countable .. subgroup of finitely .. of many the .. terms comma [where 3 phiadditive 0 i inB ] sub ) infinity . parenleftbig R sub y to the power of N to the power of parenrightbig comma u sub i in C sub per \end{ a l i g n ∗} N N N ) open parenthesisgroup Y closingR . parenthesisDefine B∞ with,R(R Y) =to Row be 1 the minus closure 1 underbar in B 1( underbarRy of the Row space 2 2 comma of functions 2 . N sub period .. This is P u N N N an H hyphenu = algebrafinite periodϕi i with ϕi ∈ B∞(R ), ui ∈ APR(R ). The space A = B∞,R(R ) \noindentExamplei .... sExample an 2 period H - algebra\ 1h 2 f periodi l l . 2 .... . 1More 0 . generally\ h f i l l commaPut .... $ Alet R = be a countable B {\ infty subgroup} of( the R additive ˆ{ N } { y }ˆ{ ) } , $group where RRemark to the $ B power2{\ . ofinfty 1 N 3 period. } ..(The Define R H Bˆ{ - sub algebrasN infinity} { y of comma}ˆ examples{ ) R}$ parenleftbig\ 2h f i l. l 1 Rdenotes 1 to the and power the 2 . of space 1 N 2 parenrightbig are of those to be the closure in Bdegenerate parenleftbig . R sub y to the power of N to the power of parenrightbig of the space of functions \noindent continuous complex functions on $ R ˆ{ N } { y }$ that converge ( to a finite number ) at infinity . u = sumExample sub finite phi i to 2 the . power14 . of u i withLet phi iA in Bbe sub infinity an parenleftbig H - algebra R to on the powerN−1, of Nand parenrightbig comma We \quad have here \quad $ M ( u )1 = $ \quad l $ im {\mid R y \mid \rightarrow u sub i in APlet sub RB parenleftbig( ; A ) be R to the the power space of of N all parenrightbig continuous period functions .. The spaceu : A =→ BA subsuch infinity that comma l R parenleftbig \ infty } u (∞ R y1 ) $ \quad f o r $ u \ in A , $ \quad Rand i t1 \quad i s \quad evident \quad that \quad R to the powerimτ→∞ of Nk parenrightbigu(τ) − ς k ∞ = 0, where ς ∈ A1 (ς depending on u). The space A = B∞(R; A1) $ Ai s $ an H\quadi hyphen s ani H s algebra -\quad algebra perioda on degenerateRemark 2 period H − 1algebra 3 period .. . The H hyphen algebras of examples 2 .. period 1 1 .. and 2 period 1 2 are degenerate period Example .. 2 period 14 period .. Let .. A sub 1 .. be .. an .. H hyphen algebra on R to the power of N minus 1 comma .. and \noindent Example \quad 2 . 1 1 . \quad Let \Nquad $ A = B {\ infty , per } ( Y ) $ .. let .. B sub infinity open parenthesis R semicolon A sub 1R closing. parenthesis .. be the \quadspacebe of all the continuous\quad closure functions u in : R $B right arrow ( A Rˆ sub{ 1N such} { thaty } lˆ im{ sub) } tau$ right\quad arrowof infinity the space bar u open of parenthesis f u n c t i o n s \quad o f the \quad form p\quadN $ u = \sum \varphi i ˆ{ u } i $ \quad with \quad a \quad summation \quad o f f i n i t e l y \quad many \quad terms , tau closing2 parenthesis . 5 . minusThe sigma spaces bar infinityXA( =Ry 0) comma (1 ≤ p < ∞). The present and next subsections where sigmaare in concerned A sub 1 open with parenthesis function sigma spaces depending of great on interest u closing in parenthesis deterministic period homogenization .. The space A = B sub infinity open parenthesis\noindenttheory Rwhere semicolon . $ A\ subvarphi 1 closingi parenthesis\ in iB s an{\ H hypheninfty algebra} ( on R ˆ{ N } { y }ˆ{ ) } , u { i }\ in C { per } ( Y )$ with $Y = \ l e f t (\ begin { array }{ cc }N −) 1 {\underline {\}} & 1 {\underline {\}}\\ R to the powerFor of each N period real p ≥ 1, we first of all introduce the space pΞ(Ry of those functions 2 , 2 \endp { arrayN ) }\ right )N { . }$ \quadε This iε s an H − algebra . 2 periodu 5∈ periodLloc(Ry .. Thefor .. which spaces the X sub sequence A to the power(u )0<ε of≤1 p parenleftbig(u defined R in sub subsection y to the power2 . 1 ) of N is parenrightbig open parenthesis 1 less or equal pp lessN infinity closing parenthesis period .. The present andp nextN ) subsections are bounded in Lloc(Rx ). This is clearly a vector subspace of Lloc(Ry . Let \noindentconcernedExample with function\ h spacesf i l l 2 of . great 1 2 interest . \ h f in i l deterministic l More generally homogenization , \ h f i l l let $ R $ be a countable subgroup of the additive Z theory period p p N ) \noindent group $ R ˆ{k uNkΞ}p =. sup $ (\quad| u(DefineHε(x)) | dx $) B {\1 infty(u ∈ pΞ(R,Ry ), } ( R ˆ{ N } ) $ to be the closure in For each real p greater equal 1 comma0<ε we≤1 firstBN of all introduce the space p Capital Xi parenleftbig R sub y to the power of N to$ Bthe power ( ofR parenrightbig ˆ{ N } { y of} thoseˆ{ ) functions}$ of the space of functions $u inu L sub = loc\sum to the{ powerf i n of i tp e parenleftbig}\varphi R subi y to ˆ{ theu power} i of $ N towith the power $ \varphi of parenrightbigi ..\ in for whichB {\ the sequenceinfty } ..( open R parenthesis ˆ{ N } ) u to the , power u { ofi epsilon}\ in closingAP parenthesis{ R } sub( 0 less R ˆepsilon{ N } less) or equal . $ 1 open\quad parenthesisThe space u to the $A power = Bof epsilon{\ infty defined in,R subsection} ( 2 period R ˆ 1{ closingN } parenthesis) $ .. is bounded in L sub loc to the power of p parenleftbig R sub x to the power of N parenrightbig period .. This is clearly a vector subspace\noindent of Li sub s loc an to H the− poweralgebra of p .parenleftbig R sub y to the power of N to the power of parenrightbig period .. Let bar u bar sub Capital Xi to the power of p = supremum 0 less epsilon less or equal 1 parenleftbigg integral sub B sub N bar u open\noindent parenthesisRemark H sub 2 epsilon . 1 3 open . \ parenthesisquad The x H closing− algebras parenthesis of closing examples parenthesis 2 \quad bar to. the 1 power 1 \quad of pand dx parenrightbigg 2 . 1 2 are degenerate . hline from p to 1 parenleftbig u in p Capital Xi parenleftbig R sub y to the power of N to the power of parenrightbig parenrightbig comma\noindent Example \quad 2 . 14 . \quad Let \quad $ A { 1 }$ \quad be \quad an \quad H − algebra on $ R ˆ{ N − 1 } , $ \quad and \quad l e t \quad $ B {\ infty } (R;A { 1 } ) $ \quad be the space of all continuous functions $ u : R \rightarrow A { 1 }$ such that l $im {\tau \rightarrow \ infty }\ parallel u ( \tau ) − \varsigma \ parallel \ infty = 0 , $ where $ \varsigma \ in A { 1 } ( \varsigma $ dependingon $u ) .$ \quad The space $ A = B {\ infty } (R;A { 1 } ) $ i s an H − algebra on

\ begin { a l i g n ∗} R ˆ{ N } . \end{ a l i g n ∗}

\noindent 2 . 5 . \quad The \quad spaces $ X ˆ{ p } { A } ( R ˆ{ N } { y } ) ( 1 \ leq p < \ infty ) . $ \quad The present and next subsections are concerned with function spaces of great interest in deterministic homogenization

\noindent theory .

For each real $ p \geq 1 , $ we first of all introduce the space $ p {\Xi } ( R ˆ{ N } { y }ˆ{ ) }$ of those functions $ u \ in L ˆ{ p } { l o c } ( R ˆ{ N } { y }ˆ{ ) }$ \quad for which the sequence \quad $ ( u ˆ{\ varepsilon } ) { 0 < \ varepsilon \ leq 1 } ( u ˆ{\ varepsilon }$ defined in subsection 2 . 1 ) \quad i s bounded in $ L ˆ{ p } { l o c } ( R ˆ{ N } { x } ) . $ \quad This is clearly a vector subspace of $ L ˆ{ p } { l o c } ( R ˆ{ N } { y }ˆ{ ) } . $ \quad Let

\ [ \ parallel u \ parallel {\Xi ˆ{ p }} = \sup { 0 < \ varepsilon \ leq 1 } ( \ int { B { N }} \mid u ( H {\ varepsilon } ( x ) ) \mid ˆ{ p } dx ) \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } ( u \ in p {\Xi } ( R ˆ{ N } { y }ˆ{ ) } ), \ ] 1 1 0 .. G period NGUETSENG comma N period SVANSTEDT \noindentwhere B sub1 1 N 0 ....\quad denotesG the . NGUETSENG open unit ball , of N R . sub SVANSTEDT x to the power of N period .... This defines a norm on p Capital Xi parenleftbig R sub y to the power of N to the power of parenrightbig comma 1 1 0 G . NGUETSENG , N . SVANSTEDT \noindentwhich makeswhere the latter $ B a Banach{ N } space$ \ h open f i l l parenthesisdenotes the the verification open unit is a routine ball of exercise $ R left ˆ{ N } { x } . $ \ h f i l l This defines a norm on where B denotes the open unit ball of N . This defines a norm on p ( N ), $ pto the{\ readerXi } closing(N Rparenthesis ˆ{ N } period{ y }ˆ{ ) } , $ Rx Ξ Ry Now commawhich .. let makes .. A be the an latter H hyphen a Banach algebra on space R to ( the the power verification of N open is parenthesis a routine for exercise H closing left parenthesis to period .. For each\noindent real pthe greaterwhich reader equal makes ) 1 . comma the .. latter we define a Banach space ( the verification is a routine exercise left N toX the sub A reader to theNow power ) . , of p let parenleftbigA be an R sub H - y algebra to the power on R of N( parenrightbigfor H). For open each parenthesis real p or≥ ....1 s, imply X sub A to the power ofwe comma define to the power of p .... or .... even X to the power of p .... when there .... is .... no .... danger .... of confusion \ hspace ∗{\pf i lN l }Now , \quad l e tp \quad $ Ap $ be an H − algebra on $ R ˆ{ N } ( $ f o r $ H ) closing parenthesisXA(Ry )( .... asor s imply XA, or even X when there is no danger of confusion ) as . $ \quad For each real $ p \geqN ) 1 , $ \quad we d ep f i n e p being thebeing closure the of closureA in p Capital of A in Xi parenleftbigpΞ(Ry . R subProvided y to the with power the of NΞ to− thenorm power, X ofA parenrightbigis a Banach period .... Provided with the Capitalspace Xi . to the power of p hyphen norm comma X sub A to the power of p is a Banach \noindentspace period$ X ˆLet{ p us} turn{ A to} the( proofs R ˆ{ ofN some} { y fundamental} ) ( $ results or \ h that f i l l weres imply pointed $ out X ˆ{ p } { A ˆ{ , }}$ \ hLet f i l l usor turnearlier\ h to f thei lin l proofs [even 3 0 ]of . $some X ˆ fundamental{ p }$ \ resultsh f i l l thatwhen were ther pointed e \ h out f i l l i s \ h f i l l no \ h f i l l danger \ h f i l l of confusion ) \ h f i l l as N earlier inProposition open square bracket 2 . 1 3 5 0 closing . The square mean bracket value periodM on R for H ( s ee subsection 2 . \noindentProposition1 )being 2viewed period the 1 as 5 closure period defined .. The of onmean $A$A, valueextends in M on $ R p to{\ the by powerXi } continuity of( N for R H ˆ to open{ N parenthesis} a{ y (} uniqueˆ{ s ee) subsection} ) . $ 2\ periodh f i l l 1Provided with the $ \Xi ˆ{ p } − $ norm $ , X ˆp{ p } { A }$ i s a Banach p closing parenthesiscontinuous viewed linear form on XA s til l denoted M. Furthermore , given u ∈ XA as .. definedand on a fixedA comma bounded .. extends open .. s by et .. continuityΩ to .. a .. open parenthesis unique closing parenthesis .. continuous linear \noindent spaceN . ε p ε form on in Rx , we have u → M(u) in L (Ω)− weak as ε → 0, where u is considered X sub Aas to the defined power on of p ..Ω. s til l denoted M period .. Furthermore comma .. given u in X sub A to the power of p .. and a fixed\ hspace bounded∗{\Prooff open i l l .} sLet et CapitalFor us turnψ Omega∈ A, towe the have proofs of some fundamental results that were pointed out in R sub x to the power of N comma .. we have u to the power of epsilon right arrow M open parenthesis u closing parenthesis \noindent earlier in [ 3Z 0 ] . 1 .. in L to the power of p open parenthesis| ψ(H Capital(x))dx Omega| ≤ closing | B | p parenthesisk ψ k p hyphen(0 < ε weak≤ 1), as epsilon right arrow 0 comma .. where ε N 0 Ξ u to the power of epsilon .. is consideredBN as \noindent Proposition 2 . 1 5 . \quad The mean value $M$ on $Rˆ{ N }$ for $H ( $ s ee subsection 2 . 1 ) viewed defined onwhere Capital| B OmegaN | stands period for the measure of BN ( with respect to Lebesgue measure on as \quad Ndefined on $A , $ \quad extends \quad− by \quad continuity to \quad a \quad ( unique ) \quad continuous linear form on Proof periodR ). .. ForAs psiε → in A0, commait follows we have | M(ψ) | ≤ | BN | 1p k ψ kΞp , from which we $vextendsingle-vextendsingle-vextendsingle-vextendsingle X ˆ{ deducep } { A the}$ first\quad parts of til the propositionl denoted byintegral $M extension sub . $ B sub\ byquad N continuity psi openFurthermore parenthesis . Now H, sub\ ,quad epsilon letgiven open parenthesis $ u \ in X ˆ{ p } { A }$ \quad and a fixed bounded open s et $ \Omega $ x closing parenthesisu and closingΩ be parenthesis as stated dxabove vextendsingle-vextendsingle-vextendsingle-vextendsingle . If u ∈ A, then it i s evident that uε → lessM(u or) equalin bar B sub N bar p 1 dividedL byp(Ω) prime− weak bar psi as barε → sub0. CapitalSo , Xiin to what the power follows of wep open assume parenthesis that 0u lessi s an epsilon arbitrarily less or equal given 1 closing parenthesis comma\noindentfunctionin $ R in ˆ{ N } { x } , $ \quad we have $ u ˆ{\ varepsilon }\rightarrow M( uwhere ) $ bar\pquad B sub Nin bar $ stands Lp0 ˆ{ for1p the} measure( \Omegap of B sub N) open− parenthesis$ weak aswith respect$ \ varepsilon to Lebesgue measure\rightarrow on 0 XA. Let ϕ ∈ L (Ω)(p0 = 1 − 1), ϕ assumed to be a nonzero function . Fix freely , $R to\quad the powerwhere of N $closing u ˆ{\ parenthesisvarepsilon period} ..$ As\quad epsilonp is right considered arrow 0 comma as .. it follows .. bar M open parenthesis psi η > 0. Thanks to the density of A in XA, we may consider some ψ ∈ A such that closingd e f i n parenthesis e d on $ bar\Omega less or equal. $ bar B sub N bar to the power of minus to the power of hline 1 p bar psi bar sub Capital Xi to R the power of p comma .. from which weε deduceε p the p η \noindent Proof . \quad For(Ω| u $− ψ\ psi| dx) \ in 1 ≤A , $ we have(0 < ε ≤ 1) first part of the proposition by extension by continuity period3 k ..ϕ NowkLp 0(Ω) comma .. let u .. and Capital Omega be as stated above period .. If u in A comma then it i s evident that u to the power of epsilon right arrow M open parenthesis u \ [ \arrowvertand \ int { B { N }}\ psi (H {\ varepsilon } ( x ) ) dx \arrowvert closing parenthesis .. inR L to the powerη of p open parenthesis Capital Omega closing parenthesis hyphen weak as \ leq \mid|M(u −Bψ) {Ω ϕdxN }\| ≤mid3 ( usep the\ f rfirst a c { part1 }{\ ofprime Proposition}\ parallel 2 . 1 5 )\ .psi On the\ parallel other {\Xi ˆ{ p }} epsilon righthand arrow , as 0pointed period .. out So commaabove ,in there what followsis some we real assume0 < that r ≤ u1 such i s an that arbitrarily given function in (X 0 sub A< to the\ varepsilon power of period\ toleq the power1 of ) p .... , Let\ ] phi in L to the power of p to the power of prime open parenthesis Z Z Capital Omega closing parenthesis open parenthesisε sub p to the power of primeη to the power of 1 underbar = 1 minus hline from p to 1 closing parenthesis comma phi assumed| toψ beϕdx a nonzero− M(ψ) functionϕdx| period ≤ .... Fix freely Ω Ω 3 \noindenteta greaterwhere 0 period $ ....\mid ThanksB to the{ densityN }\ ofmid A in$ X sub stands A to the for power the of measure comma to of the power $ B of{ pN we} may( $consider with some respect to Lebesgue measure on psi in A suchfor that all 0 < ε ≤ r. Hence , by writing \noindent $ R ˆ{ N } ) . $ \quad As $ \ varepsilon \rightarrow 0 , $ \quad i t f o l l o w s \quad parenleftbigg to the power of integralZ sub CapitalZ Omega barZ u to the power ofZ epsilon minus psi to the power of epsilon bar to the$ \ powermid ofM( p dx parenrightbigg\ psi ) hlineuε\ϕdx frommid− pM to\(uleq 1) lessϕdx or\mid equal= ( etauεB− dividedψ{ε)Nϕdx}\ by+ 3mid barψεϕdx phiˆ{ bar − subˆ{\ Lr p u prime l e {3em sub}{ open0.4 pt parenthesis}}} 1 { p } Capital\ parallel Omega closing\ psi parenthesis\ parallelΩ open parenthesis{\Xi ˆ{ 0Ωp less}} epsilon,Ω $ less\quad or equalfrom 1 closingΩ which parenthesis we deduce the first part of the proposition by extension byZ continuity . \Zquad Now , \quad l e t $ u $ \quad and and −M(ψ) ϕdx + M(ψ − u) ϕdx, $ \vextendsingle-vextendsingle-vextendsingle-vextendsingleOmega $ be M openΩ parenthesis u minusΩ psi closing parenthesis integral sub Capital Omegaas stated phiwe dx vextendsingle-vextendsingle-vextendsingle-vextendsingle above see immediately . \quad I that f $ u \ in A , $ then less orit equal i s eta evident divided thatby 3 open $uˆ parenthesis{\ varepsilon use the first} part\rightarrow of PropositionM 2 period ( 1 u .. 5 closing ) $ parenthesis\quad in period $ L ˆ{ p } ( \Omega ) − $ weak as $ \ varepsilon \rightarrow 0Z . $ \quad ZSo , in what follows we assume that $ u $ i s an arbitrarily given function in On the other hand comma as pointed out| aboveuεϕdx comma− M there(u) isϕdx some| real ≤ η 0 less r less or equal 1 such that vextendsingle-vextendsingle-vextendsingle-vextendsingleΩ integralΩ sub Capital Omega psi to the power of epsilon phi dx minus M open\noindent parenthesis$ psi X closingˆ{ p } parenthesis{ A ˆ{ integral. }}$ sub\ h Capital f i l l Let Omega $ phi\varphi dx vextendsingle-vextendsingle-vextendsingle-vextendsingle\ in L ˆ{ p ˆ{\prime }} ( \Omega less) or ( equal ˆ{ 1 eta{\ dividedunderline by 3 {\}}} { p ˆ{\prime }} = 1 − \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } ) , for\ allvarphi 0 less epsilon$ assumed less or equal to be r period a nonzero .. Hence function comma by writing . \ h f i l l Fix f r e e l y Line 1 integral sub Capital Omega u to the power of epsilon phi dx minus M open parenthesis u closing parenthesis integral sub Capital\noindent Omega$ phi\eta dx = integral> 0 sub Capital . $ \ Omegah f i l l openThanks parenthesis to the u to density the power of of epsilon $A$ minus in psi $Xˆ to the{ powerp } { ofA epsilon ˆ{ , }}$ closingwe may parenthesis consider phi some dx plus $ integral\ psi sub\ Capitalin A Omega $ such psi to that the power of epsilon phi dx Line 2 minus M open parenthesis psi closing parenthesis integral sub Capital Omega phi dx plus M open parenthesis psi minus u closing parenthesis integral sub Capital Omega\ [ ( ˆ phi{\ dxint comma} {\Omega }\mid u ˆ{\ varepsilon } − \ psi ˆ{\ varepsilon }\mid ˆ{ p } dx ) we\ seer u l immediately e {3em}{0.4 that pt } ˆ{ p } { 1 }\ leq \ f r a c {\eta }{ 3 \ parallel \varphi \ parallel { L p }\vextendsingle-vextendsingle-vextendsingle-vextendsingleprime { ( \Omega ) }} ( 0 < integral\ varepsilon sub Capital Omega\ leq u to1 the power) \ ] of epsilon phi dx minus M open parenthesis u closing parenthesis integral sub Capital Omega phi dx vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal eta \noindent and

\noindent $ \arrowvert M ( u − \ psi ) \ int {\Omega }\varphi dx \arrowvert \ leq \ f r a c {\eta }{ 3 } ( $ use the first part of Proposition 2 . 1 \quad 5 ) . On the other hand , as pointed out above , there is some real $ 0 < r \ leq 1 $ such that

\ [ \arrowvert \ int {\Omega }\ psi ˆ{\ varepsilon }\varphi dx − M( \ psi ) \ int {\Omega } \varphi dx \arrowvert \ leq \ f r a c {\eta }{ 3 }\ ]

\noindent f o r a l l $ 0 < \ varepsilon \ leq r . $ \quad Hence , by writing

\ [ \ begin { a l i g n e d }\ int {\Omega } u ˆ{\ varepsilon }\varphi dx − M ( u ) \ int {\Omega } \varphi dx = \ int {\Omega } ( u ˆ{\ varepsilon } − \ psi ˆ{\ varepsilon } ) \varphi dx + \ int {\Omega }\ psi ˆ{\ varepsilon }\varphi dx \\ − M( \ psi ) \ int {\Omega }\varphi dx + M ( \ psi − u ) \ int {\Omega } \varphi dx , \end{ a l i g n e d }\ ]

\noindent we see immediately that

\ [ \arrowvert \ int {\Omega } u ˆ{\ varepsilon }\varphi dx − M ( u ) \ int {\Omega } \varphi dx \arrowvert \ leq \eta \ ] Capital Sigma hyphen CONVERGENCE .. 1 1 1 \ hspacefor all∗{\ 0 lessf i epsilon l l } $ less\Sigma or equal− r period$ CONVERGENCE .. The proposition\quad follows1 1 thereby 1 period square Proposition .. 2 period 16 period .. The .. Gelfand transformation G : A right arrow C open parenthesis Capital Delta open Σ− CONVERGENCE 1 1 1 parenthesis\noindent Af closing o r a l parenthesis l $ 0 closing< \ parenthesisvarepsilon .. extends\ leq by r . $ \quad The proposition follows thereby for all 0 < ε ≤ r. The proposition follows thereby . Proposition 2 . 16 . $ .continuity\ square .... t $o .... a .... open parenthesis unique closing parenthesis .... continuous .... linear mapping .... of X sub A to the PropositionThe\quad Gelfand2 . transformation 16 . \quad The G\quad: AGelfand→ C(∆(A)) transformationextends by $ G : A \rightarrow power of p .... into L to the power of p open parenthesis Capital Delta open parenthesis A closingp parenthesisp closing parenthesis C( \continuityDelta ( t o A a ) ( unique ) $ )\quad continuousextends linear by mapping of XA into L (∆(A)) .... s ti l l s ti l l denoted by G period denoted by G. Proof . Let u ∈ A. Then , \noindentProof periodc o ..n t Let i n u u i t in y A\ periodh f i l l ..t Then o \ h comma f i l l a \ h f i l l ( unique ) \ h f i l l continuous \ h f i l l linear mapping \ h f i l l o f $ Xintegral ˆ{ p sub} { BA sub}$ N bar\ h u f iopen l l i parenthesis nZ t o $ L H ˆ sub{ p epsilon} ( open\Delta parenthesis( x closing A parenthesis) ) $ closing\ h f i l lparenthesiss t i l bar l to the p power of p dx less or equal bar u bar p Capital| u(H Xiε( tox)) the| dx power≤ of k u pk openpΞp parenthesis(0 < ε ≤ 1) 0. less epsilon less or equal 1 closing parenthesis B period\noindent denoted by $G .N $ Proof . \quad Let $ u \ in Ap .1 $ \quad− Then , Letting epsilonLetting rightε → arrow0, it 0 follows comma itM follows(| u | ) Mp open≤| B parenthesisN | bar1p k uu barkΞp to, hence the power of p closing parenthesis 1 divided by − p p less or equal bark GB(u sub) k NLp bar(∆( toA)) the≤ power | BN of| minus hline1 k u 1kΞ pp bar, according u bar sub to Capital Proposition Xi to the 2 power . 2 . of p The comma hence \ [ bar\ int G openproposition{ B parenthesis{ N }}\ u closingmid parenthesisu ( bar H L{\ p openvarepsilon parenthesis} Capital( x Delta ) open ) parenthesis\mid ˆ{ Ap closing} dx parenthesis\ leq \ parallel u \ parallel p {\Xi ˆ{ p }} ( 0 < p \ varepsilon \ leq 1 ) . \ ] closing parenthesisfollows less by or extension equal bar by B sub continuity N bar to the,A powerbeing of dense minus in hlineXA from.  p to 1 bar u bar sub Capital Xi to the power of p p p comma accordingRemark to2 Proposition . 1 7 . 2 periodThe mapping 2 period ..G The: X propositionA → L (∆(A)) derived from Proposition 2 . 1 follows by6 extension by continuity comma A being dense in X sub A to the power of period to the power of p square \noindent Letting $ \ varepsilon \rightarrow p 0 ,$p itfollows $M ( \mid u \mid ˆ{ p } Remarki 2 s period referred 1 7 toperiod as the .. The canonical mapping mapping G : X sub of A toXA theinto powerL (∆( of pA)) right. arrow L to the power of p open parenthesis Capital) \ f r aDelta c { 1 open}{ parenthesisp }\Theleq preceding A closing\mid parenthesis propositionB { N closing}\ has parenthesismid threeˆ{ important − derived \ r u from l corollaries e {3em Proposition}{0.4 . pt 2}} period1 1 6{ p }\ parallel u \ parallel {\Xi ˆ{ p }} , $ hence R p i s referredCorollary to as the canonical 2 . 18 mapping . We of have X subM A( tou) the = power∆(A) G( ofu) pdβ intofor L tou ∈ theXA power, where of p openM and parenthesisG Capital Delta open parenthesisdenote A closing the extension parenthesis mappings closing parenthesis constructed period in Propositions 2 . 1 5 - 2 . 1 6 , \ hspaceThe preceding∗{\ f i l l proposition} $ \ parallel has three importantG ( corollaries u ) period\ parallel L p ( \Delta (A)) \ leq \mid B respectively{ N }\mid . ˆ{ − } \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 }\ parallel u \ parallel {\Xi ˆ{ p }} CorollaryProof .... 2 . periodThis 18 period is straightforward .... We have M by open the parenthesis said propositions u closing and parenthesis use of =the integral definition sub Capital Delta open parenthesis, $ according A closing to parenthesis Proposition G open 2 parenthesis . 2 . \quad u closingThe parenthesis proposition d beta for u in X sub A to the power of comma to the of the measure β( see subsection 2.2).  Corollary 2 . 19 . Let 1 < p, q < +∞ power of p .... where M ....p and1 G 1 p p with 1 + = ≤ 1. If u ∈ X = XA and \noindentdenote thefollows extension bymappings extensionq r constructed by continuity in Propositions $2 period , A 1 $5 hyphen being 2 period dense 1 in.... 6 $ comma X ˆ{ respectivelyp } { A ˆperiod{ . }} \ squareProof period $ .... This is straightforward by the said propositions and use of the definition of the measure beta open parenthesis seethenuv subsection∈ Xr 2and periodG(uv 2) closing = G(u) parenthesisG(v). period square v ∈ Xq, \noindentCorollary ..Remark 2 period 2 19 . period 1 7 ... Let\quad 1 lessThemapping p comma q less plus $G infinity : .. Xˆ with{ hlinep } from{ A p}\ to 1 plusrightarrow 1 divided by qL = ˆ 1{ dividedp } by( r less\Delta orProof equal 1( . period AThis .. )If follows u in ) X $ to readily derivedthe power by Propositionof from p = XProposition sub A 2 .to 1 the 6 and power 2 . use 1 of 6of p .. H ando¨ lder ’ s inequality Equation:. v inCorollary X to the power 2 . of 20 q comma. The .. then fol uv lowing in X to ass the ertions power of are r and true G for open1 parenthesis≤ p < ∞ : uv closing parenthesis = G\noindent open parenthesisi s referred u closing parenthesis to as the G open canonical parenthesis mapping v closing of parenthesis $ X ˆ{ periodp } { A }$ i n t o $ L ˆ{ p } ( \DeltaProof period( .. A This ) follows ) readily . $ by Proposition 2 period 1 6 and use of H dieresis-o lder quoteright s inequality period square Corollary 2 period 20 period .. The fol lowingp ass ertions arep true for 1 less or equal p less infinity : \ centerline {The preceding( i ) If propositionu ∈ X , then has threeu ∈ X importantand G( corollariesu) = G(u). . } hline ( ii ) If u ∈ Xp, then | u |p∈ X1 and G(| u |p) =| G(u) |p . open parenthesis i closing parenthesis If u in X top the power of p commap .. then overbar u in X to the power of p .. and G open \noindent C o r o( l l aiii r y )\ Ifh f iψ l l∈ A2and . 18u . ∈\Xh f, i l lthenWehaveψu ∈ X $Mand (G( uψ)G(u )) = =G(ψu)\. int {\Delta (A parenthesis to the( power iv )of If hlineu ∈ uX closing1 and parenthesis further u =is G real open valued parenthesis , then u closingG(u) parenthesisis real valued period . If ) }openG parenthesis ( u ii closing ) d parenthesis\beta If$ u in f o X r to $ the u power\ in of p commaX ˆ{ p .. then} { barA ˆ u{ bar, }} to$ the\ powerh f i l of l pwhere in X to the$ M power $ \ h f i l l and $ G $ moreover of 1 .. and Gu ≥ open0 parenthesisa . e . bar ( almostu bar to everywhere the power of )p ,closing then parenthesisG(u) ≥ =0 bara . G e open . parenthesis u closing parenthesis bar to the power of p period ( v ) If u ∈ X1 ∩ L∞, then G(u) ∈ L∞(∆(A)) and \noindentopen parenthesisdenote iii closing the extension parenthesis If mappings psi in A and constructed u in X to the power in Propositions of p comma .. then 2 . psi 1 u 5 in− X2 to the . 1 power\ h f iof l l p ..6 and , respectively .

G open parenthesis psi closing parenthesis Gk open G(u) parenthesisk L∞(∆(A u)) closing≤ k u parenthesiskL∞ . = G open parenthesis psi u closing parenthesis period\noindent Proof . \ h f i l l This is straightforward by the said propositions and use of the definition open parenthesis iv closing parenthesis If u in X to the power of 1 and further u is real valued comma then G open parenthesis u\noindent closing parenthesisProofof the . is measure real valued( i ) period $ follows\beta .. If moreover by($ Proposition seesubsection 2 . 1 6 $2 and . use 2 of )the equality . \ square $ C o r o l l a r y \quad 2 . 19 . \quad Let $ 1 < p , q < + \ infty $ \quad with $ \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } u greaterG equal(u) 0 = ..G a( period eu period) for .. openu ∈ parenthesisA. We almost turn everywhere now to closing the proof parenthesis of ( ii comma ) . .. then Let G open parenthesis + \ f r a c { 1 }{ q } = \ f r a c { 1 }{ r }\ leq 1 . $ \quad I f $ u \ in X ˆ{ p } = X ˆ{ p } { A }$ u closing parenthesisu ∈ X greaterp. Choose equal 0 some a period sequence e period \quad and N open parenthesis(un) in v closingA such parenthesis that un If→ u inu Xin top theΞ(R power) as of 1n cap→ L to ∞ the. powerBy taking of infinitya = commaun(y) .. thenand G open parenthesis u closing parenthesisb = u(y)( inwhere L to the the power integer of infinityn > 0 openand parenthesis the point Capitaly ∈ RN Deltaare arbitrarily open parenthesis fixed A ) closingin the parenthesis closing parenthesis\ begin { a ls .. i gimple and n ∗} inequality \ tagLine∗{$ 1 bar v G\ openin parenthesisX ˆ{ q } u closing, $} parenthesisthen uv bar L\ in infinityX open ˆ{ parenthesisr } and Capital G Delta ( uv open parenthesis ) = G A closing ( u ) G ( v ) . p p p−1 parenthesis closing parenthesis less|| ora equal| − | barb | |≤ u barp | a sub− b L| infinity(| a | + period| b |) Line(a, 2 b hline∈ C) \endProof{ a l iperiod g n ∗} .. open parenthesis i closing parenthesis .. follows by Proposition 2 period 1 .. 6 .. and use of the equality G open o¨ parenthesisand u closing then parenthesis using an = obvious G open H parenthesislder ’ tos inequality the power of , hlinewe get u closing parenthesis .. for \noindent Proof . \quad This follows readily by Proposition 2 . 1 6 and use of H $ \ddot{o} $ u in A period .. We turn now to the proof of openp parenthesisp ii closing parenthesis period .. Let u in X to the power of p period k| u | − | u | k 1 ≤ c k u − u k p ..lder Choose ’ s some inequality sequence $ . \ squaren $ Ξ n Ξ Corollaryopen parenthesis 2 . u20 sub . n\ closingquad The parenthesis fol lowing .... in A such ass that ertions u sub n are right true arrow for u in p $ Capital 1 \ Xileq parenleftbigp < R to\ theinfty power of: N $ parenrightbig .... as n right arrow infinity period .... By taking a = u sub n open parenthesis y closing parenthesis .... and b = u open parenthesis y closing parenthesis open parenthesis where the integer n greater 0 and the point y in R to the power of\ [ N\ arer u l arbitrarily e {3em}{0.4 fixed pt closing}\ ] parenthesis in the s imple inequality bar bar a bar to the power of p minus bar b bar to the power of p bar less or equal p bar a minus b bar open parenthesis bar a bar\ centerline plus bar b bar{( closing i ) I f parenthesis $ u to\ in the powerX ˆ{ ofp p} minus, $1 open\quad parenthesisthen a $ comma\ overline b in C{\}{ closingu parenthesis}\ in X ˆ{ p }$ \quadand thenand using $ G an obvious ( ˆ{\ Hr o-dieresisu l e {3em}{ lder0.4 quoteright pt }} su inequality ) = comma G we ( get u ) . $ } bar bar u sub n bar to the power of p minus bar u bar to the power of p bar 1 Capital Xi less or equal c bar u sub n minus u bar\ centerline sub Capital{ Xi(to i i the )power I f $ of u p \ in X ˆ{ p } , $ \quad then $ \mid u \mid ˆ{ p }\ in X ˆ{ 1 }$ \quad and $ G ( \mid u \mid ˆ{ p } ) = \mid G ( u ) \mid ˆ{ p } . $ }

\ centerline {( i i i ) I f $ \ psi \ in A $ and $ u \ in X ˆ{ p } , $ \quad then $ \ psi u \ in X ˆ{ p }$ \quad and $ G ( \ psi ) G ( u ) = G ( \ psi u ) . $ }

\ hspace ∗{\ f i l l }( i v ) I f $ u \ in X ˆ{ 1 }$ and further $ u $ is real valued , then $G ( u )$ is real valued . \quad I f moreover

\noindent $ u \geq 0 $ \quad a . e . \quad ( almost everywhere ) , \quad then $ G ( u ) \geq 0 $ a . e .

\ centerline {( v ) I f $ u \ in X ˆ{ 1 }\cap L ˆ{\ infty } , $ \quad then $ G ( u ) \ in L ˆ{\ infty } ( \Delta ( A ) ) $ \quad and }

\ [ \ begin { a l i g n e d }\ parallel G ( u ) \ parallel L \ infty ( \Delta (A) ) \ leq \ parallel u \ parallel { L \ infty } . \\ \ r u l e {3em}{0.4 pt }\end{ a l i g n e d }\ ]

\noindent Proof . \quad ( i ) \quad follows by Proposition 2 . 1 \quad 6 \quad and use of the equality $ G ( u ) = G ( ˆ{\ r u l e {3em}{0.4 pt }} u ) $ \quad f o r $ u \ in A . $ \quad We turn now to the proof of ( ii ) . \quad Let $ u \ in X ˆ{ p } . $ \quad Choose some sequence

\noindent $ ( u { n } ) $ \ h f i l l in $A$ such that $u { n }\rightarrow u $ in $ p {\Xi } ( R ˆ{ N } ) $ \ h f i l l as $ n \rightarrow \ infty . $ \ h f i l l By taking $ a = u { n } ( y ) $ \ h f i l l and

\noindent $b = u ( y ) ($ wheretheinteger $n > 0 $ and the point $ y \ in R ˆ{ N }$ are arbitrarily fixed ) in the s imple inequality

\ [ \mid \mid a \mid ˆ{ p } − \mid b \mid ˆ{ p }\mid \ leq p \mid a − b \mid ( \mid a \mid + \mid b \mid ) ˆ{ p − 1 } ( a , b \ in C) \ ]

\noindent and then using an obvious H $ \ddot{o} $ lder ’ s inequality , we get

\ [ \ parallel \mid u { n }\mid ˆ{ p } − \mid u \mid ˆ{ p }\ parallel 1 {\Xi } \ leq c \ parallel u { n } − u \ parallel {\Xi ˆ{ p }}\] 1 1 2 G . NGUETSENG , N . SVANSTEDT p−1 p p with c = p supm>0 k| um | − | u |kΞp < ∞. We deduce that | un | → | u | in 1 p 1 p Ξ as n → ∞, hence | u | ∈ X , s ince | un | ∈ A( Proposition 2 . 2 ) . On the other hand , according to Proposition 2 . 1 6 , we have in the L1(∆(A))− norm ,

p p p p G(| un | ) → G(| u | ) and | G(un) | →| G(u) | asn → ∞. Therefore the rest of ( i i ) fo llows by Proposition 2 . 2 , once again . Assertion ( ii i ) being straightforward , let us next verify ( iv ) . For this purpose , fix freely u ∈ X1.

Suppose u i s real valued . Then , by ( i ) we have G(u) = G(u) and so G(u) i s real valued too . Suppose further that u ≥ 0 a . e . Let ψ ∈ A with ψ ≥ 0. Then ψu ∈ X1 with ψu ≥ 0 a . e . , hence M(ψu) ≥ 0( use Proposition 2 . 1 5 ) . Consequently Z G(ψ)G(u)dβ ≥ 0, ∆(A) R as is straightforward by ( i ii ) and use of Corollary 2 . 1 8 . Thus , ∆(A) ϕG(u)dβ ≥ 0 for all ϕ ∈ C(∆(A)) with ϕ ≥ 0. This shows that G(u) ≥ 0 a . e . ( see , e . g . , [ 5 , p . 47 , Corol . 3 ] ) . We will finally establish ( v ) . Let u ∈ X1 ∩ L∞. S ince | u |≤ k u kL∞ a . e . , we have | ψu |≤ k u k L∞ | ψ | a . e . for all ψ ∈ A. Thus M(| ψu |) ≤ k u kL∞ M(| ψ |) for all ψ ∈ A ( see Proposition 2 . 1 5 ) . We deduce by Corollary 2 . 1 8 and use of parts ( ii ) and ( i ii ) that Z Z | G(ψ)G(u)dβ| ≤ k u k L∞ | G(ψ) | dβ ∆(A) ∆(A) for all ψ ∈ A, or equivalently , Z | ϕG(u)dβ| ≤ k u kL∞k ϕ kL1(∆(A)) ∆(A) for all ϕ ∈ C(∆(A)). Hence ( v ) follows .  N  −1 1  Remark 2 . 2 1 . Let A = C (Y ) with Y = ( see subsection per 2, 2 p p 2 . 3 ) . Then XA = Lper(Y ) (1 ≤ p < ∞), where the r ight - hand s ide denotes the p N space of Y − periodic functions in Lloc(R ). Indeed , this follows immediately by two N p N facts : 1 ) the space pΞ(R ) is continuously embedded in Lloc(R ); 2) the space p N Lper(Y ) i s continuously embedded in pΞ(R ), as i s straightforward by [ 2 6 , Lemma 1 ] . 2 . 6 . Sobolev spaces W m,p(∆(A)). Let A be an H - algebra on RN ( for H). Be - fore we can define so - called Sobolev spaces on ∆(A), we need to introduce the notion of a partial derivative on ∆(A). This will be achieved by carrying over the usual derivatives on RN . Specifically , for any integer m ≥ 1, let

m m N ) α N A = {ψ ∈ C (Ry : Dy ψ ∈ Aforα ∈ N , | α |≤ m} and

α m k ψ k m =| sup kDy ψk∞ (ψ ∈ A ), α|≤m

| α ∂ α| m where Dy = α1 αN Provided with the norm k · km,A is a Banach ∂y1 ...∂yN . space . Furthermore , put 1 1 2 .. G period NGUETSENG comma N period SVANSTEDT \noindentwith c = p1 supremum 1 2 \quad subG m greater . NGUETSENG 0 bar bar , u N sub . m SVANSTEDT bar minus bar u bar bar sub Capital Xi to the power of p to the power of p minus 1 less infinity period .. We deduce that .. bar u sub n bar to the power of p right arrow bar u bar to the power of p in \noindent with $ c = p \sup { m > 0 }\ parallel \mid u { m }\mid − \mid Capital Xi to the power of 1 .. as m u n right\mid arrow\ parallel infinity commaˆ{ hencep − .. bar1 u} bar{\ toAXi∞ the=ˆ power{ ∩p }} ofA p in< X to\ infty the power of. 1 $ comma\quad s inceWe .. deduce bar u sub that n bar\ toquad the . power$ \mid of p inu A{ openn parenthesis}\mid ˆ Proposition{ p }\rightarrow 2 .. period 2 closing\mid parenthesisu \ periodmid ˆ ..{ Onp the}$ other in hand $ \Xi commaˆ{ 1 }$ \quad as $according n \rightarrow to Proposition ..\ infty 2 period 1, .. $ 6 comma hence we\quad have inm$ the≥\1mid L to theu power\mid of 1ˆ{ openp }\ parenthesisin X Capital ˆ{ 1 Delta} , open $ parenthesiss i n c e \quad A closing$ parenthesis\mid u closing{ n }\ parenthesismid ˆ hyphen{ p }\ normin commaA ( $ Proposition 2 \quad . 2 ) . \quad On the other hand , accordingG open parenthesis to Proposition bar u sub n bar\quad to the2 power . 1 \ ofquad p closing6 , parenthesis we have right in the arrow $Lˆ G open{ parenthesis1 } ( bar\Delta u bar to the(A power of)) p closing− parenthesis$ norm and , bar G open parenthesis u sub n closing parenthesis bar to the power of p right arrow bar G open parenthesis u closing parenthesis bar to the power of p as n right arrow infinity period \ [G(Therefore the\mid rest of openu { parenthesisn }\mid i i closingˆ{ p } parenthesis) \rightarrow .. fo llows by PropositionG( 2 period\mid 2 commau \mid .. onceˆ{ againp } period) .. and Assertion\mid G .. open ( parenthesis u { n ii} i closing) \ parenthesismid ˆ{ p }\rightarrow \mid G ( u ) \mid ˆ{ p } as n being\rightarrow straightforward\ commainfty let. us\ next] verify open parenthesis iv closing parenthesis period .. For this purpose comma fix freely u in X to the power of 1 period hline \noindentSuppose uTherefore i s real valued the period rest .... Then of ( comma i i ) by\ openquad parenthesisfo llows i closing by Proposition parenthesis we 2 have . 2 G open, \quad parenthesisonce u again closing . \quad A s s e r t i o n \quad ( i i i ) parenthesisbeing straightforward = G open parenthesis , u let closing us parenthesisnext verify and so ( G iv open ) . parenthesis\quad For u closing this parenthesis purpose i, s fix real freely $ u \ invaluedX too ˆ{ period1 } ... Suppose $ further that u greater equal 0 a period e period .. Let psi in A with psi greater equal 0 period .. Then psi u in X to the power of 1 \ [ with\ r u lpsi e {3em u greater}{0.4 equal pt }\ 0] a period e period comma hence M open parenthesis psi u closing parenthesis greater equal 0 open parenthesis use Proposition .. 2 period 1 5 closing parenthesis period .. Consequently integral sub Capital Delta open parenthesis A closing parenthesis G open parenthesis psi closing parenthesis G open parenthesis u\noindent closing parenthesisSuppose d beta $ greater u $ i equal s real 0 comma valued . \ h f i l l Then,by(i)wehave $G ( u ) = Gas ( is straightforward u )$ andso by open parenthesis $G ( i ii u closing )$ parenthesis isreal and use of Corollary .. 2 period 1 8 period .. Thus comma integral sub Capital Delta open parenthesis A closing parenthesis phi G open parenthesis u closing parenthesis d beta greater equal 0\noindent valued too . \quad Suppose further that $ u \geq 0 $ a . e . \quad Let $ \ psi \ infor allA phi $ in with C open $ parenthesis\ psi \geq Capital0 Delta . open $ parenthesis\quad Then A closing $ \ psi parenthesisu closing\ in parenthesisX ˆ{ 1 }$ .. with phi greater equalwith 0 period $ \ psi .. Thisu shows\ thatgeq G open0$ parenthesis a.e. u ,henceclosing parenthesis $M greater ( \ psi equal 0 au period ) e period\geq .. open0 parenthesis ( $ use see Proposition \quad 2 . 1 5 ) . \quad Consequently comma e period g period comma .. open square bracket 5 comma \ [ p\ periodint {\ 47 commaDelta Corol(A) period 3 closing} G( square bracket\ psi closing) parenthesis G ( period u .. ) We dwill finally\beta establish\geq open parenthesis0 , \ ] v closing parenthesis period .. Let u in X to the power of 1 cap L to the power of infinity period .. S ince bar u bar less or equal bar u bar sub L infinity \noindenta period eas period is comma straightforward we have bar psi by u bar ( iless ii or ) equal and bar use u bar of L Corollary infinity bar psi\quad bar a2 period . 1 e 8 period . \quad .. forThus all psi in $ A , period\ int ..{\ ThusDelta M open(A) parenthesis bar}\ psivarphi u bar closingG parenthesis ( u less ) or equal d bar\beta u bar sub\geq L infinity0 $ M open parenthesis bar psif o bar r a closing l l $ parenthesis\varphi \ in C( \Delta ( A ) ) $ \quad with $ \varphi \geq 0 . $for all\quad psi inThisshowsthat A open parenthesis see $G Proposition ( ..u 2 period ) \ 1geq 5 closing0 parenthesis $ a . e period . \quad .. We( deduce see by , e Corollary . g . 2 , period\quad 1 [ 5 , 8p. .. and 47 use , of Corol . 3 ] ) . \quad We will finally establish ( v ) . \quad Let $ u \ in X ˆ{ 1 } \capparts openL ˆ{\ parenthesisinfty } ii closing. $ parenthesis\quad S andi n c e open $ parenthesis\mid u i ii closing\mid parenthesis\ leq that\ parallel u \ parallel { L \ inftyvextendsingle-vextendsingle-vextendsingle-vextendsingle}$ integral sub Capital Delta open parenthesis A closing parenthesis G opena . parenthesis e . ,wehave psi closing $ parenthesis\mid \ Gpsi open parenthesisu \mid u closing\ leq parenthesis\ parallel d beta vextendsingle-vextendsingle-vextendsingle-u \ parallel L \ infty \mid vextendsingle\ psi \mid less$ or a equal . e bar . \ uquad bar Lf o infinity r a l l integral $ \ psi sub Capital\ in DeltaA open . $ parenthesis\quad Thus A closing $ M parenthesis ( \mid bar G\ openpsi parenthesisu \mid psi) closing\ leq parenthesis\ parallel bar d beta u \ parallel { L \ infty } M( \mid \ psi \mid ) $ f ofor r all a l l psi in $ A\ psi comma\ orin equivalentlyA ( comma $ see Proposition \quad 2 . 1 5 ) . \quad We deduce by Corollary 2 . 1 8 \quad and use o f vextendsingle-vextendsingle-vextendsingle-vextendsingle integral sub Capital Delta open parenthesis A closing parenthesis phi G\noindent open parenthesisparts u ( closing ii ) parenthesis and ( i d ii beta ) that vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal bar u bar sub L infinity bar phi bar sub L to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis\ [ \arrowvert \ int {\Delta (A) } G( \ psi ) G ( u ) d \beta \arrowvert \ leqfor all\ phiparallel in C open parenthesisu \ parallel Capital DeltaL open\ infty parenthesis\ int A closing{\ parenthesisDelta closing(A) parenthesis}\mid period ....G( Hence open\ psi parenthesis) \mid v closingd \ parenthesisbeta \ ] follows period square Remark .. 2 period 2 1 period .. Let A = C sub per open parenthesis Y closing parenthesis .. with Y = Row 1 minus 1 underbar 1 underbar Row 2 2 comma 2 . to the power of N open parenthesis see subsection 2 period 3 closing parenthesis period .. Then \noindentX sub A tof theo r power a l l of $ p\ =psi L sub\ perin to theA power , $ of p or open equivalently parenthesis Y closing , parenthesis open parenthesis 1 less or equal p less infinity closing parenthesis comma where the r ight hyphen hand s ide denotes the space of Y hyphen \ [ periodic\arrowvert functions\ inint L sub{\ locDelta to the power(A) of p parenleftbig}\varphi R to the powerG of( N parenrightbig u ) d period\beta .. Indeed\arrowvert comma this follows\ leq immediately\ parallel by twou facts\ parallel : .. 1 closing{ parenthesisL \ infty }\ parallel \varphi \ parallel { L ˆ{ 1 } ( \Deltathe space(A)) p Capital Xi parenleftbig}\ ] R to the power of N parenrightbig .. is continuously embedded in L sub loc to the power of p parenleftbig R to the power of N parenrightbig semicolon 2 closing parenthesis the space L sub per to the power of p open parenthesis Y closing parenthesis \noindenti s continuouslyf o r embeddeda l l $ \varphi in p Capital\ Xiin parenleftbigC( R to\Delta the power of( N parenrightbig A ) ) comma . $ as\ ih sf straightforward i l l Hence ( by v open ) follows square$ . bracket\ square 2 .. $6 comma Lemma 1 closing square bracket period 2 period 6 period .... Sobolev spaces W to the power of m comma p open parenthesis Capital Delta open parenthesis A closing parenthesis\noindent closingRemark parenthesis\quad 2 period . 2 .... 1 Let. \quad A be anLet H hyphen $ A algebra = C on{ R toper the} power( of Y N open ) $ parenthesis\quad with for H closing $ Y =parenthesis\ l e f t ( period\ begin ....{ array Be hyphen}{ cc } − 1 {\underline {\}} & 1 {\underline {\}}\\ 2 , 2 \end{ array }\ right )ˆ{ N } ( $fore see we can subsection define so hyphen 2 . called3 ) . Sobolev\quad spacesThen on Capital Delta open parenthesis A closing parenthesis comma we need to introduce$ X ˆ{ thep } { A } = L ˆ{ p } { per } ( Y ) ( 1 \ leq p < \ infty ) , $ where the r ight − hand s ide denotes the space of $ Ynotion− of$ a partial derivative on Capital Delta open parenthesis A closing parenthesis period .. This will be achieved by carrying overperiodic functions in $ L ˆ{ p } { l o c } ( R ˆ{ N } ) . $ \quad Indeed , this follows immediately by two facts : \quad 1 ) thethe space usual derivatives $ p {\ onXi R to} the( power R of ˆ{ NN period} ) .. $ Specifically\quad commais continuously for any integer embedded m greater equal in $1 comma L ˆ{ p let} { l o c } (A R to theˆ{ powerN } of) m = ; braceleftbig 2 )$ psi in thespace C to the power $Lˆ of m parenleftbig{ p } { per R sub} y to( the Y power ) of $ N to the power of parenrightbig : Di sub s continuously y to the power of embedded alpha psi in in A for $ alpha p {\ inXi N to} the( power R of ˆ{ NN comma} ) bar alpha , $ bar as less i s or straightforward equal m bracerightbig by [ 2 \quad 6 , Lemma 1 ] . and \noindentbar psi bar2 m . = 6 bar . \ supremumh f i l l Sobolev alpha bar spaces less or equal $Wˆ m{ vextenddouble-vextenddoublem , p } ( \Delta D sub( y to A the power ) ) of alpha . $ psi\ h f i l l Let vextenddouble-vextenddouble$ A $ be an H − algebra infinity on open $ R parenthesis ˆ{ N } psi( $in A f too r the $ power H of ) m closing . $ parenthesis\ h f i l l Be comma− where D sub y to the power of alpha = partialdiff to the power of bar alpha bar divided by partialdiff y sub 1 to the power of alpha\noindent sub 1 periodfore period we can period define partialdiff so − y subcalled N to the Sobolev power of spaces alpha sub on N sub $ period\Delta .. Provided( A with ) the norm , $ .. weneed bar times to introduce the bar sub m comma A to the power of m .. is .. a Banach space period \noindentFurthermorenotion comma of put a partial derivative on $ \Delta ( A ) . $ \quad This will be achieved by carrying over theLine usual 1 A to derivatives the power of infinity on = $ cap R ˆ Case{ N 1} m Case. $ 2\ periodquad LineSpecifically 2 m greater equal , for 1 any integer $ m \geq 1 , $ l e t

\ [ A ˆ{ m } = \{\ psi \ in C ˆ{ m } ( R ˆ{ N } { y }ˆ{ ) } : D ˆ{\alpha } { y } \ psi \ in A f o r \alpha \ in N ˆ{ N } , \mid \alpha \mid \ leq m \}\ ]

\noindent and

\ [ \ parallel \ psi \ parallel m = \mid \sup {\alpha \mid \ leq m }\Arrowvert D ˆ{\alpha } { y }\ psi \Arrowvert \ infty ( \ psi \ in A ˆ{ m } ), \ ]

\noindent where $ D ˆ{\alpha } { y } = \ f r a c {\ partial ˆ{\mid }\alpha \mid }{\ partial y ˆ{\alpha { 1 }} { 1 } ... \ partial y ˆ{\alpha { N }} { N }} { . }$ \quad Provided with the norm \quad $ \ parallel \cdot \ parallel { m } , A ˆ{ m }$ \quad i s \quad a Banach space . Furthermore , put

\ [ \ begin { a l i g n e d }\ l e f t . A ˆ{\ infty } = \cap A\ begin { a l i g n e d } & m \\ &. \end{ a l i g n e d }\ right . \\ m \geq 1 \end{ a l i g n e d }\ ] Capital Sigma hyphen CONVERGENCE .. 1 1 3 $ We\Sigma provide A− to$ the CONVERGENCE power of infinity\quad with the1 1lo cally3 convex topology defined by the family of norms Webar provide times bar m $ openA ˆ{\ parenthesisinfty } m$ greater with equal the 1 lo closing cally parenthesis convex comma topology which definedmakes it a by Fr acute-e the family chet space of period norms .. Σ− CONVERGENCE 1 1 3 We provide A∞ with the lo cally convex topology defined Finally commaby the s et family of norms \noindentD to the power$ \ parallel of m open parenthesis\cdot Capital\ parallel Delta openm parenthesis ( m A closing\geq parenthesis1 ) closing ,$ parenthesis whichmakes = braceleftbig it aFr k · k m(m ≥ 1), which makes it a Fr e´ chet space . Finally , s et phi$ \ inacute C open{e} parenthesis$ chet Capital space Delta . \quad open parenthesisFinally A , closing s et parenthesis closing parenthesis : G to the power of minus 1 open parenthesis phi closing parenthesis in A to the power of m bracerightbig open parenthesis m greater equal 1 closing parenthesis D \ begin { a l i g n ∗} open parenthesis Capital Deltam open parenthesis A closing parenthesis−1 m closing parenthesis = braceleftbig phi in C open parenthesis D ˆ{ m } ( \DeltaD (∆((A)) A = {ϕ )∈ C(∆( )A)) = : G \{\(ϕ) ∈ Avarphi} (m ≥ 1)\ in C( \Delta (A)) Capital Delta open parenthesis A closing parenthesis closing parenthesis−1 : G to∞ the power of minus 1 open parenthesis phi closing parenthesis: G ˆ{ in− A to1 the} power( \ ofvarphi infinityD(∆( bracerightbig)A)) =\ in{ϕ ∈ period CA(∆( ˆA{ Equation:))m : G}\}(ϕ) Remark∈ A(}. 2 m period\geq 22 period1 .. D ) to\\ theD( power of\ mDelta open parenthesis( A ) Capital ) Delta = open\{\ parenthesisvarphiDm(∆( AA closing))\ =inG parenthesis(AmC()andD(∆( closing\ADelta)) = parenthesisG(A∞(). = A G open ) parenthesis ) Remark : GA2.22 m ˆ{. closing − parenthesis1 } ( and\varphi D open parenthesis) \ in CapitalA ˆ{\ Deltainfty open}\} parenthesis. A\\\ closingtag ∗{ parenthesis$Remark closing 2 parenthesis . 22 = G .$ open} D parenthesis ˆ{ m } A( infinity\Delta closing(A) parenthesis )=G(Am) periodWe are now is a position to andD( define partial\ derivativesDelta ( on A∆(A). ) ) = G ( A \ infty ).We are nowDefinition is a position 2 . to 23 define . partialBy derivatives the partial on derivative Capital Delta of openindex parenthesisi (1 ≤ i A≤ closingN) on parenthesis∆(A) we period \end{ a l i g n ∗} Definitionshall 2 period understand 23 period the .. unboundedBy the partial l derivativeinear operator of index∂ i openfrom parenthesisC(∆(A)) 1 lessto orC(∆( equalA)) i less or equal N closing 1 parenthesisdefined on Capital as DeltaD( open∂i) parenthesis = D (∆( AA)) closing (D(∂ parenthesisi) stands we for the domain of ∂i), ∂iϕ = \ centerlineshall understand{Weare the unbounded now is a l inearposition operator to partialdiff define subpartial i .. from derivatives C open parenthesis on $ Capital\Delta Delta( open A parenthesis ) . $ } A closing parenthesis closing parenthesis .. to C∂ open parenthesis Capital Delta open parenthesis A closing parenthesis closing (G ◦ ◦ G−1)ϕforϕ ∈ D1(∆(A)). parenthesis ∂yi \noindentdefined ....Definition as .... D open 2 parenthesis . 23 . partialdiff\quad By sub the i closing partial parenthesis derivative = D to the of power index of 1 $ open i parenthesis ( 1 Capital\ leq Deltai \ leq N ) $ on $ \Delta ( A ) $ we open parenthesisMore A closing generally parenthesis , the closing partial parenthesis derivative open parenthesis of index D open parenthesisα ∈ N partialdiffon sub∆(A i) closing parenthesis shall understand the unbounded l inear operator $ \ partial { i }$ N \quad from $ C ( \Delta .... stands ....i s for .... defined the .... domainto be the .... of unbounded partialdiff sub linear i closing operator parenthesis∂α commafrom partialdiffC(∆(A)) subto i phiC(∆( =A)) (open A parenthesis )such ) that $ G\ circquad partialdiffto $ divided C ( by partialdiff\Delta y i( circ G A to the ) power ) $ of minus 1 closing parenthesis phi for phi in D to the powerD of(∂α 1) open = parenthesisD|α|(∆(A)) Capitaland Delta∂α openϕ = parenthesis (G ◦ Dα A◦ G closing−1)ϕ parenthesisfor ϕ ∈ closing D|α parenthesis|(∆(A)). We period \noindent d e f i n e d \ h f i l l as \ h f i l l $ D ( \ partialy { i } ) = D ˆ{ 1 } ( \Delta ( More generally commam .. the .. partial .. derivative .. of index .. alpha in N toα the power of Nm .. on .. Capital Delta open equip D (∆(A)) with the norm k ϕ k m = sup|α|≤m k ∂ ϕ k∞ (ϕ ∈ D (∆(A))), A))(D(parenthesis A closing parenthesis\ ..partial i s .. defined{ i } ) $ \ h f i l l stands \ h f i l l f o r \ h f i l l the \ h f i l l domain \ h f i l l o f $ \ partialand {D(∆(i }A))),with\ partial the family{ i }\ of normsvarphi k= · k $m (m ≥ 1). It is easily to be theseen unbounded that D linearm(∆(A operator)) is a Banach partialdiff space to the and powerD(∆( ofA alpha)) is a.. Fr frome´ chet C open space parenthesis . Furthermore Capital Delta open parenthesis A closing parenthesis closing parenthesis .. to C openm parenthesis Capital Delta open parenthesis A closing parenthesis closing \ begin { a l, iG gmaps n ∗} Am is ometrically onto D (∆(A)) and A∞ isomorphically onto D(∆(A)). The parenthesis .. such that 0 0 (G topological\ circ \ f r dual a c {\ ofpartialD(∆(A))}{\is denotedpartial by yD (∆( i A}\)). circWe assumeG ˆ{ that − D1(∆(} A))) is\varphi f o r D open parenthesis partialdiff to the power of alpha closing parenthesis = D to the0 power of bar alpha bar open parenthesis \varphi provided\ in D with ˆ{ 1 the} ( strong\Delta dual topology(A)). . Each T ∈ D (∆(A)) is called a Capital Deltadistribution open parenthesis on ∆( AA); closingthe value parenthesis of T at closing some parenthesisϕ ∈ D(∆(A ....)) is and denoted .... partialdiff to the power of alpha phi = parenleftbig\end{ a l i g n G∗} circ D sub y to the power of alpha circ G to the power of minus 1 parenrightbig phi .... for .... phi in D to the power by hT, ϕi. The derivative of index α ∈ NN of T is defined to be the distribution of bar alpha barα open parenthesis Capital Delta openα parenthesis A| closingα| parenthesis closing parenthesis period .... We More generally∂ T on , \∆(quadA) thegiven\quad by ph a∂ r tT, i a ϕ li \quad= (−d e1) r i vh aT, t iϕ vi e \forquad ϕo f index∈ D(∆(\quadA)). $It \alpha \ in equip Dis to the an power easy of exercisem open parenthesis to verify Capital that the Delta transformation open parenthesisT A→ closing∂αT maps parenthesis continuously closing parenthesis .... with Nthe ˆ norm{ N } ....$ bar\quad phi baron m\quad = supremum$ \Delta sub bar( alpha A bar ) less $ or\ equalquad mi bar s \ partialdiffquad d e f to i n thee d power of alpha phi bar sub to be theand unbounded linear operator $ \ partial ˆ{\alpha }$ \quad from $ C ( \Delta ( infinity open parenthesis0 phi in D to the power of m open parenthesis Capital Delta open parenthesis A closing parenthesis closing A ) )l inearly $ \quadD (∆(toA)) $into C it ( self\ .Delta ( A ) ) $ \quad such that parenthesis closing parenthesisIn passing comma we wish to draw attention to one basic result . and D open parenthesis Capital Delta open parenthesis A closingm parenthesis closing parenthesis .. with .. the .. family .. of \noindentProposition$ D ( 2\ partial . 24 . ˆ{\Foralpha any} ϕ)∈ D =(∆( DA))( ˆ{\m ≥mid1) and\ anyalpha multi\ -mid index} (α \Delta ( norms .. barwith times bar sub m open parenthesis m greater equal 1 closing parenthesis period .. It .. is .. easily .. seen that AD )to the ) power $ \ ofh mf i lopen l and parenthesis\ h f i l l Capital$ \ partial Delta openˆ{\ parenthesisalpha A}\ closingvarphi parenthesis= closing ( Gparenthesis\ circ is a BanachD ˆ{\ spacealpha } { y } and\ circ D openG parenthesis ˆ{ − 1 Capital} ) Delta\varphi open parenthesis$ \ h f i A l l closingf o r \ parenthesish f i l l $ closing\varphi parenthesis\ in is aD Fr ˆ acute-e{\mid chet\ spacealpha period\mid } ..( Furthermore\Delta comma( A G ) ) . $ \ h f i l l We Z | α |≤ m, wehave ∂αϕ(s)dβ(s) = 0. 1 ≤ maps A m is ometrically onto D to the power of m open parenthesis∆(A) Capital Delta open parenthesis A closing parenthesis closing parenthesis\noindent andequip A infinity $ D isomorphically ˆ{ m } ( onto\Delta D open parenthesis( A Capital) ) $Delta\ h open f i l l parenthesiswith the A norm closing\ h parenthesis f i l l $ \ closingparallel parenthesis\varphi Proof period\ parallel . Clearlym it = i s enough\sup {\ tomid assume\ thatalpham =\ 1mid. Thus\ leq , the problemm }\ parallel reduces \ partial ˆ{\alpha } \varphi \ parallel {\ inftyR } ( \varphi 1 \ in D ˆ{ m } ( \Delta (A))) The topologicalto verifying dual of that D open parenthesis∂iϕdβ = Capital 0 for Deltaϕ ∈ D open(∆(A parenthesis)) and 1 A≤ closingi ≤ N. parenthesisWe will closing need parenthesis .. is denoted , $ ∆(A) by D to thethe power equality of prime open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis period .. We assume that \noindent and $ D ( \Delta ( A ) ) $Z \quad with \quad the \quad family \quad o f norms \quad D to the power of prime open parenthesis CapitalM(g Delta∗ u) = openM(u parenthesis) g(y)dy A closing parenthesis closing parenthesis .. is provided with$ \ parallel the .. strong dual\cdot topology\ parallel period .. Each{ m T} in D( to the m power\geq of prime1 open ) parenthesis . $ \quad CapitalI t Delta\quad openi s parenthesis\quad e a A s i l y \quad seen that closing$ D parenthesis ˆ{ m } closing( \Delta parenthesis( .. is A ) )$ isaBanachspaceand $D ( \Delta (A) for u ∈ Π∞ and g ∈ L1( N )( see [ 3 1 , Proposition 4 . 1 ] ) , where ) $called i s a a distribution Fr $ \acute on Capital{e} $ Delta chet open space parenthesisR . \quad A closingFurthermore parenthesis semicolon $ , G$ the value of T at some phi in D open maps $A∗ denotes m$ theis ometrically convolution on ontoRN . $DˆSo ,{ lettm } ing( ψ =\DeltaG−1(ϕ), (where Aϕ )is as ) above $ and $ A \ infty $ parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis is denoted∂ψ isomorphically, we see onto that $ the D proposition ( \Delta is proved( if A we )can check ) . that $ M( ∂yi ) = 0, 1 ≤ by angbracketleft T comma phi right angbracket period .... The derivative of index alphaN in N∞ to theN power of N of T is defined toThe be the topological distributioni ≤ N. dualBut of this $ i Ds straightforward ( \Delta .( Indeed A ,) let )f ∈ $ D(\Rquad) =isC0 denoted(R ) with by $ D ˆ{\prime } ( \DeltaR f(y)dy(= A 1. ) ) . $ \quad We assumeM that( ∂ψ ) = M(f ∗ ∂ψ ). partialdiff to the powerBy of alpha the above T .. on equality Capital Delta, we openhave parenthesis∂yi A closing∂yi parenthesisRecalling .. given that by .. angbracketleft partialdiff$ D ˆ{\ tof the∗prime∂ψ power= ψ}∗ of∂f( alphaand\ TDelta appealingcomma phi( right to A the angbracket above) ) equality $ = open\quad parenthesis , onceis provided again minus , we 1 closingwith get on the parenthesis the\quad other tostrong the power dual of bar topology . \quad Each $ T \ in ∂yiD ˆ{\prime∂yi , } ( \Delta ( A ) ) $ \quad i s alpha bar angbracketleft T∂ψ comma phi rightR ∂f angbracket .. for .. phi in D open parenthesis Capital Delta open parenthesis A closing hand M(f ∗ ) = M(ψ) dy = 0. Hence the proposition follows .  parenthesiscalled a closing distribution parenthesis∂yi period on $ ..\ ItDelta ..∂yi is .. an( A ) ;$ thevalueof $T$ atsome $ \varphi \ ineasyD( exercise to\ verifyDelta that the( transformation A ) )$ T right isdenoted arrow partialdiff to the power of alpha T maps continuously and l inearly D to the power of prime open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis into it\noindent self period by $ \ langle T, \varphi \rangle . $ \ h f i l l The derivative of index $ \alpha \ inIn passingN ˆ{ weN wish}$ to of draw $ attentionT $ is to defined one basicto result be period the distribution Proposition 2 period 24 period .... For any phi in D to the power of m open parenthesis Capital Delta open parenthesis A closing parenthesis\noindent closing$ \ parenthesispartial ˆ open{\alpha parenthesis} T m $ greater\quad equalon 1 closing $ \Delta parenthesis( and A any ) multi $ hyphen\quad indexgiven alpha by with\quad $ \Equation:langle 1 less\ partial or equal ..ˆ{\ baralpha alpha bar} lessT, or equal\ mvarphi comma we\ haverangle integral= sub Capital ( − Delta1 open ) parenthesis ˆ{\mid A\ closingalpha parenthesis\mid }\ partialdifflangle toT, the power of\varphi alpha phi open\rangle parenthesis$ \ squad closingf o parenthesis r \quad d$ beta\varphi open parenthesis\ in s closingD( parenthesis\Delta =( 0 period A ) ) . $ \quad I t \quad i s \quad an easyProof exercise period .. Clearly to verify it i s enough that to the assume transformation that m = 1 period $ .. T Thus\ commarightarrow the problem\ partial reduces toˆ{\alpha } T $ mapsverifying continuously that integral and sub Capital Delta open parenthesis A closing parenthesis partialdiff sub i phi d beta = 0 for phi in D to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis and 1 less or equal i less or\noindent equal N periodl inearly .. We will $Dˆneed {\prime } ( \Delta ( A ) )$ intoitself. the equality \ centerlineM open parenthesis{ In passing g * u closing we wish parenthesis to draw = M attention open parenthesis to one u closing basic parenthesis result integral . } g open parenthesis y closing parenthesis dy \noindentfor u in CapitalProposition Pi to the power 2 . 24 of infinity . \ h f .. i l and l For g inany L to the $ power\varphi of 1 parenleftbig\ in D R ˆ to{ m the} power( of\ NDelta parenrightbig(A open parenthesis) ) ( see .. m open\geq square bracket1 ) 3$ 1 commaand any .. Proposition multi − index 4 period $ 1\ closingalpha square$ with bracket closing parenthesis comma .. where .. * .. denotes the \ beginconvolution{ a l i g n on∗} R to the power of N period .. So comma .. lett ing psi = G to the power of minus 1 open parenthesis phi closing parenthesis\ tag ∗{$ 1 comma\ leq .. where$}\ phimid is as\ abovealpha comma\mid .. we see\ leq that m , we have \ int {\Delta (A) } \ partialthe propositionˆ{\alpha is proved}\ if wevarphi can check( that s .. M open) d parenthesis\beta partialdiff( s psi divided ) = by partialdiff 0 . y i closing parenthesis =\end 0 comma{ a l i g n1∗} less or equal i less or equal N period .. But this i s straightforward period .. Indeed comma let f in D parenleftbig R to the power of N parenrightbig = C sub 0 to the power of\noindent infinity parenleftbigProof . R\ toquad the powerClearly of N parenrightbig it i s enough .. with to integral assume f open that parenthesis $m y= closing 1 parenthesis . $ \quad dy =Thus 1 period , the problem reduces to verifyingBy the above that equality $ comma\ int we{\ haveDelta M open(A) parenthesis partialdiff}\ partial psi divided{ i by}\ partialdiffvarphi y i closingd parenthesis\beta = M open 0 $ parenthesisf o r $ \varphi f * partialdiff\ in psi dividedD ˆ{ 1 by} partialdiff( \Delta y i closing( parenthesis A ) period ) $ .. Recallingand $ 1 that\ fleq * partialdiffi psi\ leq dividedN by partialdiff. $ \quad y i =We w i l l need psi * partialdiff f divided by partialdiff y i sub comma and appealing to the above equality comma once again comma we get on the\noindent other handthe equality M open parenthesis f * partialdiff psi divided by partialdiff y i closing parenthesis = M open parenthesis psi closing parenthesis integral\ [ M partialdiff ( g f divided∗ u by partialdiff) = My i dy ( = 0 period u ) .. Hence\ int the propositiong ( y follows ) period dy square\ ]

\noindent f o r $ u \ in \Pi ˆ{\ infty }$ \quad and $ g \ in L ˆ{ 1 } ( R ˆ{ N } ) ( $ see \quad [ 3 1 , \quad Proposition 4 . 1 ] ) , \quad where \quad $ ∗ $ \quad denotes the convolution on $ R ˆ{ N } . $ \quad So , \quad l e t t ing $ \ psi = G ˆ{ − 1 } ( \varphi ) , $ \quad where $ \varphi $ is as above , \quad we s ee that the proposition is proved if we can check that \quad $ M ( \ f r a c {\ partial \ psi }{\ partial y i } ) = 0 , 1 \ leq i \ leq N . $ \quad But this i s straightforward . \quad Indeed , let $ f \ in D ( R ˆ{ N } ) = C ˆ{\ infty } { 0 } ( R ˆ{ N } ) $ \quad with $ \ int f ( y ) dy = 1 . $ By the above equality , we have $M ( \ f r a c {\ partial \ psi }{\ partial y i } ) = M ( f ∗ \ f r a c {\ partial \ psi }{\ partial y i } ) . $ \quad Recalling that $ f ∗ \ f r a c {\ partial \ psi }{\ partial y i } = $ $ \ psi ∗ \ f r a c {\ partial f }{\ partial y i } { , }$ and appealing to the above equality , once again , we get on the other hand $ M ( f ∗ \ f r a c {\ partial \ psi }{\ partial y i } ) = M ( \ psi ) \ int \ f r a c {\ partial f }{\ partial y i } dy = 0 . $ \quad Hence the proposition follows $ . \ square $ 1 14 .. G period NGUETSENG comma N period SVANSTEDT \noindentThroughout1 the 14 rest\quad .. ofG the . study NGUETSENG it .. i s .. , assumed N . SVANSTEDT that .. A to the power of infinity .. i s dense in A open parenthesis this amounts .. to .. saying .. that .. D open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis .. 1 14 G . NGUETSENG , N . SVANSTEDT iThroughout s .. dense .. in the .. C rest open parenthesis\quad of Capital the study Delta open it \ parenthesisquad i s A\quad closingassumed parenthesis that closing\quad parenthesis$ A closing ˆ{\ infty parenthesis}$ Throughout the rest of the study it i s assumed that A∞ i s dense in period\quad ..i It sdense .. i s .. worth in .. $A noting ($ this amounts A\quad( thisto amounts\quad saying to\ sayingquad that that\quadD(∆($A)) Di ( s dense\Delta in(C(∆( AA))) ). )It $ \quad i s \quad dense \quad in \quad that thisi hypothesis s worth is always noting satisfied that in this practice hypothesis period .. isThen always comma satisfied it becomes in practice possible to . Then , $ Cidentify ( any\Delta given function( u A in L to ) the ) power ) of 1 . open $ parenthesis\quad I t Capital\quad Deltai s open\quad parenthesisworth A\quad closingnoting parenthesis closing it becomes possible to identify any given function u ∈ L1(∆(A)) with the distribution parenthesisthat this with hypothesis the distribution is T always sub u in satisfied D to the power in practiceof prime open . \ parenthesisquad Then Capital , it Delta becomes open parenthesis possible A to closing T ∈ D0(∆(A)) defined by parenthesisidentify closingu any parenthesisgiven function $ u \ in L ˆ{ 1 } ( \Delta ( A ) ) $ with the distribution $ Tdefined{ u by}\ in D ˆ{\prime } ( Z \Delta ( A ) ) $ d eangbracketleft f i n e d by T sub u comma phihTu right, ϕi = angbracketu(s)ϕ =(s integral)dβ(s)( subϕ ∈ Capital D(∆(A Delta))). open parenthesis A closing parenthesis u ∆(A) open parenthesis s closing parenthesis phi open parenthesis s closing parenthesis d beta open parenthesis s closing parenthesis open parenthesis\ [ \ langleHence phi inT DL open{p(∆(u parenthesisA} )) ,⊂ D0\(∆(varphi CapitalA)) Delta (1 ≤\rangle openp ≤ parenthesis∞) with= continuous\ Aint closing{\ parenthesisDelta embedding closing(A) . parenthesis Con} -u closing ( parenthesis s ) period\varphi sequently( s , )given d a real\betap ≥ 1 and( an s integer ) ( m ≥\varphi1, we may define\ in D( \Delta (A)) ).Hence\ ] L to the power of p open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis subset D to the power of prime openW parenthesism,p(∆(A)) Capital = {u ∈ DeltaLp(∆( openA)) : parenthesis∂αu ∈ Lp(∆( AA closing)) for parenthesis| α |≤ m} closing, parenthesis open parenthesis 1 less or equal p less or equal infinity closing parenthesisα with continuous embedding period .. Con hyphen \noindentwhereHence the $ partial L ˆ{ derivativesp } ( \Delta∂ u are computed(A)) in the distribution\subset s enseD ˆ{\ onprime∆(A), }of ( \Delta sequentlycourse comma . given Provided a real p greater with the equal norm 1 and an integer m greater equal 1 comma we may define (W A to the ) power ) of ( m comma 1 \ pleq open parenthesisp \ leq Capital\ infty Delta open) $ parenthesis with continuous A closing parenthesis embedding closing . \quad parenthesisCon − = sequently , given a real $ p \geq 1 $ and an integer $m \geq 1 , $ wemay define open brace u in L to the power of p open parenthesis P Capitalk ∂α Deltau kp open parenthesis 1 A closing parenthesis closing parenthesis : k u k W m, p(∆(A)) = Lp(∆(A)) (u ∈ W m,p(∆(A))), partialdiff to the power of alpha u in L to the power| α |≤ ofm p open parenthesis Capitalp Delta open parenthesis A closing parenthesis closing\ [ W ˆ parenthesis{ m , for p bar} alpha( bar\Delta less or equal( m A closing ) brace ) comma = \{ u \ in L ˆ{ p } ( \Delta ( A)):where theW partialm,p(∆(A derivatives\))partialis a Banach partialdiffˆ{\ spacealpha to the (} in power particularu of\ in alphaWL um, are ˆ2{(∆( computedp A})) i( s ain Hilbert the\Delta distribution space( ) s A . ense on ) Capital ) Deltaf o r open\mid parenthesis\alpha A\mid closingHowever parenthesis\ leq , in practicem comma\} the, appropriate\ ] space i s not the whole W m,p(∆(A)) but it s of courseclosed period subspace .. Provided with the norm bar u bar W m comma p open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis = Row 1 \noindent where the partial derivatives $ \ partial ˆ{\Z alpha } u $ are computed in the distribution s ense on sum bar partialdiff to the power ofW alpham,p(∆( uA bar))/ sub= { Lu p∈ W openm,p(∆( parenthesisA)) : Capitaludβ = Delta 0} open parenthesis A closing parenthesis $ \Delta ( A ) , $ C closing parenthesis to the power of p Row 2 bar alpha bar less or equal m .∆( 1A divided) by p open parenthesis u in W to the power of o f course . \quad Provided with the norm m comma pequipped open parenthesis with the Capital seminorm Delta open parenthesis A closing parenthesis closing parenthesis closing parenthesis comma W to the power of m comma p open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis is a Banach\ [ \ parallel space open parenthesisu \ parallel in particularW W m to the , power p of ( m comma\Delta 2 open( parenthesis A ) Capital ) = Delta\ l opene f t (\ parenthesisbegin { array A }{ cc }\sum &closing\ parallel parenthesis closing\ partial parenthesisˆ{\alpha i s a HilbertP} u spacek ∂\α closingparallelu kp parenthesisˆ{ p } period{ L p ( \Delta (A)) }\\ \mid \alpha k u k\mid \ leq= m \end{ array }\ rightLp(∆(A))) \ f r a c { 1p }{(u ∈p W} m,p((∆(A u))/ )\. in W ˆ{ m , p } However comma inW practice m,p(∆(A the))/C appropriate| α |= spacem i s not the whole W to the1 power of m commaC p open parenthesis Capital Delta( \ openDelta parenthesis(A))), A closing parenthesis closing\ ] parenthesis but it s closedUnfortunately subspace ,W m,p(∆(A))/C so topologized i s in general non - separated and W to thenon power - complete of m comma ( see p open subsection parenthesis 2 . Capital 7 ) . Delta open parenthesis A closing parenthesis closing parenthesis slash \noindent $ W ˆ{ m , p } ( \Deltam,p ( A ) ) $ is a Banachm,p space ( in particular C = braceleftbiggDefinition u in W to 2 the. 25 power . of mLet commaW# p open(∆(A parenthesis)) be separated Capital completionDelta open parenthesis of W (∆( A closingA))/C parenthesis closing parenthesis$ W ˆ{ m : integral , 2 sub} Capital( \ Delta open( parenthesis A ) A closingm,p )$ parenthesis isaHilbertspace). ud betam,p = 0 bracerightbigg and J to be the canonical mapping of W (∆(A))/C into W# (∆(A)). equipped withWe the refer seminorm to , e . g . , [ 7 , chap . I I , § 3, n◦7], [8, chap . I , § 1, n◦4] and [ 1 Howeverbar u bar , subin practiceW m comma the p open appropriate parenthesis Capital space Delta i s not open the parenthesis whole A $Wˆclosing{ parenthesism , closing p } parenthesis( \Delta slash ( A )8 , pp ) $. 6 but1 - 62 ] , for the basic notions involved in the above definition . C = Row 1 sum bar partialdiff tom,p the power of alpha u bar sub L p open parenthesis Capital Delta open parenthesis A closing it s closedRemark subspace2.26.W# (∆(A)) is a Banach space and further the fo llowing classical parenthesisassertions closing parenthesis hold true to the . power of p Row 2 bar alpha bar = m . hline from p to 1 open parenthesis u in W to the power of m comma p open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis slash C closing parenthesis 1) J i s l inear period\ [ W ˆ{ m , p } ( \Delta ( A ) ) / C = \{ u \ in W ˆ{ m , p } ( 2) J(W m,p(∆(A))/ ) is dense in W m,p(∆(A)) \DeltaUnfortunately(A)): comma W to the power\ int of m{\ commaDelta pC open(A) parenthesis# Capital} ud Delta\beta open parenthesis= 0 A\}\ closing] parenthesis closing parenthesis slash C .... so topologized i s in general non hyphen separated and 3) k J(u) k W m,p(∆(A)) =k u k (u ∈ W m,p(∆(A))/ ) non hyphen complete open parenthesis# see subsection 2 periodW m,p(∆( 7A closing))/C parenthesis period C \noindent equipped with the seminorm Definition 24 period ) If F 25i period s a Banach .... Let space W sub and hashL tois the apower continuous of m comma linear p map open of parenthesisW m,p(∆( CapitalA))/C into Delta open parenthesis A closing parenthesisF, closing parenthesis be separated completion of W to the powerL0 ofW mm,p comma(∆(A)) p open parenthesis Capital Delta \ [ \ parallelthen thereu \ existsparallel a unique{ W continuous m , l p inear ( mapping\Delta of(A))/C# } = \ l e f t (\ begin { array }{ cc }\sum open parenthesis A closing parenthesis0 closing parenthesis slash C & \ parallelinto F such\ partial that Lˆ{\= Lalpha◦ J. } u \ parallel ˆ{ p } { L p ( \Delta (A)) }\\ and J to be theThe canonical preceding mapping remark of W leads to the us power immediately of m comma to p openthe fo parenthesis llowing proposition Capital Delta . open parenthesis A closing parenthesis\mid \alpha closing parenthesis\mid slash= C m into\end W{ subarray hash}\ toright the power) \ ofr u m l e comma{3em}{ p0.4 open pt parenthesis} ˆ{ p } Capital{ 1 } Delta( open u parenthesis\ in W ˆ{ m A, closing p } parenthesis( \Delta closing parenthesis(A))/C). period \ ] We refer to comma e period g period comma .. open square bracket 7 comma chap period I I comma S 3 comma n to the power of circ 7 closing square bracket comma open square bracket 8 comma chap period I comma S 1 comma n to the power of circ 4 closing\noindent squareUnfortunately bracket .. and open $ square , Wˆbracket{ m 1 .. 8 , comma p } pp( period\Delta 6 1 hyphen( 62 closing A ) square ) bracket / commaC $ \ h f i l l so topologized i s in general non − separated and for the basic notions involved in the above definition period \noindentRemark 2non period− 26complete period W sub ( see hash subsection to the power of 2 m . comma 7 ) . p open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis is a Banach space and further the fo llowing classical \noindentassertions holdDefinition true period 2 . 25 . \ h f i l l Let $ W ˆ{ m , p } {\# } ( \Delta ( A ) ) $ be1 separated closing parenthesis completion J i s l inear of $W ˆ{ m , p } ( \Delta ( A ) ) / C $ 2 closing parenthesis J open parenthesis W to the power of m comma p open parenthesis Capital Delta open parenthesis A closing parenthesis\noindent closingand parenthesis $ J $ to slash be C the closing canonical parenthesis mapping is dense in of W $Wˆ sub hash{ m to the , power p } of m( comma\Delta p open parenthesis(A) Capital) / Delta C$ open into parenthesis $Wˆ A{ closingm ,parenthesis p } {\ closing# } parenthesis( \Delta ( A ) ) . $ 3 closing parenthesis bar J open parenthesis u closing parenthesis bar W sub hash to the power of m comma p open parenthesis WereferCapital Delta to open , e parenthesis . g . , A\ closingquad [7 parenthesis ,chap. closing I parenthesis I , \S $ = 3bar u , bar sub n ˆ W{\ mcirc comma} p7 open parenthesis] , [ Capital 8 Delta, $ open chap parenthesis . I , \S A closing$ 1 parenthesis , n ˆ closing{\ circ parenthesis} 4 slash ] $ C open\quad parenthesisand [ 1 u in\quad W to8 the , power pp . of 6 m 1 comma− 62 p ] open , parenthesisfor the Capital basic Delta notions open parenthesisinvolved A in closing the parenthesisabove definition closing parenthesis . slash C closing parenthesis 4 closing parenthesis If F i s a Banach space and L is a continuous linear map of W to the power of m comma p open parenthesis Capital\noindent DeltaRemark open parenthesis $2 A . closing 26 parenthesis . Wˆ closing{ m parenthesis , p } slash{\# C } ( \Delta ( A ) ) $ is a Banach space and further the fo llowing classical into F comma then there exists a unique continuous l inear mapping L to the power of prime of W sub hash to the power of m comma\noindent p openassertions parenthesis Capital hold Delta true open . parenthesis A closing parenthesis closing parenthesis into F such that L = L to the power of prime circ J period \ centerlineThe preceding{ $1 remark leads) J$ us immediately islinear to the fo} llowing proposition period \ centerline { $ 2 ) J ( W ˆ{ m , p } ( \Delta ( A ) ) / C )$ isdensein $ W ˆ{ m , p } {\# } ( \Delta ( A ) ) $ }

\ [ 3 ) \ parallel J ( u ) \ parallel W ˆ{ m , p } {\# } ( \Delta (A ) ) = \ parallel u \ parallel { W m , p ( \Delta (A))/C } ( u \ in W ˆ{ m , p } ( \Delta (A))/C) \ ]

4 ) If $F$ i s a Banach space and $ L $ is a continuous linear map of $Wˆ{ m , p } ( \Delta ( A ) ) / C $ into $ F , $ then there exists a unique continuous l inear mapping $ L ˆ{\prime }$ o f $ W ˆ{ m , p } {\# } ( \Delta ( A ) ) $

\noindent into $F$ suchthat $L = Lˆ{\prime }\ circ J . $

\ centerline {The preceding remark leads us immediately to the fo llowing proposition . } Capital Sigma hyphen CONVERGENCE .. 1 1 5 \ hspaceProposition∗{\ f i2 l period l } $ \ 27Sigma period Let− the$ distribution CONVERGENCE derivative\quad partialdiff1 1 5 to the power of alpha open parenthesis alpha in N to the power of N comma bar alpha bar greater equal 1 closing parenthesis be viewed Σ− CONVERGENCE 1 1 5 \noindentas .. a .. mappingProposition .. of W 2 to . the 27 power . Let of m the comma distribution p open parenthesis derivative Capital Delta $ \ openpartial parenthesisˆ{\alpha A closing} parenthesis( \alpha Proposition 2 . 27 . Let the distribution derivative ∂α (α ∈ N , | α |≥ 1) be viewed closing\ in parenthesisN ˆ{ N } slash, C ..\mid into L to\alpha the power of\mid p open parenthesis\geq 1 Capital ) $ Delta be open viewedN parenthesis A closing parenthesis closing as a mapping of W m,p(∆(A))/ into Lp(∆(A)). Then there exists parenthesisas \quad perioda \quad .. Thenmapping .. there ..\quad exists ..o f a .. $ unique W ˆ{ mC , p } ( \Delta ( A ) ) / C $ \quad i n t o a unique continuous linear mapping , s ti l l denoted by ∂α, of W m,p(∆(A)) $ Lcontinuous ˆ{ p } linear( mapping\Delta comma( .. A s ti l ) l denoted ) by . $ partialdiff\quad toThen the power\quad of alphathe re comma\quad ..# ofe x W i s subt s hash\quad to thea \ powerquad unique continuousinto linearLp(∆(A mapping)) , \quad s ti l l denoted by $ \ partial ˆ{\alpha } , $ \quad o f $ W ˆ{ m of m comma p open parenthesisα Capitalα Delta openm,p parenthesis A closing parenthesis closing parenthesis .. into L to the power of p, open p parenthesis} {\such# that} Capital( ∂ \ DeltaJDelta(v) = open∂ v parenthesisfor( Av ∈ W )A closing(∆( ) $A)) parenthesis/\Cquad. Furthermorei n closing t o $ parenthesis L , ˆ{ p } ( \Delta ( A ) ) $ such that partialdiff to the power of alpha J open parenthesis v closing parenthesis = partialdiff to the power of alpha v for v \noindent such that $ \ partial ˆ{\alpha } J ( v ) = \ partial ˆ{\alpha } v $ f o r in W to the power of m comma p open parenthesis Capital DeltaP openk ∂ parenthesisαu kp A closing 1 parenthesis closing parenthesis slash $ v \ in W ˆ{ m ,k pu k} (m, p \Delta= ( A ) )Lp(∆( /A)) C . $ \quad Furthermore , C period .. Furthermore comma W# (∆(A)) | α |= m p bar u bar sub W sub hash m comma p sub open parenthesis Capital Delta open parenthesis A closing parenthesis closing \ begin { a l i g n ∗} m,p parenthesis = Row 1 sum bar partialdiff to the power of alpha u barforu sub L∈ pW open# (∆( parenthesisA)). Capital Delta open parenthesis A closing\ parallel parenthesisu closing\ parallel parenthesis{ toW the{\ power# }} of pm Row 2 , bar p alpha{ ( bar =\Delta m . 1 divided(A)) by p for u in W} sub= hash\ l e to f t the(\ begin { array }{ cc }\sum m &power\ parallel of m2 comma . 7 . p\ openpartialSobolev parenthesisˆ{\ spacesalpha Capital} DeltauH open(∆(\ parallelA parenthesis)) withˆ A{ closingpA} {an parenthesisL almostp closing ( \Delta parenthesis periodic(A)) period }\\ \mid2 period\alphaH 7 period- algebra\ ..mid Sobolev . =We .. m spaces\end consider ..{ array H to the}\ hereright power the) of particular\ mf r a open c { 1 parenthesis}{ casep }\\ where Capitalf o r DeltaA u openis\ in parenthesis anW ˆ{ m A closing , p } {\# } parenthesis( \Deltaalmost closing(A)). parenthesis periodic .. with H .. - A algebra .. an .. almost( see subsection .. periodic .. 2 H . hyphen 3 ) . algebra So we period have here \end{ a l i g n ∗} We .. consider .. here the particular .. case where .. A .. is .. anN almost .. periodic .. H hyphen algebra open parenthesis see subsection 2 period 3 closing parenthesisA = APR(R period), .. So we have here \noindent 2 . 7 . \quad Sobolev \quad spaces \quad $ H ˆ{ m } ( \Delta ( A ) ) $ \quad with \quad A = APwhere sub R parenleftbigR i s a countable R to the subgroup power of N of parenrightbigN ( viewed comma as an additive group ) . In this $ A $ \quad an \quad almost \quad p e r i o dR i c \quad H − algebra . where Rsetting i s a countable , we subgroup suppose ofp R= to the 2, powerso that of N the open parenthesis Sobolev spaces viewed as under an additive consideration group closing are parenthesis period We \quad c o n s i d e r \quad here the particular \quad case where \quad $ A $ \quad i s \quad an almost \quad p e r i o d i c \quad H − algebra .. In this Hm(∆(A)) = W m,2(∆(A)) ( integers m ≥ 1). In this context we will be able to point (setting see subsection commaout afew .. we interesting suppose 2 . 3 p) = . results 2\ commaquad bySo .. means so we that have of the Fourier .. here Sobolev analysis spaces under . consideration are ∞ H to the powerTo begin of m open , we observeparenthesis that CapitalA Deltai s dense open parenthesis in A ( indeed A closing, Γ parenthesisR i s dense closing in A, parenthesisas = W to the power\ [ A of =mis comma pointed AP 2{ openR out} parenthesis in( subsection R ˆ{ CapitalN 2} . Delta 3), ) and open so\ ] parenthesis we are j ustified A closing in parenthesisintroducing closing the preceding parenthesis open parenthesis integers m greaterSobolev equal spaces 1 closing . parenthesis Now , we period recall .. that In this∆( contextA) i s herewe will a be compact able to Abelian group and point outβ ais few nothing interesting but results the Haar by means measure of Fourier on analysis∆(A)( periodProposition 2 . 6 ) . The dual \noindent where $ R $ i s a countable subgroup of $ R ˆ{ N } ( $ viewed as an additive group ) . \quad In t h i s To begingroup comma of we∆( observeA) i s that the A discrete to the power group of infinity .. i s dense in A open parenthesis indeed comma Capital Gamma subs e Rt t i i n s g dense , \ inquad A commawesuppose .. as is $p = 2 ,$ \quad so that the \quad Sobolev spaces under consideration are $pointed H ˆ{ outm } in subsection( \Delta 2 period( 3 closing A parenthesis ) ) = and W so we ˆ{ arem j ustified , 2 in} introducing( \NDelta the preceding( A ) ) ( $ ∆(b A) = {γkb : k ∈ R} (withγkb = G(γk), γk(y) = exp(2iπk · y)(y ∈ R )) i n tSobolev e g e r s spaces $ m period\geq .. Now1 comma ) we . $ recall\quad that CapitalIn this Delta context open parenthesis we will A be closing able parenthesis to i s here a compact Abelianpoint group outwhich anda few may interesting be identified results by with meansR of( Fourierthe reader analysis is . referred to beta is nothingsubsection but the Haar2 . 3 measure and on in Capital particular Delta open to parenthesis Remark A closing 2 . 8 parenthesis ) . Thus open , parenthesis the Proposition .. 1 To2 period begin 6 closingFourier , we parenthesis observe transform periodthat .. $ of The A a ˆ dual{\ function groupinfty }$ u \quad∈ L (∆(iA s)) densemay be in viewed $A as ($ a mapping indeed $ , \Gamma { R }$ isdenseinof Capital, Delta $A open parenthesis ,$ \quad A closingas i parenthesis s i s the discrete group pointed out in subsection 2 . 3 ) and so we are j ustified in introducing the preceding Delta-hatwider open parenthesis A closing parenthesisZ = open brace hatwide-gamma k : k in R closing brace open parenthesis Sobolev spaces . \quad Now , we recall that $ \Delta ( A ) $ i s here a compact Abelian group and with gamma-hatwide k = G open parenthesisk gamma→ ak(u) k = closing parenthesisu(s) γbk(s)dβ comma(s), gamma k open parenthesis y closing parenthesis = exponent$ \beta open$ is parenthesis nothing 2 buti pi k the times Haar y closing measure parenthesis∆(A on) $ parenleftbig\Delta y( in R A to the ) power ($ of Proposition N parenrightbig\quad closing2 . 6 ) . \quad The dual group o f $ \Delta ( A ) $ i s the discrete group parenthesis of R into C. The complex numbers ak(u)(k ∈ R) are the so - called Fourier which ..coefficients may be .. identified of u .. with∈ ..L R1(∆( openA)) parenthesis. According the .. reader to .. a is .. referredclassical .. to ..result subsection ( see .. 2 period 3 .. and \ [ \widehat{\Delta} ( A ) = \{\widehat{\gamma} k : k \ in R \} ( with in particular .. to .. Remark .. 2 period. 856]) closing, ∆(A) parenthesis period .. Thus comma .. the .. Fourier .. transform .. of a .. \widehat{\,gamma e.g.,} k [2 = G 0,p ( \gammab is ank orthonormal ) , \gamma basis ofk the ( Hilbert y space) = \exp ( function .. uL2 in(∆(A)). 2 i \ pi k \Thereforecdot y ) ( y \ in R ˆ{ N } )) \ ] L to thewe power have of 1 , openfor any parenthesisu ∈ L2(∆( CapitalA)), Delta open parenthesis A closing parenthesis closing parenthesis may be viewed as a mapping comma P 2 u = ak(u)γkb ( in the L (∆(A))− norm ) , ( 2 . 3 ) k right arrow a sub k open parenthesis u closing parenthesis = integral sub Capital Delta open parenthesis A closing parenthesis u \noindent which \quad may be \quad i d e n t i f i e d \quad with \quad $ R ( $ the \quad reader \quad i s \quad r e f e r r e d \quad to \quad s u b s e c t i o n \quad 2 . 3 \quad and open parenthesis s closing parenthesis overbar gamma-hatwidek ∈ subR k open parenthesis s closing parenthesis d beta open parenthesis sin closing particular parenthesis\ commaquad to \quad Remark \quad 2 . 8 ) . \quad Thus , \quad the \quad Fourier \quad transform \quad o f a \quad f u n c t i o n \quad $ uof R\ ....inhence into$ .... C period .... The .... complex .... numbers .... a sub k open parenthesis u closing parenthesis open parenthesis $ L ˆ{ 1 } ( \Delta ( A ) ) $ maybe viewed as amapping , k in R closing parenthesis .... are .... the .... so hyphen called ....X Fourier 2 coefficients .. of u in L to the power of 1 openk u parenthesisk 2L2(∆(A)) Capital = Delta| ak(u open) | . parenthesis A closing parenthesis closing parenthesis \ [ k \rightarrow a { k } ( u ) = \ int {\Delta (A) } u ( s ) \ overline {\}{\widehat{\gamma}} { k period .. According .. to .. a .. classical .. result .. open parenthesis see commak ∈ R .. e period g period comma .. open square bracket 2( ..} 0 commas ) d \beta ( s ) , \ ] N p period 56At closing the present square bracket t ime , closing for k parenthesis= (k1, ..., kN comma) ∈ R Delta-hatwiderand α = (α1, open ..., αN parenthesis) ∈ N , it A is closing not parenthesis is an orthonormalhard .. basis to of see the that Hilbert space L to the power of 2 open parenthesis Capital Delta open parenthesis A closing parenthesis closing\noindent parenthesiso f $period R $ .. Therefore\ h f i l l i n t o \ h f i l l $ C . $ \ h f i l l The \ h f i l l complex \ h f i l l numbers \ h f i l l $ awe have{ k comma} ( for u any )u in L ( to the k power\ in of 2R open ) parenthesis $ \ h f i Capital l l are Delta\ h f iopen l l the parenthesis\ h f i l lA closingso − c parenthesis a l l e d \ h closing f i l l Fourier ∂αγk = (2iπ)|α|kαγk, (2.4) parenthesis comma b b \noindentu = sum acoefficients sub k open parenthesis\quad uo fclosing $ u parenthesis\ in gamma-hatwideL ˆ{ 1 } ( k open\Delta parenthesis( in A the ) L to )the power . $ of\quad 2 openAccording \quad to \quad a \quad c l a s s i c a l \quad r e s u l t \quad ( see , \quad e . g . , \quad [ 2 \quad 0 , parenthesisp $ . Capital 56 Delta ] ) open , parenthesis\widehat A closing{\Delta parenthesis} ( closing A ) parenthesis $ is an hyphen orthonormal norm closing\quad parenthesisbasis comma of the .. Hilbert space open$ L parenthesisˆ{ 2 } ( 2 period\Delta 3 closing( parenthesis A ) ) . $ \quad Therefore k in R \noindenthence we have , for any $ u \ in L ˆ{ 2 } ( \Delta ( A ) ) , $ Line 1 bar u bar 2 L sub 2 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis = sum bar a\ hspace sub k open∗{\ parenthesisf i l l } $ u u closing = parenthesis\sum a bar{ tok the} power( ofu 2 period ) \ Linewidehat 2 k in{\ Rgamma} k ($ inthe $Lˆ{ 2 } ( At\ theDelta present(A)) t ime comma for k =− open$ parenthesis norm ) , k\ subquad 1 comma( 2 . period 3 ) period period comma k sub N closing parenthesis in R and alpha = open parenthesis alpha sub 1 comma period period period comma alpha sub N closing parenthesis in N to the power\ [ k of N\ in commaR it\ is] not hard to see that Equation: open parenthesis 2 period 4 closing parenthesis .. partialdiff to the power of alpha gamma-hatwide k = open parenthesis 2\noindent i pi closing parenthesishence to the power of bar alpha bar k to the power of alpha hatwide-gamma k comma \ [ \ begin { a l i g n e d }\ parallel u \ parallel 2{ L } { 2 } ( \Delta ( A ) ) = \sum \mid a { k } ( u ) \mid ˆ{ 2 } . \\ k \ in R \end{ a l i g n e d }\ ]

Atthe present time , for $k = ( k { 1 } , . . . , k { N } ) \ in R $ and $ \alpha = ( \alpha { 1 } ,..., \alpha { N } ) \ in N ˆ{ N } , $ i t i s not hard to see that

\ begin { a l i g n ∗} \ partial ˆ{\alpha }\widehat{\gamma} k = ( 2 i \ pi ) ˆ{\mid \alpha \mid } k ˆ{\alpha }\widehat{\gamma} k , \ tag ∗{$ ( 2 . 4 ) $} \end{ a l i g n ∗} 1 1 6 .. G period NGUETSENG comma N period SVANSTEDT \noindentLine 1 where1 1 k to6 \ thequad powerG of . NGUETSENGalpha = k sub , 1 N to . the SVANSTEDT power of alpha to the power of 1 k sub 2 to the power of alpha to the power of 2 period period period k sub N to the power of alpha to the power of N period Hence Line 2 a sub k open parenthesis 1 1 6 G . NGUETSENG , N . SVANSTEDT partialdiff\ [ \ begin to{ a the l i g powern e d } ofwhere alpha u closing k ˆ{\ parenthesisalpha } == open k parenthesis ˆ{\alpha 2 i pi} closing{ 1 } parenthesisˆ{ 1 } k to ˆ the{\ poweralpha of bar} { alpha2 }ˆ bar{ 2 k } . . . k ˆ{\alpha } { N }ˆ{ N } . Hence \\ to the power of alpha a sub k open parenthesiswhere u closingkα = k parenthesisα1kα2...kα N open. Hence parenthesis 2 period 5 closing parenthesis afor{ anyk u} in H( to\ thepartial power ofˆ{\ m openalpha parenthesis} u Capital1 )2 = DeltaN ( open 2 parenthesis i \ pi A closing) ˆ{\ parenthesismid \alpha closing parenthesis\mid } α |α| α periodk ˆ{\ ..alpha Having} madea these{ k preliminaries} ( u commaa )k(∂ let (u) us = 2 (2 turniπ) now.k a to 5k( theu) ) (2\.end5) { a l i g n e d }\ ] proof offor the any followingu ∈ H propositionm(∆(A)). periodHaving made these preliminaries , let us turn now to the proof Propositionof the 2 period following 28 period proposition .. The fol . lowing ass ertions are true : \noindentopen parenthesisf o r any i closing $ u parenthesis\ in barH ˆ times{ m bar} sub( H\ mDelta open parenthesis( A Capital ) Delta) . open $ parenthesis\quad Having A closing made paren- these preliminaries , let us turn now to the proof ofProposition the following 2 . 28proposition . The fol . lowing ass ertions are true : thesis closing parenthesis slash C is a norm on H to the power of m open parenthesism Capital Delta open parenthesis A closing ( i ) k · kHm(∆(A))/ is a norm on H (∆(A))/C. parenthesis closing parenthesis slash C period C ( ii )D(∆(A)) is dense in Hm(∆(A)). \noindentopen parenthesisProposition ii closing parenthesis 2 . 28 . D\ openquad parenthesisThe fol Capital lowing Delta ass open ertions parenthesis are A closing true :parenthesis closing parenthesis Proof . ( i ) Let u ∈ Hm(∆(A))/ with k u k = 0. Then ∂αu = 0 for all .. is dense in H to the power of m open parenthesis CapitalC Delta openHm parenthesis(∆(A))/C A closing parenthesis closing parenthesis period α ∈ N with | α |= m. Fix freely k = (k , ..., k ) ∈ R with k 6= ω (ω the origin \ centerlineProof period{N( .... i open $ ) parenthesis\ parallel i closing\ parenthesiscdot \ parallel Let1 u in HN to{ theH power m of ( m open\Delta parenthesis(A))/ Capital Delta open in N ). Consider an integer 1 ≤ n ≤ N such that k 6= 0, and let α = (α ) ∈ N with Cparenthesis}$ is A anormon closingR parenthesis $Hˆ{ closingm } parenthesis( \Delta slash C( with A .... bar )n u ) bar sub / H C m open . $j parenthesis}N Capital Delta open α = m, α = 0 if j 6= n. Then kα = km 6= 0; hence a (u) = 0, according to ( 2 . 5 ) ; parenthesis An closing parenthesisj closing parenthesis slashn C = 0 period ....k Then partialdiff to the power of alpha u = 0 for all and so u = 0( use ( 2 . 3 ) ) , s ince a = 0. This shows ( i ) . \ centerlinealpha in N to{( the i i power $ ) of N D with ( .... bar\Delta alpha bar( =ω m A period ) .... ) Fix $ freely\quad k =is open dense parenthesis in $Hˆ k sub{ 1m comma} ( period\Delta ( A )( ) ii ) . Consider $ } a sequence of nonempty finite sets Rn ⊂ R(n ranging over the period periodpositive comma integers k sub N closing ) such parenthesis that in R with k negationslash-equal omega open parenthesis omega the origin in R to the power of N closing parenthesis period .. Consider an integer 1 less or equal n less or equal N such that k sub n equal-negationslash\noindent Proof 0 comma . \ h f i and l l let( i alpha ) Let = open $ parenthesis u \ in alphaH ˆ{ subm j} closing( parenthesis\Delta in( N to A the power ) of ) N / C $ R ⊂ R , R = ∪n R . withwith\ h alpha f i l l sub$ n\ =parallel m comma alphau sub\ parallel j = 0n if j negationslash-equaln{+1 H m≥1 ( n n\ periodDelta .. Then(A))/C k to the power of alpha =} k sub= n to 0 the. $ power\ h fofLet i l m l negationslash-equalThenu ∈ Hm $(∆(\Apartial)). For 0 semicolonˆ each{\alpha integer hence} an subu≥ k1, open=put parenthesis 0 $ f o r u a closing l l parenthesis = 0 comma according to open parenthesis 2 period 5 closing parenthesis semicolon and so u = 0 open parenthesis use open parenthesis 2 .. period 3 \noindent $ \alpha \ in N ˆ{ N }$ with X\ h f i l l $ \mid \alpha \mid = m . $ \ h f i l l Fix f r e e l y closing parenthesis closing parenthesis comma s inceun a= sub omegaak(u =)γk. 0 period .. This shows open parenthesis i closing parenthesis $ k = ( k { 1 } , . . . , k { Nb} ) \ in R $ with $ k \not= \omega period k ∈ R ( open\omega parenthesis$ the ii closing o r i g i parenthesis n .. Consider .. a sequence ofn nonempty finite sets R sub n subset R open parenthesis n ranging overThis the gives a s equence (un)n≥1 with un ∈ D(∆(A)) and further , thanks to ( 2 . 4 ) - \noindentpositive( integers 2in . 5 $ ) closing R, ˆ{ parenthesisN } ) such . $ that\quad Consider an integer $ 1 \ leq n \ leq N $ such that $ kR sub{ n n subset}\ne R sub0 n plus , 1 $ comma and R l e = t cup $ n\alpha sub greater= equal ( 1 R\ subalpha n period{ j } ) \ in N ˆ{ N }$ with $ \alpha { n } = mα , \alphaX { α j } = 0 $ i f $ j \not= n . $ \quad Then Let u in H to the power of m open parenthesis∂ un = Capitalak Delta(∂ u)γkb open( parenthesis| α |≤ m). A closing parenthesis closing parenthesis period ..$ For k ˆ each{\alpha integer n} greater= equalk ˆ{ 1m comma} { n put}\not= 0 ;$ hence $a { k } ( u ) = 0 , $ k ∈ R accordingLine 1 u sub to n = sum a sub k open parenthesis u closing parenthesis gamma-hatwiden k period Line 2 k in R sub n (2.5);andso $u = 0 ($ use(2 \quadm .3)) ,since $a {\omega } = 0 This givesHence a s equence , by ( open2 . 3 parenthesis ) it follows u sub that n closingun parenthesis→ u in H sub(∆( nA greater)) as equaln → ∞ 1, withwhich u sub shows n in D open parenthesis Capital. $ \ Deltaquad openThis parenthesis shows ( A i closing ) . parenthesis closing parenthesis and further comma thanks to open parenthesis 2 period .. 4 closing parenthesis hyphen (ii).  ( iopen i ) parenthesis\quad Consider 2 period 5\quad closinga parenthesis sequence comma of nonempty finite sets $ R { n }\subset R ( n $ rangingLine 1 partialdiff overAs an the immediate to the power consequence of alpha u sub of this n = , sum there a sub i s k the open fo parenthesis llowing corollary partialdiff . toCorollary the power of alpha u closing positive integers ) such that D(∆(A))/ = {ϕ ∈ D(∆(A)) : R ϕdβ = 0} parenthesis2 gamma-hatwide . 29 . The k open space parenthesis barC alpha bar less or equal∆( mA) closing parenthesisis period Line 2 k in R sub n Hence comma by open parenthesis 2 period 3 closing parenthesis .... it follows that u sub n right arrow u in H to the power of m\ [R open parenthesis{ n }\subset Capital DeltaR open{ n parenthesis + 1 A} closing, parenthesis R = \ closingcup { parenthesisn } {\geq .... as n1 right} arrowR { infinityn } comma. \ ] .... denseinHm(∆(A))/ . which shows C open parenthesisThus i i closing , according parenthesis to part period square ( i ) of Proposition 2.28,Hm(∆(A))/ i s a separated \noindentAs an immediateLet $ consequence u \ in of thisH ˆ comma{ m } there( i s\Delta the fo llowing( corollary A ) period ) .C $ \quad For each integer $ n preHilbert space ; so that Hm(∆(A)) = W m,2(∆(A)) in the present s ett ing co - \geqCorollary1 2 , period $ put 29 period .. The space D# open parenthesis# Capital Delta open parenthesis A closing parenthesis closing m parenthesisincides slash C = with open thebrace completion phi in D open of parenthesisH (∆(A Capital))/C. DeltaAs openwe will parenthesis see in A a closing l itt parenthesis le while , closing parenthesis m :\ integral[ \ begin sub{Ha lCapital i(∆( g n eA d))} Delta/Cui s open{ notn parenthesis} necessarily= \sum Acomplete closinga parenthesis{ .k } For( phi s d implicity u beta ) = 0 closing we\widehat assume brace{\ ..ingamma is the} sequelk . \\ kdense\ inthat H toR them {= power 1n. }\ ofWeend m will{ opena l i needg parenthesis n e d one}\ ] preliminary Capital Delta result open parenthesis . A closing parenthesis closing parenthesis slash C period Lemma 2 . 30 . The fol lowing two assertions are equivalent . Thus comma according to part( i .. ) open There parenthesis exists i closing a constant parenthesisc > of0 Propositionsuch that 2 period 2 8 comma H to the power of m open\noindent parenthesisThis Capital gives Delta a opens equence parenthesis $ A ( closing u { parenthesisn } ) closing{ n parenthesis\geq slash1 }$ C i with s a separated $ u { n }\ in D( \Delta ( A ) )$k andfurtheru k 2 ≤ c k ,u thanksk 1 to (2 . \quad 4 ) − preHilbert space semicolon .... so that H sub hashL (∆( toA the)) power ofH m(∆( openA))/C parenthesis Capital Delta open parenthesis A closing ( 2 . 5 ) , parenthesis closing parenthesis1 = W sub hash to the power of m comma 2 open parenthesisN Capital Delta open parenthesis A closing for al l u ∈ H (∆(A))/C. ( ii )R is a dis crete subgroup of R ( s ee [ 6 , TGVII . parenthesis2 closing ] ). parenthesis .... in the present s ett ing co hyphen \ [ \incidesbegin with{ a l i theg n e completion d }\ partial of H toˆ{\ the poweralpha of} m openu { parenthesisn } = Capital\sum Deltaa open{ parenthesisk } ( A\ partial closing parenthesisˆ{\alpha closing} parenthesisu ) \ slashwidehat C period{\gamma .. As} we willk see ( in a\ lmid itt le while\alpha comma \mid \ leq m ) . \\ kH to\ thein powerR of{ mn open}\end parenthesis{ a l i g n e dCapital}\ ] Delta open parenthesis A closing parenthesis closing parenthesis slash C i s not necessarily complete period .. For s implicity we assume in the sequel that m = 1 period .. We will need one preliminary result period \noindentLemma 2 periodHence 30 , period by ( .. 2 The . 3 fol ) lowing\ h f i two l l assertionsit follows are equivalent that $ period u { n }\rightarrow u $ in $ H ˆ{ m } ( open\Delta parenthesis( i closing A ) parenthesis ) $ \ ..h There f i l l existsas $ a constant n \rightarrow c greater 0 such\ thatinfty , $ \ h f i l l which shows bar u bar sub L to the power of 2 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis less\ [ or( equal i c bari u ) bar sub . H\ tosquare the power\ ] of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis slash C for al l u in H to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis slash C\noindent period As an immediate consequence of this , there i s the fo llowing corollary . Corollaryopen parenthesis 2 . ii29 closing . \quad parenthesisThe space R is a dis $D crete subgroup ( \Delta of R to the( power A of N ) open ) parenthesis / C s ee = open\{\ square bracketvarphi 6\ in commaD( TGVII period\Delta 2 closing(A)): square bracket closing parenthesis\ int {\ periodDelta (A) }\varphi d \beta = 0 \} $ \quad i s

\ begin { a l i g n ∗} dense in H ˆ{ m } ( \Delta (A))/C. \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }Thus , according to part \quad (i)ofProposition $2 . 2 8 , Hˆ{ m } ( \Delta ( A ) ) / C$ isaseparated

\noindent preHilbert space ; \ h f i l l so that $ H ˆ{ m } {\# } ( \Delta ( A ) ) = W ˆ{ m , 2 } {\# } ( \Delta ( A ) ) $ \ h f i l l in the present s ett ing co −

\noindent incides with the completion of $ H ˆ{ m } ( \Delta ( A ) ) / C . $ \quad As we will see in a l itt le while , $ H ˆ{ m } ( \Delta ( A ) ) / C$ i snot necessarily complete . \quad For s implicity we assume in the sequel that $ m = 1 . $ \quad We will need one preliminary result .

\noindent Lemma 2 . 30 . \quad The fol lowing two assertions are equivalent .

\ centerline {( i ) \quad There exists a constant $ c > 0 $ such that }

\ [ \ parallel u \ parallel { L ˆ{ 2 } ( \Delta (A)) }\ leq c \ parallel u \ parallel { H ˆ{ 1 } ( \Delta (A))/C }\ ]

\noindent f o r a l l $ u \ in H ˆ{ 1 } ( \Delta ( A ) ) / C . $ ( ii $ ) R$ is a dis crete subgroup of $Rˆ{ N } ($ see [6 ,TGVII.2] ) . Capital Sigma hyphen CONVERGENCE .. 1 1 7 $ Proof\Sigma period− ..$ Let CONVERGENCE u in H to the power\quad of1 1 1open 7 parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesisProof . slash\quad C periodLet .. $ It u i s clear\ in thatH ˆ{ 1 } ( \Delta ( A ) ) / C . $ \quad It i s clear that Σ− CONVERGENCE 1 1 7 Proof . Let u ∈ H1(∆(A))/ . It i s clear that Line 1 bar u bar 2 L sub 2 open parenthesis Capital Delta open parenthesis AC closing parenthesis closing parenthesis = sum bar a\ [ sub\ begin k open{ a parenthesis l i g n e d }\ uparallel closing parenthesisu \ barparallel to the power2{ ofXL 2} Line{ 2 omega} ( negationslash-equal\Delta ( k A in R ) ) = \sum k u k 2L (∆(A)) = | a (u) |2 \midand a { k } ( u ) \mid ˆ{ 22 }\\ k \Lineomega 1 bar u\not bar= 2 H sub k 1\ openin parenthesisR \end{ Capitala l i g n e d Delta}\ ] open parenthesisω 6= k ∈ R A closing parenthesis closing parenthesis slash C = 4 pi to theand power of 2 sum bar k bar to the power of 2 bar a sub k open parenthesis u closing parenthesis bar to the power of 2 comma Line 2 omega equal-negationslash k in R \noindentwhere .. barand k bar .. i s the Euclidean norm of k and omega theX origin in R to the power of N period .. Thus comma .. assuming k u k 2H (∆(A))/ = 4π2 | k |2| a (u) |2, open parenthesis i closing parenthesis implies1 at onceC k \ [ \Equation:begin { a lopen i g n e parenthesis d }\ parallel 2 period 6u closing\ parallel parenthesis2 ..{ sumH } bar{ a1 subω}6= kk( open∈ R\ parenthesisDelta (A))/C u closing parenthesis bar to the =power 4 of 2 less\ pi orˆ{ equal2 }\ 4 pi tosum the power\mid of 2 ck to the\mid powerˆ{ of2 2 sum}\ barmid a suba k open{ k parenthesis} ( u u closing ) parenthesis\mid ˆ{ bar2 } to the, \\ | k | k ω N . power\omega of 2 barwhere\ kne bar tok the poweri\ sin the of EuclideanR 2 omega\end{ equal-negationslasha lnorm i g n e d of}\ ] and kthe in R origin omega in equal-negationslashR Thus , k assuming in R and that( for i ) any implies u in H atto the once power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis slash C period .. Hence \noindent where \quad $ \midX k \mid $ \quadX i s the Euclidean norm of $ k $ and $ \omega $ Equation: open parenthesis 2 period 7 closing| a (u) | parenthesis2≤ 4π2c2 .. bar| a k( baru) |2 greater| k |2 equal r greater 0 open(2.6) parenthesis omega equal-negationslashthe origin in k $Rˆ in R closing{ N } parenthesis. $ \quadk Thus , \quad assumingk (with i ) r implies = 1 divided at by once 2 pi c sub comma which means thatω R6= ....k ∈ i R s a discreteω 6= k ∈ subgroupR of R to the power of N comma and so open parenthesis i closing parenthesis implies 1 \ beginopen{ parenthesisa land i g n ∗} that i i for closing any parenthesisu ∈ H (∆( periodA))/C ..... ConverselyHence suppose .... open parenthesis ii closing parenthesis .... holds period ....\sum This amounts\mid toa saying{ k that} ....( open u parenthesis ) \mid 2 periodˆ{ 2 7}\ closingleq parenthesis4 \ ....pi holdsˆ{ 2 for} c ˆ{ 2 }\sum \mid a some{ k suitable} ( constant u ) r greater\mid 0ˆ period{ 2 }\ .... Immediatelymid k we\mid see thatˆ{ i2 f u}\ ltag ies in∗{ H$ to ( the 2power . of 1 6open )parenthesis $}\\\omega Capital | k |≥ r > 0 (ω 6= k ∈ R) (2.7) Delta\ne openk parenthesis\ in R A closing\omega parenthesis\ne closingk parenthesis\ in R slash C comma \endthen{ a openl i g n ∗} parenthesis 2 period 6 closing parenthesis holds with c = 1 divided by 2 pi r sub period .... Hence open parenthesis i with r = 1 which means that R i s a discrete subgroup of N , and so ( i ) implies closing parenthesis follows2πc , period .... This completes the proof period square R \noindentWe are now( iand i able ) . that to Conversely j ustify for anyour allegation suppose $ u about\ (in ii ) theH holds completeness ˆ{ 1 .} This( of amounts H\Delta to the power to( saying of A1 open that ) parenthesis ( ) 2 . / 7 ) Capital C Delta. $ open\quad Hence parenthesisholds A closing for parenthesis closing parenthesis slash C period 1 \ beginProposition{ a lsome i g n ....∗} suitable 2 period constant 31 period Hr to> 0 the. powerImmediately of 1 open parenthesis we see that Capital i f Deltau l ies open in parenthesisH (∆(A))/ AC, closing parenthesis \mid k \mid \geq r > 01 ( \omega \ne k \ in R) \ tag ∗{$ ( 2 . 7 closing parenthesisthen ( slash 2 . 6 C ) open holds parenthesis with c = with2πr . theHence .... norm ( i ) .... follows bar times . This bar sub completes H to the power the proof of 1 open.  parenthesis Capital ) $} 1 Delta openWe parenthesis are now A closingable to parenthesis j ustify our closing allegation parenthesis about slash the C closing completeness parenthesis of ....H is(∆( ....A complete))/C. .... if 1 \end{ a l i g n ∗} 1 and onlyProposition if R is a dis crete2 subgroup.31.H (∆( of RA)) to/C the( powerwith of the N period norm k · kH (∆(A))/C) is complete if Proof periodand Honly to the if powerR is of a 1 dis open crete parenthesis subgroup Capital of DeltaRN . open parenthesis A closing parenthesis closing parenthesis slash \noindent with $ r1 = \ f r a c { 1 }{ 2 \ pi c } { , }$1 which means that $ R $ \ h f i l l i s a discrete subgroup of C being a closedProof vector.H subspace(∆(A))/ ofC Hbeing to the a powerclosed of vector 1 open subspace parenthesis of CapitalH (∆( DeltaA)), by open the parenthesis open A closing parenthesis $ R ˆ{ N } , $ and so ( i ) implies 1 closing parenthesismapping comma theorem by the ( open see , e . g . , [ 1 0 , p . 1 9 ] ) we see that H (∆(A))/C with 1 mappingthe theorem norm openk · parenthesiskH1(∆(A))/ seeis complete comma e period i f and g period only commai f the ..two open norms squarek bracket · k H (∆( 1 ..A 0)) comma/C and p period 1 9 closing \noindent ( i i ) . \ h f i l l CConversely suppose \ h f i l l ( i i ) \ h f i l l holds . \ h f i l l This amounts to saying that \ h f i l l ( 2 . 7 ) \ h f i l l holds f o r square bracketk · k closingH1(∆(A)) parenthesis we see .. that H to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesisare closing equivalent parenthesis . slash But C thiswith happens the norm if and only if condition ( i ) of Lemma 2 . \noindent some suitable constant $ r > 0 . $ \ h f i l l Immediately we see that i f $ u $ bar times3 bar 0 sub is H fulfilled to the power . ofTherefore 1 open parenthesis the proposition Capital Delta follows open by parenthesis the same A lemma closing parenthesis.  closing parenthesis l i e s in $ H ˆ{ 1 } ( \Delta N( A ) ) / C , $ slash C is complete i f and only i f the two norms bar times bar H to the power of 1 open parenthesis1 Capital Delta open parenthesis Thus , if for example R = Q (Q the rationals ) , then the norm k · kH (∆(A))/C on A closing parenthesisH1(∆(A)) closing/C is not parenthesis complete slash and C and hence .. bar the times latter bar space sub H tois thenot power a Hilbert of 1 open space parenthesis . Con- Capital Delta open parenthesis\noindentsequently Athen closing ( parenthesis 2, in . general 6 ) closing holds the parenthesis passage with $c to the = completion\ f r a c { 1 i s}{ necessary2 \ pi . r } { . }$ \ h f i l l Hence ( i ) follows . \ h f i l l This completes the proof $ .are equivalent\ square period $ .. But this happens if and only if condition .. open parenthesis i closing parenthesis .. of Lemma 2 .. period 3 0 .. is 3. Σ − convergenceinLp (1 ≤ p < ∞) \noindentfulfilled periodWe are .. Therefore now able the proposition to j ustify follows our by allegationthe same lemma about period the square completeness of $ H ˆ{ 1 } ( Throughout the present section , Ω denotes an open s et in N (Ω independent of \DeltaThus comma( if A for example ) ) R = / Q to C the power . $ of N open parenthesis Q the rationalsRx closing parenthesis comma then the norm ε > 0) and H = (Hε)ε>0 is as above ( see ( 2 . 1 ) ) . The letter E will .. bar timesdenote bar sub a H to family the power of positive of 1 open real parenthesis numbers Capital admitting Delta open 0 parenthesis as an A closing accumulation parenthesis closing parenthesis slash\noindent C on Proposition \ h f i l l $ 2 . 31 . H ˆ{ 1 } ( \Delta (A))/C point . In the particular case where E = (ε ) with 0 < ε ≤ 1 and ε → 0 ( $H to with the power the \ ofh 1 f i open l l norm parenthesis\ h f i l Capital l $ \ Deltaparallel open parenthesis\cdotn n∈N A\ closingparallel parenthesisn { H closing ˆ{ 1 } parenthesisn ( \Delta slash C is( not A))/Cas n → ∞, we} ) $ \ h f i l l i s \ h f i l l complete \ h f i l l i f complete and hence the latter space is not a Hilbert space period 1 N ) will refer to E as a fundamental s equence . For ψ ∈ Lloc(Ω × Ry , it is customary Consequentlyto put comma in general the passage to the completion i s necessary period \noindent3 period Capitaland only Sigma if hyphen $R$ convergence is a dis in L crete to the subgroup power of p of open $Rˆ parenthesis{ N } 1 less. $ or equal p less infinity closing parenthesis \noindent Proof $ . H ˆ{ 1 } ( \Delta ( A ) ) / C$ being a closed vector subspace of Throughout the present section comma Capitalε Omega denotes an open s et in R sub x to the power of N open parenthesis ψ (x) = ψ(x, Hε(x)) (x ∈ Ω) (3.1) Capital$ H ˆ{ Omega1 } independent( \Delta ( A ) ) ,$ bytheopen of epsilon greater 0 closing parenthesis .. and H = open parenthesis H sub epsilon closing parenthesis sub epsilonψ greater 0 .. is \noindentwhenevermapping the theorem r ight - (hand see s ide, e makes . g . s ense, \quad . This[ 1 \ willquad be0 the ,p.19] case if in particular )wesee \quad that $ H ˆ{ 1 } .. as above .. open parenthesisK( Ω; seeL∞ ..( openN )) parenthesis( Ω 2 period 1 closingΩ parenthesisN ) closingLp(Ω; parenthesisA) (1 ≤ p period≤ ∞), .. The letter E will ( \Deltalies in( A ) )Ry / C$ withthenormthe closure of in Rx or denote a N ) $ \ parallelwhere A \iscdot any closed\ parallel vector subspace{ H ˆ{ of1 }B(Ry( \equippedDelta with(A))/C the supremum norm }$ is complete i f and only i f the two norms family of( positive see [ real 2 6 numbers ] , and .. admitting observe 0that .. as Lemma .. an .. accumulation 2 and Proposition point period 3 therein .. In the , together $ \particularparallel case where\cdot E =\ openparallel parenthesisH epsilon ˆ{ 1 sub} n( closing\Delta parenthesis( sub A n in ) N with ) 0less / epsilon C $ sub and n\ lessquad or equal$ \ parallel 1\cdot .. and epsilon\ parallel sub n right{ arrowH ˆ{ 01 as} n right( arrow\Delta infinity(A)) comma we }$ will refer to E as a fundamental s equence period .. For psi in L sub loc to the power of 1 parenleftbig Capital Omega times R sub\noindent y to the powerare equivalentof N to the power . \ ofquad parenrightbigBut this comma happens it is customary if and only if condition \quad ( i ) \quad o f Lemma 2 \quad . 3 0 \quad i s f uto l f put i l l e d . \quad Therefore the proposition follows by the same lemma $ . \ square $ Equation: open parenthesis 3 period 1 closing parenthesis .. psi to the power of epsilon open parenthesis x closing parenthesis =\ hspace psi open∗{\ parenthesisf i l l }Thus x comma , if Hfor sub example epsilon open $R parenthesis = Qˆ x closing{ N } parenthesis( Q$ closing the parenthesis rationals open ) ,parenthesis then the x in norm \quad Capital$ \ parallel Omega closing\cdot parenthesis\ parallel { H ˆ{ 1 } ( \Delta (A))/C }$ on whenever the r ight hyphen hand s ide makes s ense period .. This will be the case if in particular \noindentpsi lies in K$ parenleftbig H ˆ{ 1 } to the( power\Delta of hline( Capital A Omega ) ) semicolon / C$ L to the is power not of complete infinity parenleftbig and hence R sub the y to latter the space is not a Hilbert space . powerConsequently of N to the power , in of general parenrightbig the parenrightbig passage to open the parenthesis completion to the i power s necessary of hline Capital . Omega the closure of Capital Omega in R sub x to the power of N closing parenthesis .. or L to the power of p open parenthesis Capital Omega semicolon A closing\ [ 3 parenthesis . \Sigma open parenthesis− convergence 1 less or equal p in less or L equal ˆ{ p infinity} ( closing 1 parenthesis\ leq p comma< \ infty ) \ ] where A is any closed vector subspace of B parenleftbig R sub y to the power of N to the power of parenrightbig .. equipped with the supremum Throughoutnorm .. open the parenthesis present see section .. open square $ , bracket\Omega 2 6$ closing denotes square an bracket open comma s et in.. and $Rˆ observe{ N that} { Lemmax } 2( and\Omega $ Propositionindependent 3 therein comma together o f $ \ varepsilon > 0 ) $ \quad and $ H = ( H {\ varepsilon } ) {\ varepsilon > 0 }$ \quad i s \quad as above \quad ( see \quad ( 2 . 1 ) ) . \quad The letter $ E $ will denote a family of positive real numbers \quad admitting 0 \quad as \quad an \quad accumulation point . \quad In the particular case where $ E = ( \ varepsilon { n } ) { n \ in N }$ with $ 0 < \ varepsilon { n } \ leq 1 $ \quad and $ \ varepsilon { n }\rightarrow 0 $ as $ n \rightarrow \ infty , $ we

\noindent will refer to $ E $ as a fundamental s equence . \quad For $ \ psi \ in L ˆ{ 1 } { l o c } ( \Omega \times R ˆ{ N } { y }ˆ{ ) } , $ it is customary to put

\ begin { a l i g n ∗} \ psi ˆ{\ varepsilon } ( x ) = \ psi ( x , H {\ varepsilon } ( x ) ) ( x \ in \Omega ) \ tag ∗{$ ( 3 . 1 ) $} \end{ a l i g n ∗}

\noindent whenever the r ight − hand s ide makes s ense . \quad This will be the case if in particular $ \ psi $ liesin $K (ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; L ˆ{\ infty } ( R ˆ{ N } { y }ˆ{ ) ) } ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega $ the closure of $ \Omega $ in $ R ˆ{ N } { x } ) $ \quad or $ L ˆ{ p } ( \Omega ; A ) ( 1 \ leq p \ leq \ infty ) , $ where $A$ is any closed vector subspace of $B ( Rˆ{ N } { y }ˆ{ ) }$ \quad equipped with the supremum norm \quad ( see \quad [ 2 6 ] , \quad and observe that Lemma 2 and Proposition 3 therein , together 1 1 8 .. G period NGUETSENG comma N period SVANSTEDT \noindentwith their1 proofs 1 8 comma\quad ..G remain . NGUETSENG r igorously , valid N . when SVANSTEDT Capital Omega .. i s .. unbounded provided C .. is replaced with K closing parenthesis period 1 1 8 G . NGUETSENG , N . SVANSTEDT \noindentFinally commawith .. their in .. the proofs .. s equel , ..\quad A .. denotesremain .. a r .. igorously given .. H hyphen valid algebra when .. on$ \ ..Omega R to the$ power\quad of Ni .. s for\quad .. H ..unbounded provided with their proofs , remain r igorously valid when Ω i s unbounded provided C with$ C .. $ the\quad i s is replaced with K). replacedassumption with that A $K infinity ) is dense . $ in A period .. The basic notation attached to A i s as before Finally , in the s equel A denotes a given H - algebra on N open parenthesis see section 2 closing parenthesis period R for H with the assumption that A∞ is dense in A. The basic notation attached \noindent3 period 1F period i n a l l .. y The , \ weakquad Capitalin \quad Sigmathe hyphen\quad convergences equel in L\ toquad the power$ A of $ p open\quad parenthesisdenotes Capital\quad Omegaa \quad closinggiven \quad H − algebra \quad on \quad to A i s as before ( see section 2 ) . parenthesis$ R ˆ{ N period}$ \ ..quad Let ..f o 1 r less\quad or equal$ p H less $ infinity\quad periodwith \quad the 3 . 1 . The weak Σ− convergence in Lp(Ω). Let 1 ≤ p < ∞. assumptionDefinition .. 3 that period $ 1 Aperiod\ infty .. A .. sequence$ isdensein .. open parenthesis $A u .$sub epsilon\quad closingThe parenthesis basic notation sub epsilon attached in E comma to u Definition 3 . 1 . A sequence (u ) , u ∈ Lp(Ω), i s said to sub$A$ epsilon i in s L as to beforethe power of p open parenthesis Capital Omegaε ε closing∈E parenthesisε comma .. i s .. said .. to .. be .. weakly .. ( see sectionbe weakly 2 ) . Σ− Capital Sigma hyphen p p p p convergent in L (Ω) to some u0 ∈ L (Ω; L (∆(A))) = L (Ω × ∆(A)) if convergentas .. in .. L to the power of p open parenthesis Capital Omega closing parenthesis .. to .. some .. u sub 0 in L to the\noindent power of p3 open . 1 parenthesis . \quad The Capital weak Omega $ \ semicolonSigma L− to$ the convergence power of p open in parenthesis $ L ˆ{ Capitalp } ( Delta\Omega open parenthesis) . $ A\quad closingLet parenthesis\quad closing$ 1 parenthesis\ leq p closing< parenthesis\ infty = L. to $ the power of p open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis .. if as \noindent D e f i n i t i o n \quad 3 . 1 . \quad A \quad sequenceE 3 ε \→quad0, wehave$ ( u {\ varepsilon } ) {\ varepsilon E ni epsilon right arrow 0 commaZ we have integralZZ sub Capital Omega u sub epsilon open parenthesis x closing parenthesis psi \ in E } , u {\ varepsilonε }\ in L ˆ{ p } ( \Omega ) , $ \quad i s \quad s a i d \quad to \quad be \quad weakly \quad to the power of epsilon open parenthesisuε(x)ψ x(x closing)dx → parenthesisu dx0(x, right s)ψb( arrowx, s)dxdβ integral(s) (3 integral.2) sub Capital Omega times Capital Delta$ \Sigma open parenthesis− $ A closingΩ parenthesis u sub 0 openΩ×∆( parenthesisA) x comma s closing parenthesis psi-hatwide open parenthesis x comma s closing parenthesis dxd beta open parenthesis s closing parenthesis open parenthesis 3 period 2 closing parenthesis \noindent convergentp0 \quad 1in \quad $p L ˆ{ p } ( \Omega ) $ \quad to \quad some \quad $ u { 0 } for all psifor in all L toψ the∈ L power(Ω; A of)( p top0 = the 1 − power of1 prime), where openψb parenthesis= G ◦ ψ( usual Capital composition Omega semicolon ) . Remark A closing parenthesis open \ in L ˆ{ p } ( \Omega ;0 L ˆ{ p } ( \Delta ( A ) ) ) = L ˆ{ p } ( \Omega parenthesis3 sub.2. pψb toi the s the power function of prime in toL thep (Ω; powerC(∆( ofA))) 1 underbargiven by =ψ 1b( minusx) = G hline(ψ(x)) fromfor p to 1 closing parenthesis comma where psi-hatwide\times =\Delta G circ psi( open A parenthesis ) ) usual $ \ compositionquad i f as closing parenthesis period Remark 3 period 2 period psi-hatwide i s the function in L to the power of p to the power of prime open parenthesis Capital Omega\ begin semicolon{ a l i g n ∗} C open parenthesis Capital Delta open parenthesisx ∈ Ω. A closing parenthesis closing parenthesis closing parenthesis givenE by\ ni hatwide-psi\ varepsilon open parenthesis\rightarrow x closing parenthesis0 ,= G weopen parenthesis have \\\ psiint open{\ parenthesisOmega x} closingu {\ parenthesisvarepsilon closing} parenthesis( x ) for We\ psi willˆ{\ brieflyvarepsilon express the} above( x notion ) of dx convergence\rightarrow by writing\ intuε →\uint0 in {\Omega \times \Deltax in Capital(A) Omega period} u { 0 } ( x , s ) \widehat{\ psi } ( x , s ) dxd \beta (We s will ) briefly ( express 3 the . above 2 notion ) of convergence by writing u sub epsilon right arrow u sub 0 in Lp(Ω) − weakΣ. \endL{ toa the l i g n power∗} of p open parenthesis Capital Omega closing parenthesis hyphen weak Capital Sigma period Before we proceed any further comma .. let us prove a result from which we will next \noindentBeforef o r a we l l proceed $ \ psi any further\ in L , ˆ{ letp us ˆ{\ proveprime a result}} ( from\ whichOmega we will;next A ) derive ( ˆ{ 1 {\underline {\}}} { p ˆ{\prime }} derive one fundamental example of a weakly Capital Sigma hyphen convergent sequence inp L to the power of p open parenthesis = 1 −one fundamental\ r u l e {3em}{0.4 example pt } ˆ of{ p a weakly} { 1 }Σ−)convergent , $ where sequence $ \ inwidehatL (Ω).{\ psi } = G \ circ \ psi Capital OmegaProposition closing parenthesis 3 . 3 period . Let u ∈ Lp(Ω; A). We have uε → u in Lp(Ω)− weak as ( $Proposition usual composition 3 period 3 period ) .. . Let u in L to the power of p open parenthesis Capitale Omega semicolon A closing parenthesis Remarkε $3→ 0, . 2 . \widehat{\ psi } $ i s the function in $Lˆ{ p ˆ{\prime }} ( \Omega period .. We have u toε the power of epsilon right arrow u-tildewide in L to the power of p open parenthesis Capital Omega closing ;C(where\Deltau is( defined A as ) in ) ( )$ 3 . 1 givenby) and ue(x) $ =\Mwidehat(u(x)) for{\ psix }∈ Ω. ( x ) = G ( \ psi parenthesis hyphen weak as epsilon right arrow 0 comma N ( x )Proof ) . $ f oLet r K( Ω) ⊗ A denote the space of complex functions ψ on Ω × Ry of where uthe to the power of epsilon .. is defined as in .. open parenthesis 3 period 1 closing parenthesis and u-tildewide open parenthesisform x closing parenthesis = M open parenthesis u open parenthesis x closing parenthesis closing parenthesis for x in Capital Omega\ begin period{ a l i g n ∗} x Proof\ in period\Omega .. Let K parenleftbig. to the powerX of hline Capital Omega parenrightbig oslash A denote the space of complex ψ(x, y) = ϕi(x)w (y)(x ∈ Ω y ∈ N ) functions\end{ a li psi g n on∗} overbar Capital Omega times R sub y to thei power of N of, theR p form with a summation of finitely many t erms , ϕi ∈ L (Ω), wi ∈ A. Having regard to axiom \ centerlinepsi open parenthesis{We will x comma briefly y closing express parenthesis the above= sum phi notion i open ofparenthesis convergence x closing by parenthesis writing w sub $ ui open{\ parenthesisvarepsilon } \rightarrow( HA )u 4 {of0 Definition}$ in } 2 . 1 , it is clear that the claimed convergence property y closing parenthesisholds true parenleftbig if u i s taken x in overbar in K( CapitalΩ) Omega⊗ A, subhence comma in K y( in R toΩ; theA) power, thanks of N to parenrightbig the fact with a summationthat K( of finitelyΩ) ⊗ manyA is t dense erms comma in K( phi i inΩ; LA to)( the powersee , ofp e open . g . parenthesis , [ 4 , Capital p . 46 Omega ] ) . closing parenthesis comma\ begin w{ a sub l i g i n in∗} A period .. Having regard to L ˆ{ p }Therefore( \Omega the proposition) − weak \Sigma . axiom open parenthesis HA closing parenthesisK( 4Ω; ofA Definition) Lp(Ω; 2 periodA)( 1 comma it is clear that the claimed convergence property \end{ a l i gfollows n ∗} by the density of in the way of proceeding i s a routine holds trueexercise if u i s left taken to inthe K reader parenleftbig).  to the power of hline Capital Omega parenrightbig oslash A comma hence in K parenleftbig to the power of hline Capital Omega semicolon A parenrightbig comma thanks to the fact that \noindent BeforeThis we yields proceed the claimed any further fundamental , \quad examplelet us through prove the a result next result from . which we will next K parenleftbig to the power of hline Capital Omegau parenrightbig∈ Lp(Ω; A oslash). A is dense in K parenleftbig to the power of hline derive oneCorollary fundamental 3 .example 4 . of aLet weakly $ \Sigma −Then$ convergent , the sequence s equence in $ L ˆ{ p } Capital Omega(uε) semicolon A parenrightbigΣ open− parenthesis see comma .. e period g period comma .. open square bracket 4 comma ( \Omega ε>0)is . $ weakly .. p period 46convergent closing square in bracketLp(Ω) closingto u parenthesis= G ◦ u. period .. Therefore the proposition b 0 follows byProof the density . ofFor K parenleftbig each ψ to the∈ powerLp (Ω; ofA hline), Capitalwe have Omega semicolonuψ ∈ AL parenrightbig1(Ω; A); hence in L to the power of p open\noindent parenthesisProposition Capital Omega 3 .semicolon 3 . \quad A closingLet parenthesis $ u \ openin parenthesisL ˆ{ p } the( way of\Omega proceeding; i s a A routine ) . $ \quad We have $ u ˆ{\ varepsilonthe corollary}\rightarrow \ widetilde {u} $ in $ L ˆ{ p } ( \Omega ) − $ weak as exercise left to the reader closing parenthesis period3.3. square $ \ varepsilonfollows readily\rightarrow by Proposition0 , $  This yields the claimedThe next fundamental result isexample very s through imple theand next the result proof period is therefore omitted . Corollary .. 3 period 4 period .. Let u in L to the power of p open parenthesis(u ) , uCapital∈ Lp(Ω) Omega, semicolonΣ A− closing parenthesis \noindentPropositionwhere $ u 3 ˆ{\ . 5varepsilon . Suppose}$ \quad a s equenceis definedε ε as∈E inε \quad (is 3 weakly . 1 ) and $ \ widetilde {u} period .. Thenconvergent comma .. the .. s equence .. open parenthesis u to the power of epsilon closing parenthesis sub epsilon greater 0 .. ( x ) =M(p up ( x ) )$for$x \ in \Omega . $ is .. weaklyin .. CapitalL (Ω) Sigmato hyphenu0 ∈ L (Ω × ∆(A)). Then : convergent in L to the power of p open parenthesis Capital Omega closing parenthesis .. to u-hatwide = G circ u period \noindentProof periodProof .. For . ..\quad each ..Let psi in $ L K to the ( ˆ power{\ r u of l e p{3em to the}{0.4 power pt }}\ of primeOmega open parenthesis) \otimes Capital OmegaA $ semicolon denote the space of complex functions A$ closing\ psi $ parenthesis on $\ overline comma .. we{\}{\ .. haveOmega .. u psi}\ in Ltimes to the powerR ˆ{ ofN 1 open} { y parenthesis}$ o f Capital the Omega semicolon A closing parenthesis semicolon .. hence .. the .. corollary \noindentfollows readilyform by Proposition .. 3 period 3 period square The next result is very s imple and the proof is therefore omitted period \ [ Proposition\ psi ( 3 period x 5 , period y .... ) Suppose = a\ ssum equence\ openvarphi parenthesisi u ( sub epsilon x ) closing w parenthesis{ i } ( sub yepsilon ) in E ( comma x u\ in sub epsilon\ overline in L to{\}{\ the powerOmega of p} open{ parenthesis, } y Capital\ in R Omega ˆ{ N closing} ) parenthesis\ ] comma is weakly Capital Sigma hyphen convergent in L to the power of p open parenthesis Capital Omega closing parenthesis .. to u sub 0 in L to the power of p open parenthesis Capital\noindent Omegawith times a Capital summation Delta open of finitely parenthesis many A closing t erms parenthesis $ , closing\varphi parenthesisi period\ in .. ThenL ˆ :{ p } ( \Omega ) , w { i }\ in A . $ \quad Having regard to axiom ( HA ) 4 of Definition 2 . 1 , it is clear that the claimed convergence property

\noindent holds true if $u$ i s taken in $K (ˆ{\ r u l e {3em}{0.4 pt }}\Omega ) \otimes A ,$ hencein $K (ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; A ) ,$ thanks to the fact that $ K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ) \otimes A$ is densein $K (ˆ{\ r u l e {3em}{0.4 pt }} \Omega ; A ) ( $ see , \quad e . g . , \quad [ 4 , \quad p . 46 ] ) . \quad Therefore the proposition

\noindent follows by the density of $K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; A ) $ in $ L ˆ{ p } ( \Omega ; A ) ( $ the way of proceeding i s a routine

\noindent exercise left to the reader $ ) . \ square $

\ centerline { This yields the claimed fundamental example through the next result . }

\noindent C o r o l l a r y \quad 3 . 4 . \quad Let $ u \ in L ˆ{ p } ( \Omega ; A ) . $ \quad Then , \quad the \quad s equence \quad $ ( u ˆ{\ varepsilon } ) {\ varepsilon > 0 }$ \quad i s \quad weakly \quad $ \Sigma − $

\noindent convergent in $ L ˆ{ p } ( \Omega ) $ \quad to $ \widehat{u} = G \ circ u . $

\noindent Proof . \quad For \quad each \quad $ \ psi \ in L ˆ{ p ˆ{\prime }} ( \Omega ; A ) , $ \quad we \quad have \quad $ u \ psi \ in L ˆ{ 1 } ( \Omega ; A ) ; $ \quad hence \quad the \quad c o r o l l a r y

\noindent follows readily by Proposition \quad $ 3 . 3 . \ square $

\ centerline {The next result is very s imple and the proof is therefore omitted . }

\noindent Proposition 3 . 5 . \ h f i l l Suppose a s equence $ ( u {\ varepsilon } ) {\ varepsilon \ in E } , u {\ varepsilon }\ in L ˆ{ p } ( \Omega ) ,$ isweakly $ \Sigma − $ convergent

\noindent in $ L ˆ{ p } ( \Omega ) $ \quad to $ u { 0 }\ in L ˆ{ p } ( \Omega \times \Delta ( A ) ) . $ \quad Then : Capital Sigma hyphen CONVERGENCE .. 1 1 9 $ open\Sigma parenthesis− $ i closing CONVERGENCE parenthesis\quad u sub1 epsilon 1 9 right arrow u-tildewide sub 0 .. in L to the power of p open parenthesis Capital( iOmega $ ) closing u {\ parenthesisvarepsilon hyphen}\ weakrightarrow as E ni epsilon\ rightwidetilde arrow 0{u comma} { ..0 where}$ \quad in $ L ˆ{ p } ( \Omega Σ− CONVERGENCE 1 1 9 ( i )u → u in Lp(Ω)− weak as E 3 ε → 0, where ) u-tildewide− $ weak sub as0 open $ parenthesis E \ ni x closing\ varepsilon parenthesisε e0 =\rightarrow integral sub Capital0 Delta , $ open\quad parenthesiswhere A closing parenthesis u sub 0 open parenthesis x comma s closing parenthesisZ d beta open parenthesis s closing parenthesis open parenthesis x in Capital \ [ \ widetilde {u} { 0 } ( x ) = \ int {\Delta (A) } u { 0 } ( x , s Omega closing parenthesis period ue0(x) = u0(x, s)dβ(s)(x ∈ Ω). ∆(A) )open d parenthesis\beta ii( closing s parenthesis ) ( If x E is\ ain fundamental\Omega s equence). comma\ ] .. then open parenthesis u sub epsilon closing p parenthesis sub( ii epsilon ) If inE Eis .. a is fundamental bounded in L to s theequence power , of p open then parenthesis(uε)ε∈E Capitalis bounded Omega in closingL (Ω) parenthesis. period r,∞ r ∞ N ∞ Now commaNow for , for each each real real number number r greaterr ≥ equal1, let 1 commaXA = letXA X∩ L sub(R Ay to). theEquipped power of r with comma the infinityL − = X sub A to the power\ centerline of r cap L{( to ii ther, )∞ power If of $E$ infinity parenleftbig is a fundamental R sub y to s the∞ equence powerN ofN , parenrightbig\quad then period $ ( .. Equipped u {\ varepsilon with the } norm , XA is a Banach space ( note that L (R ) is continuously embedded in ) {\ varepsilonN \ in E }$ \quad is bounded in $ L ˆ{ p } ( \Omega ) . $ } L to ther powerΞ(R )) of. infinityFor hyphen future norm purposes comma X we sub wish A to to the power show of that r comma i f infinity a s equence is a Banach(uε) spaceε∈E open parenthesis p p note that Li to s the weakly powerΣ of− infinityconvergent parenleftbig in RL to(Ω) the powerto u of0 N parenrightbig∈ L (Ω × ∆( ..A is)) continuously, then as embeddedE 3 \noindent Now , for each real number $ r \geq 10 , $ l e t $ X ˆ{ r , \ infty } { A } in r Capital Xi parenleftbig R to the power of N parenrightbig closingp ,∞ parenthesis period .. For .. future purposes .. we wish to = X ˆ{ εr }→{ A0, (}\ 3 . 2cap ) holdsL ˆ{\ forinftyψ ∈} K(( RΩ; ˆ{XAN }) { providedy } ) .1 $ <\ pquad < ∞Equipped. It with the .. show that .. i f a s equence .. open parenthesis u sub epsilon closing parenthesis0 sub epsilon in E may be remarked in passing that i f ψ ∈ K( Ω; Xp ,∞), then ψ ∈ K( Ω; L∞( N )) $i s L weakly ˆ{\ infty Capital} Sigma − $ hyphen norm convergent $ , .. X in ˆ L{ tor the ,power\ infty of pA open} parenthesis{ A }$ isCapital a Banach OmegaR closing space parenthesis ( note that .. to $ L ˆ{\ inftyand therefore} ( Rψε ˆis{ N well} -) defined $ \quad by (is 3 . continuously 1 ) . We willembedded also need the following u sub 0 in Lobvious to the power remark of p . open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesisin $ r comma{\Xi ..} then( as .. R E ˆ ni{ epsilonN } right) arrow ) . 0 comma $ \quad For \quad future purposes \quad we wish to \quad show that \quad i f a s equence \quad Remark 3 . 6 . Given ζ0 ∈ and a sequence of complex numbers (ζ ) , we $ (open u parenthesis{\ varepsilon 3 period 2} closing) {\ parenthesisvarepsilonC .. holds for\ in psi inE K}$ open parenthesis to the powerε ε∈E of hline Capital Omega have ζ → ζ0 as E 3 ε → 0 i f and only i f , for any sequence (ε ) with 0 < ε ≤ 1, semicoloni s weakly X sub A $ ε to\Sigma the power− of$ p to convergent the power of prime\quad commain $ infinity L ˆ{ closingp } parenthesisn( n∈N\Omega .. provided)n $ ..\ 1quad less pto less infinity$ u { 0 } \ in L ˆε{n ∈pE,} εn (→ 0 as\Omegan → ∞, \wetimes have ζε\nDelta→ ζ0 as n(→∞ A. ) ) , $ \quad then as \quad $ E \ ni period .. It may beHaving remarked made in this point , let us now concentrate on proving the claimed result . \ varepsilonpassing that i f psi\rightarrow in K open parenthesis0 ,to $ the power of hline Capital Omega semicolon X sub A to the power of p to the power ( 3 . 2Proposition ) \quad holds f 3o r . 7$ .\ psi Assume\ in K that ( ˆ1{\ 0. A Xp , closing$ \ varepsilon parenthesisLet be sub as{ epsilon aboven }\ in . Ein comma LetE, u sub epsilon\Invarepsilon view in of the density{ n }\ ofrightarrowin A we may0 $ consider as $ n \rightarrow p0 \ inftyL to thesome power, $ ofw we p∈ open haveA such parenthesis $ that\zeta Capitalk v{\− wvarepsilon Omegak Ξ closing≤ η. parenthesisLet{ n }}\ commarightarrow .. is weakly Capital\zeta Sigma0 $ hyphen as convergent $ n \rightarrow in \ infty . $ L to the power of p open parenthesis Capital Omega closing parenthesis .. to sN ome u sub 0 in L to the power of p open parenthesis Capital Omega times Capital Delta open parenthesisf(x, y) = ϕ A(x) closingw(y)( parenthesisx ∈ Ω,y ∈ closingR ), parenthesis period .. Then comma .. as \ hspace ∗{\ f i l l }Having made this point , let us now concentrate on proving the claimed result . E ni epsilonwhich right gives arrow a 0 function comma ..f we∈ have K( .. openΩ; A parenthesis). Now 3 , periodwe can 2 closingwrite parenthesis for al l psi in K open parenthesis to the power of hline Capital Omega semicolon X sub A to the power of p to the power of prime comma infinity closing parenthesis \noindent Proposition \quad 3 . 7 . \quad RAssume that $ 1 < p < \ infty . $ \quad Suppose a s equence period Z ZZ Z Ω×∆(A)u0ψb $ ( u {\ varepsilonε } ) {\ varepsilon d \ in E } =, u {\)dβdx,varepsilonε }\ in $ uεψ dx+ − +R βdxfbΩ ×− ∆(A )u0fdβdxb R ε − f )dx Proof period .... According to Remark .... 3 period 6 commaR ( we may assumeψb without lossuε(ψ of generality that Ω Ω u f εR dx − Ω $E L is a ˆ{ fundamentalp } ( s\ equenceOmega period) .... , According $ ε \quadΩ×∆( toA) u parti0 s weakly open parenthesis $ \Sigma ii closing− parenthesis$ convergent of Proposition in $ .... L 3 ˆ{ periodp } 7 ( \Omega ) $ \quad to s ome $ u { 0 }\ in L ˆ{ p } ( \Omega \times \Delta ( comma it followsthe object being to establish that the left - hand s ide goes to zero as E 3 ε → 0. First A ) ) . $ \quad Then , \quad as that the, sequence by H o¨ openlder parenthesis ’ s inequality u sub we epsilon have closing parenthesis sub epsilon in E is bounded in L to the power of p open parenthesis$ E \ Capitalni \ Omegavarepsilon closing parenthesis\rightarrow period .. With0 this , $ in mind\quad commawe have let us\ beginquad (3.2)forall $ \ psi \ in K ( ˆ{\ r u l e {3emZ}{0.4 pt }}\Omega ; X ˆ{ p ˆ{\prime } , \ infty } { A } ) . $ by showing that open parenthesis 3 periodε ε 2 closing parenthesis holds forε psiε in K parenleftbig to the power of hline Capital | uε(ψ − f )dx| ≤ k uε k Lp(Ω) k ψ − f kLp 0(Ω). Omega parenrightbig oslash X subΩ A to the power of p to the power of prime comma infinity period .. But then it clearly suffices to \noindentverify thatProof open parenthesis . \ h f i l l 3 periodAccording 2 closing to parenthesis Remark \ holdsh f i l true l 3 for . each6 , psi we of may the assumeform without loss of generality that psi open parenthesis x comma y closing parenthesis = phi open parenthesis x closing parenthesis v open parenthesis y closing parenthesis\noindent parenleftbig$ E $ x is in overbar a fundamental Capital Omega s equence sub comma . y\ inh fR i lto l theAccording power of N to parenrightbig part ( ii comma ) of phi Proposition in K parenleftbig\ h f i l l 3 . 7 , it follows to the power of hline Capital Omega parenrightbig comma v in X sub A to the power of p to the power of prime comma infinity period\noindent that the sequence $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ is bounded in $ LLet ˆ{ psip be} as( above\Omega period .. Let) eta . greater $ \quad 0 periodWith .. In this view of in the mind density , letof A inus X begin sub A to the power of p to the power ofby prime showing comma that we may ( consider 3 . 2 ) holds for $ \ psi \ in K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ) \otimessome w inX A ˆsuch{ p that ˆ{\ ..prime bar v minus} , w bar\ infty Capital Xi} { toA the} power. $ of\ pquad to theBut power then of prime it less clearly or equal suffices eta period to .. Let verifyf open parenthesis that ( 3 x . comma 2 ) holds y closing true parenthesis for each = phi $ open\ psi parenthesis$ o f the x closing form parenthesis w open parenthesis y closing parenthesis parenleftbig x in overbar Capital Omega sub comma y in R to the power of N parenrightbig comma \ [ which\ psi gives( a function x , f in Ky parenleftbig ) = to\ thevarphi power of( hline x Capital ) Omega v semicolon ( y A ) parenrightbig ( x period\ in ..\ overline Now comma{\}{\Omega } { , } wey can\ in write R ˆ{ N } ), \varphi \ in K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ) , v \ in X ˆintegral{ p ˆ{\ subprime Capital} Omega, u\ subinfty epsilon} { psiA to} the. power\ ] of epsilon dx plus minus integral integral sub Capital Omega plus sub integral integral u sub epsilon f to the power of epsilon integral sub Capital Omega times Capital Delta open parenthesis A to the power of dx sub closing parenthesis to the power of minus sub u sub 0 to the power of integral to the power of Capital Omega times\noindent CapitalLet Delta open$ \ psi parenthesis$ be A as closing above parenthesis . \quad uLet 0psi-hatwideintegral $ \eta > sub0 open . parenthesis $ \quad toIn the view power of d the beta density dx of hatwide-f$ A $ inCapital $ X Omega ˆ{ p times ˆ{\ subprime minus}} Capital{ A Delta} , open $ weparenthesis may consider A sub psi-hatwide to the power of = closing parenthesis u subsome 0 hatwide-f $ w d\ betain dxA from $ closing such parenthesis that \quad d beta$ dx\ parallel comma to integralv sub− Capitalw \ Omegaparallel u sub epsilon\Xi openˆ{ p parenthesis ˆ{\prime }} psi\ leq to the\ powereta of. epsilon $ \ minusquad fLet to the power of epsilon closing parenthesis dx the object being to establish that the left hyphen hand s ide goes to zero as E ni epsilon right arrow 0 period \ [First f comma ( x by H , o-dieresis y ) lder =quoteright\varphi s inequality( we x have ) w ( y ) ( x \ in \ overline {\}{\Omega } { , } y vextendsingle-vextendsingle-vextendsingle-vextendsingle\ in R ˆ{ N } ), \ ] integral sub Capital Omega u sub epsilon open parenthesis psi to the power of epsilon minus f to the power of epsilon closing parenthesis dx vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal bar u sub epsilon bar L p open parenthesis Capital Omega closing parenthesis bar psi to the power of epsilon minus f to the\noindent power of epsilonwhich bar gives sub L a p functionprime sub open $ parenthesisf \ in CapitalK ( Omega ˆ{\ r closing u l e {3em parenthesis}{0.4 pt period}}\Omega ;A) . $ \quad Now , we can write

\ [ \ int {\Omega } u {\ varepsilon }\ psi ˆ{\ varepsilon } dx { + } − \ int {\ int } {\Omega }{ + } {\ int } \ int { u {\ varepsilon } f ˆ{\ varepsilon }{\ int }ˆ{ dx } {\Omega \times \Delta (A }ˆ{ − } { ) } { u }ˆ{\ int } { 0 }}ˆ{\Omega \times \Delta ( A ) u{0{\widehat{\ psi }{\ int }}}}ˆ{ d } { ( }\beta dx {\widehat{ f } {\Omega }} \times { − } \Delta ( A ˆ{ = } {\widehat{\ psi }} ) u { 0 }\widehat{ f } d \beta dx ˆ{ ) d \beta dx , } {\ int {\Omega } u {\ varepsilon } ( \ psi ˆ{\ varepsilon }} − f ˆ{\ varepsilon } ) dx \ ]

\noindent the object being to establish that the left − hand s ide goes to zero as $ E \ ni \ varepsilon \rightarrow 0 . $ F i r s t , by H $ \ddot{o} $ lder ’ s inequality we have

\ [ \arrowvert \ int {\Omega } u {\ varepsilon } ( \ psi ˆ{\ varepsilon } − f ˆ{\ varepsilon } ) dx \arrowvert \ leq \ parallel u {\ varepsilon }\ parallel L p ( \Omega ) \ parallel \ psi ˆ{\ varepsilon } − f ˆ{\ varepsilon }\ parallel { L p }\prime { ( \Omega ) } . \ ] 1 20 .. G period NGUETSENG comma N period SVANSTEDT \noindentOn the other1 20 hand\quad commaG . NGUETSENG , N . SVANSTEDT OnLine the 1 bar other psi to hand the power , of epsilon minus f to the power of epsilon bar L p to the power of prime sub open parenthesis Capital 1 20 G . NGUETSENG , N . SVANSTEDT On the other hand , Omega closing parenthesis less or equal bar phi bar sub infinity parenleftbigg integral sub K bar v to the power of epsilon minus w \ [ \ begin { a l i g n e d }\ parallel \ psi ˆ{\ varepsilon } − f ˆ{\ varepsilon }\ parallel L p ˆ{\prime } { ( to the power of epsilon bar to the power of p to the power of primeZ dx parenrightbigg0 1 p 1 divided by prime comma Line 2 hline \Omega ) }\ leq \ parallelε ε \varphi0 \ parallel ε {\εinftyp } ( \ int { K }\mid v ˆ{\ varepsilon } where K i s a compact set in Capitalk ψ − Omegaf k Lp containing(Ω) ≤ k ϕ thek∞ support( | v of− phiw period| dx)p .., But K 0 − parenleftbiggw ˆ{\ varepsilon to the power}\ of integralmid ˆ sub{ p K ˆ bar{\ vprime to the power}} dx of epsilon ) minus p \ wf rto a c the{ 1 power}{\ ofprime epsilon} bar, to\\ the power of p to the\ r u power l e {3em of}{ prime0.4 ptdx}\ parenrightbiggend{ a l i g n 1 e ddivided}\ ] by p to the power of prime less or equal c open parenthesis K closing parenthesis bar v minuswhere w bar subK i Capital s a compact Xi to the set power in Ω ofcontaining p prime open the parenthesis support epsilon of ϕ. inBut E closing parenthesis comma where the constant c open parenthesis K closing parenthesis greater 0 depends solely on K period .. From all that we deduce \noindent where $K$ iR s a compact0 set in $ \Omega $ containing the support of $ \varphi vextendsingle-vextendsingle-vextendsingle-vextendsingleε ε p 1 integral sub Capital Omega u sub epsilon open parenthesis psi to the ( | v − w | dx) ≤ c(K) k v − w kΞp 0 (ε ∈ E), power. $ of\quad epsilonBut minus f to the powerK of epsilon closingp0 parenthesis dx vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal c etawhere open the parenthesis constant epsilonc(K in) > E0 closingdepends parenthesis solely on commaK. From all that we deduce \ [where ( ˆ{\ .. cint .. is} ..{ a ..K positive}\mid .. realv .. numberˆ{\ varepsilon .. independent} .. − of bothw ˆ ..{\ etavarepsilon .. and .. epsilon}\ periodmid ..ˆ{ Inp .. anotherˆ{\prime }} dx ) \ f r a c { 1 }{ p ˆ{\prime }}\Z leq c ( K ) \ parallel v − w \ parallel {\Xi ˆ{ p }} connection comma again by H dieresis-o lder quoterightε ε s inequality and use of Proposition 2 period 1 6 comma we have \prime ( \ varepsilon \ in | E),uε(ψ − f )dx\ ]| ≤ cη (ε ∈ E), vextendsingle-vextendsingle-vextendsingle-vextendsingleΩ integral integral sub Capital Omega times Capital Delta open paren- thesis A closing parenthesis u sub 0 open parenthesis hatwide-f minus psi-hatwide closing parenthesis d beta dx vextendsingle- where c is a positive real number independent of both η and vextendsingle-vextendsingle-vextendsingle less or equal c bar u sub 0 bar L p open parenthesis Capital Omega times Capital Delta \noindentε. whereIn the another constant connection $c , again ( K by H ) o¨ lder> ’ s0 inequality $ depends and solely use of Proposition on $K . $ \quad From all that we deduce open parenthesis2 . 1 A6 , closing we have parenthesis closing parenthesis bar phi bar L p to the power of prime sub open parenthesis Capital Omega closing parenthesis bar v minus w bar sub Capital Xi to the power of p prime comma \ [ hence\arrowvert \ int {\Omega } u {\ varepsilon } ( \ psi ˆ{\ varepsilon } − f ˆ{\ varepsilon } ) dx \arrowvertZZ \ leq c \eta ( \ varepsilon \ in E), \ ] vextendsingle-vextendsingle-vextendsingle-vextendsingle integral integral sub Capital0 Omega times Capital Delta open paren- | u (f − ψ)dβdx| ≤ c k u k Lp(Ω × ∆(A)) k ϕ k Lp k v − w k p 0, thesis A closing parenthesis u sub0 b 0 openb parenthesis to0 the power of hatwide-f minus(Ω) psi-hatwide closingΞ parenthesis d beta dx Ω×∆(A) vextendsingle-vextendsingle-vextendsingle-vextendsingle less or equal c eta comma \noindentwhere chence i swhere a positive\quad real independent$ c $ \quad of bothieta s \ andquad epsilona \quad periodp .. o Considering s i t i v e \quad that r e a l \quad number \quad independent \quad o f both \quad $ \integraleta $ sub\quad Capitaland Omega\quad u sub$ epsilon\ varepsilon f to the power. $of epsilon\quad dxIn right\quad arrowanother integral integral sub Capital Omega times connection , again by H $ \ddotZZ{o} $ lder ’ s inequality and use of Proposition 2 . 1 6 , we have Capital Delta open parenthesis A closing parenthesis u sub 0fb f-hatwide d beta dx | u0( −ψb)dβdx| ≤ cη, as E ni epsilon right arrow 0 comma we haveΩ in× the∆(A) end \ [ l\ Earrowvert ni epsilon right\ arrowint 0 im\ int vextendsingle-vextendsingle-vextendsingle-vextendsingle{\Omega \times \Delta (A) integral} u sub{ Capital0 } Omega( \ uwidehat sub epsilon{ f } where c i s a positive real independent of both η and ε. Considering that −psi to \ thewidehat power{\ ofpsi epsilon} dx) right d arrow\beta integraldx integral\arrowvert sub Capital Omega\ leq timesc Capital\ parallel Delta openu parenthesis{ 0 }\ Aparallel closing Lparenthesis p ( u sub\Omega 0 hatwide-psi\times d beta dx\ vextendsingle-vextendsingle-vextendsingle-vextendsingleDeltaZ (A))ZZ \ parallel \varphi less or\ equalparallel c eta commaL p ˆ{\prime } { ( ε \Omegawhere c i) s a}\ positiveparallel real independentv − of etawu periodεf \dxparallel→ .... Therefore{\u the0Xifdβdxb desiredˆ{ p result}}\ followsprime by , \ ] Ω Ω×∆(A) the arbitrariness of eta period .... Finally comma if psi is considered in K open parenthesis to the power of hline Capital Omega semicolon Xas subE A3 toε → the0, powerwe have of p into the the power end of prime comma infinity closing parenthesis comma then comma based on \noindent hence the density of K parenleftbig to the power ofZ hline CapitalZZ Omega parenrightbig oslash X sub A to the power of p to the power of prime comma infinity in K openl parenthesisim | tou theψεdx power→ of hline Capitalu ψdβdx Omega| ≤ semicoloncη, X sub A to the power of p to the \ [ \arrowvert \ int \ intE3ε→0{\Omegaε \times \Delta0 b (A) } u { 0 } ( ˆ{\widehat{ f }} power of prime comma infinity closing parenthesisΩ comma the sameΩ×∆( lineA) of argument as fo llowed − \widehat{\ psi } ) d \beta dx \arrowvert \ leq c \eta , \ ] before showswhere thatc wei s again a positive arrive at real open independent parenthesis 3 of periodη. 2Therefore closing parenthesis the desired comma result thereby follows completing by the proof period square p0,∞ the arbitrariness of η. Finally , if ψ is considered in K( Ω; XA ), then , based on As a consequence of this comma there is the0 following corollary period0 \noindent where $ c $ i s a positivep ,∞ real independentp ,∞ of both $ \eta $ and $ \ varepsilon Corollarythe 3 period density 8 period of K( .... ForΩ) u in⊗ KXA parenleftbigin K( to theΩ; powerXA of), hlinethe same Capital line Omega of argument semicolon Xas sub fo A to the power of p. comma $ \quad infinityllowedConsidering parenrightbig that open parenthesis 1 less p less infinity closing parenthesis comma the s equence open parenthesis u to the powerbefore of epsilon shows closing that parenthesis we again sub arrive epsilon at greater ( 3 . 2 0 ) is , weakly thereby completing the proof .  \ [ Capital\ int Sigma{\Omega hyphen} As convergentu a consequence{\ varepsilon in L to the of power this} ,f of there ˆ p{\ open isvarepsilon parenthesisthe following Capital} corollarydx Omega\rightarrow . closing parenthesis\ int .. t o\ u-hatwideint {\Omega \times \Delta (A) } u { 0 }\widehatp,∞ { f } d \beta dx \ ] ε sub period Corollary 3 . 8 . For u ∈ K( Ω; XA ) (1 < p < ∞), the s equence (u )ε>0 is Proof periodweakly .. Endeed comma this follows immediately by combining Proposition 3 period 7 .. with Corol hyphen p lary .. 3Σ period− convergent 4 period square in L (Ω) t o ub. \noindentThe nextProof resultas . $ is EtheEndeed corner\ ni hyphen ,\ thisvarepsilon stone follows of Capital immediately\rightarrow Sigma hyphen by combining convergence0 , $ Proposition theory wehave period 3 in . 7 the with end TheoremCorol 3 period - lary 9 period3. ..4. Assume that 1 less p less infinity period .. Suppose E .. is a fundamental s equence \ [and l let{ E a s equence\ ni The open\ varepsilon next parenthesis result u is sub\ therightarrow epsilon corner closing - stone0 parenthesis} of imΣ− convergence sub\arrowvert epsilon in theory E ..\ beint . bounded{\Omega in L to} theu power{\ ofvarepsilon p } open\ psi parenthesisˆ{\Theoremvarepsilon Capital 3 Omega .} 9dx . closing\Assumerightarrow parenthesis that period1\ int ..< Then p < ∞\ commaint. Suppose{\ .. aOmega subsequenceE is\times Ea tofundamental the power\Delta of s prime(A) .. can be } p u extracted{ 0 }\equence fromwidehat E su and{\ chpsi let that a} opens equenced parenthesis\beta(uε)ε u∈E subdx be epsilon bounded\arrowvert closing in parenthesisL (Ω)\ leq. Then subc epsilon ,\eta a in subsequence E to, the\ ] power of prime .. is 0 weakly CapitalE Sigmacan be hyphen convergent in L to the power of p open parenthesis Capital Omega closing parenthesis period p extracted from E su ch that (uε)ε∈E0 is weakly Σ− convergent in L (Ω). \noindent where $ c $ i s a positive real independent of $ \eta . $ \ h f i l l Therefore the desired result follows by

\noindent the arbitrariness of $ \eta . $ \ h f i l l Finally , if $ \ psi $ is considered in $K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; X ˆ{ p ˆ{\prime } , \ infty } { A } ) , $ then , based on

\noindent the density of $K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ) \otimes X ˆ{ p ˆ{\prime } , \ infty } { A }$ in $ K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; X ˆ{ p ˆ{\prime } , \ infty } { A } ) , $ the same line of argument as fo llowed

\noindent before shows that we again arrive at ( 3 . 2 ) , thereby completing the proof $ . \ square $

\ centerline {As a consequence of this , there is the following corollary . }

\noindent Corollary 3 . 8 . \ h f i l l For $ u \ in K ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; X ˆ{ p , \ infty } { A } ) ( 1 < p < \ infty ) ,$ thesequence $( uˆ{\ varepsilon } ) {\ varepsilon > 0 }$ i s weakly

\noindent $ \Sigma − $ convergent in $ L ˆ{ p } ( \Omega ) $ \quad t o $ \widehat{u} { . }$

\noindent Proof . \quad Endeed , this follows immediately by combining Proposition 3 . 7 \quad with Corol − l a r y \quad $ 3 . 4 . \ square $

\ centerline {The next result is the corner − stone o f $ \Sigma − $ convergence theory . }

\noindent Theorem 3 . 9 . \quad Assume that $ 1 < p < \ infty . $ \quad Suppose $ E $ \quad is a fundamental s equence and let a s equence $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ \quad be bounded in $ L ˆ{ p } ( \Omega ) . $ \quad Then , \quad a subsequence $ E ˆ{\prime }$ \quad can be

\noindent extracted from $E$ su ch that $ ( u {\ varepsilon } ) {\ varepsilon \ in E ˆ{\prime }}$ \quad i s weakly $ \Sigma − $ convergent in $ L ˆ{ p } ( \Omega ) . $ Capital Sigma hyphen CONVERGENCE .. 1 2 1 \noindentProof period$ ..\ ForSigma any epsilon− $ in CONVERGENCE E comma put \quad 1 2 1 ProofF sub .epsilon\quad openFor parenthesis any $ psi\ varepsilon closing parenthesis\ in = integralE , sub $ Capital put Omega u sub epsilon open parenthesis x closing Σ− CONVERGENCE 1 2 1 Proof . For any ε ∈ E, put parenthesis psi open parenthesis x comma H sub epsilon open parenthesis x closing parenthesis closing parenthesis dx open paren- thesis\ [F psi{\ invarepsilon L to the power} of( p to the\ psi powerZ) of prime = open\ int parenthesis{\Omega Capital} u Omega{\ varepsilon semicolon A closing} ( parenthesis x ) closing\ psi p0 parenthesis( x , comma H {\ varepsilonFε(}ψ) =(u xε(x)ψ )(x, H )ε(x))dx (ψ (∈ L \(Ω;psiA)), \ in L ˆ{ p ˆ{\prime }} ( \Omega Ω ;A)),where p 1 divided by prime\ ] = 1 minus hline from p to 1 sub period .. This .. yields a sequence .. open parenthesis F sub epsilon 1 p p0 closing parenthesiswhere subp 0 epsilon= 1 in− E in .. bracketleftbig1. This L yields to the a power sequence of p to the( powerFε)ε∈E ofin prime[ openL (Ω; parenthesisA)]0 ( Capital Omega 0 semicolon Atopological closing parenthesis dual bracketrightbigof Lp (Ω; A)) which prime open is bounded parenthesis ( in topological the latter space ) . Hence , \noindent where $ p \ f r0 a c { 1 }{\prime } = 1 − \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } { . }$ \quad This \quad yields a sequence \quad dual of Lobserving to the power that ofL pp to(Ω; theA) poweri s a ofseparable prime open Banach parenthesis space Capital ( thanks Omega to the semicolon separability A closing of parenthesis closing $ ( F {\ varepsilon } ) {\ varepsilon \ in E }$ in \quad $ [ L ˆ{ p ˆ{\prime }} ( parenthesisA, whichas stated is bounded open parenthesis in the latter space closing parenthesis period .. Hence comma observing that \Omega ;A)] \prime ( $ topological L to thein power point of p to ( AH the power ) 1 ofof prime Definition open parenthesis 2 . 1 Capital ! ) Omega , we semicolon can A closing extract parenthesis a i s a separable dual o f $ L ˆ{ p ˆ{\prime0 }} ( \Omega ; A ) ) $0 which is bounded ( in the latter space ) . \quad Hence , observing that Banach spacesubsequence open parenthesisE thanksfrom to theE separabilityin such a of way A comma that , as stated as E 3 ε → 0,Fε → $ L ˆ{ p ˆ{\primep0 }} ( \Omega ; A ) $ i s a separable Banach spacep0 ( thanks to the separability of in pointF ..0 openin parenthesis[L (Ω; A AH)]0− closingweak∗, parenthesisthat i 1 s .. , ofFε Definition(ψ) → ..F 20( periodψ) for 1 .. any ! closingψ parenthesis∈ L (Ω; A comma). .. we .. can .. extract$A .. ,$ aThe .. subsequence as next stated point .. E to is the to power characterize of prime the .. from .. E in such afunctional way that commaF0. ..However as .. E to , as the will power presently of prime become ni epsilon apparent right arrow , it 0 comma is more F appropriate sub epsilon right arrow F sub 0 \noindent in point \quad ( AH ) 1 \quad of Definition \quadp0 2 . 1 \quad !), \quad we \quad can \quad e x t r a c t \quad a \quad subsequence \quad .. in .. bracketleftbigto characterize L to the the power closely of p to connected the power of functional prime open parenthesisG0 : L (Ω; CapitalC(∆(A))) Omega→ C semicolongivenby A closing parenthesis $ E ˆ{\prime }$ \−quad1 from p\0 quad $ E $ bracketrightbigG0(ϕ prime) = F0 sub(G hyphen◦ ϕ), ϕ weak∈ L *(Ω; commaC(∆(A ..))) that. ..Prior i s comma to this , let ψ ∈ K(Ω; A). inF subsuch epsilonClearly a way open that parenthesis , \quad psi closingas \quad parenthesis$ E rightˆ{\ arrowprime F}\ sub 0ni open parenthesis\ varepsilon psi closing\rightarrow parenthesis .. for0 any psi , Fin L{\ to thevarepsilon power of p to}\ therightarrow power of prime openF { parenthesis0 }$ \quad Capitalin Omega\quad semicolon$ [ A L closing ˆ{ p parenthesis ˆ{\prime period}} ..( The next\Omega ;A)] \prime { − weakZ } ∗ , $ \quad that \quad i s , point .. is to characterize the p0 1 0 $ F {\ varepsilon } (| Fε(ψ\)psi|≤ c( ) | ψ(x,\rightarrow Hε(x)) | χK(x)Fdx)p{ 0 (ε}∈ E( ), \ psi ) $ \quad f o r any $ \ psi functional F sub 0 period .. However commaΩ as will presently become apparent0 comma it is more appropriate \ into characterizeL ˆ{ p ˆ the{\ closelyprime connected}} ( functional\Omega G sub; 0 A : L to ) the power . $ of\quad p to theThe power next of pointprime open\quad parenthesisis to Capital characterize the where c is a positive constant ( independent of ε and ψ, as well ),K i s a compact set Omega semicolon C open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis closing parenthesis \noindentinfunctionalΩ containing $ the F support{ 0 } of.ψ, $ and\quadχKHoweveri s the characteristic , as will presently function of becomeK in Ω apparent. , it is more appropriate right arrow C given 0 By lett ing E0 3 ε → 0 and applying Proposition 3 . 3 ( with u(x, y) =| ψ(x, y) |p , toby characterize G sub 0 open parenthesis the closely phi closing connected parenthesis functional = F sub 0 open $ G parenthesis{ 0 } : G to L the ˆ{ powerp ˆ{\ of minusprime 1 circ}} phi( closing\Omega parenthesis;C( comma\Delta phi in L to(A))) the power of p to the power\rightarrow of prime open parenthesisC $ given Capital Omega semicolon C open parenthesis by $ G { 0 } ( \varphi ) = F { 0 } ( G ˆ{ − 1 }\ circ \varphi ), \varphi Capital Delta open parenthesis A closing parenthesis closing parenthesisN closing parenthesis period .. Prior to this comma let psi in x ∈ Ω, y ∈ R ), weget K\ in open parenthesisL ˆ{ p ˆ{\ Capitalprime Omega}} semicolon( \Omega A closing;C( parenthesis period\Delta ( A ) ) ) . $ \quad Prior to this , let $ \ psi \ in K( \Omega ; A ) . $ 0 Clearly | F0(ψ) |≤ ckψbkLp(Ω×∆(A)) bar F sub epsilon open parenthesis psi closing parenthesis bar less or equal c parenleftbigg integral sub Capital Omega bar psi open\noindent parenthesisandC l that e x a rcomma l y for any H subψ ∈ epsilon K(Ω; openA), where parenthesis it is x worth closing recalling parenthesis that closingψb = parenthesisG ◦ψ, and bar further to the power of p to the power of primeψb has chi support K open parenthesis in K. Thus x closing , parenthesis dx parenrightbigg p 1 divided by prime open parenthesis epsilon in E to\ [ the\mid power ofF prime{\ varepsilon closing parenthesis} ( comma\ psi ) \mid \ leq c ( \ int {\Omega }\mid \ psi ( x , H {\ varepsilon } ( x ) ) \mid ˆ{ p ˆ{\prime }}\ chi K ( x ) dx where c is a positive constant open parenthesis| G0( independentϕ) |≤ c k ϕ k ofLp epsilon0(Ω×∆(A and)) psi comma as well closing parenthesis comma K i s ) p \ f r a c { 1 }{\prime } ( \ varepsilon \ in E ˆ{\prime } ), \ ] a compact set 0 0 for all ϕ ∈ K(Ω; C(∆(A))) = K(Ω×∆(A)). Using the density of K(Ω; C(∆(A))) in Lp (Ω; Lp (∆(A))) = in Capitalp0 Omega containing the support of psi comma and chi K i s the characteristic function of K in Capital Omega period L (Ω × ∆(A)), we can extend G0 by continuity to an element of By lett ingp0 E to the power of primep ni epsilon right arrow 0 and applying Proposition .. 3 period 3 .. open parenthesis with u open\noindent parenthesis[Lwhere(Ω x× comma∆( $A))] c y0 $closing= isL parenthesis a(Ω positive× ∆(A =)). bar constantHence psi open parenthesis (there independent exists x comma ofu y0 closing $∈\ varepsilon parenthesis bar$ to and the power $ \ psi of p to the,$ power as of well prime $) comma , K$ i sacompactset in $ \Omega $ containing the support of $ \ psi , $ and $ \ chi K $ i s the characteristic function of x in Capital Omega comma y in R to the power of Np closing parenthesis comma we get bar F sub 0 open parenthesis $ K $ in $ \Omega . $ L (Ω × ∆(A))suchthat psi closing parenthesis bar less or equal c vextenddouble-vextenddouble-vextenddoubleZZ hatwide-psi vextenddouble-vextenddouble- vextenddouble L p to the power of primeG ( subϕ) = open parenthesisu (x, Capital s)ϕ(x, s Omega)dxdβ(s times) Capital Delta open parenthesis A closing \noindent By lett ing $Eˆ{\0prime }\ ni 0\ varepsilon \rightarrow 0 $ and applying Proposition \quad 3 . 3 \quad ( with parenthesis closing parenthesis Ω×∆(A) $ uand that ( for x any , psi iny K open ) =parenthesis\mid Capital\ psi Omega( semicolon x , A closing y ) parenthesis\mid ˆ{ commap ˆ{\ whereprime it is}} worth, recalling $ that psi-hatwidefor all = Gϕ circ∈ K(Ω; psiC comma(∆(A))) ... andThus , \ begin { a l i g n ∗} further psi-hatwide has support in K period ..ZZ Thus comma x \ in \Omega , y \ in R ˆ{ N } ) , we get \\\mid F { 0 } ( \ psi ) bar G sub 0 open parenthesis phi closingF0(ψ parenthesis) = bar lessu0(x, or s equal)ψb(x, sc) bardxdβ phi(s) bar sub L p prime sub open parenthesis Capital Omega\mid times\ leq Capitalc Delta\Arrowvert open parenthesis\widehat A closingΩ{\× parenthesis∆(psiA)}\Arrowvert closing parenthesisL p ˆ{\prime } { ( \Omega \times \Delta (A)) } 0 for all phifor in all Kψ open∈ K(Ω; parenthesisA) and therefore Capital Omega for all semicolonψ ∈ Lp (Ω; CA open), thanks parenthesis to the Capital density Delta of open parenthesis A closing \end{ a l i g n ∗} 0 parenthesis closing parenthesisp closing parenthesis = K open parenthesis Capital Omega times Capital Delta open parenthesis K(Ω; A) in L (Ω; A). The theorem follows .  A closing parenthesis closing parenthesis period .. Using the density of K open parenthesis Capital Omega semicolon C open \noindentRemarkand that for3 . 1any 0 . $ \ psiThe\ abovein K( compactness\Omega theorem; is A the main ) , reason $ where for it is worth recalling that parenthesisrequiring Capital Delta a homogenization open parenthesis A algebra closing parenthesis to be separable closing parenthesis . closing parenthesis $ \inwidehat .. L to the{\ psi power} of= p to the G power\ circ of prime\ psi parenleftbig, $ Capital\quad Omegaand semicolon L to the power of p to the power of prime openf u r tparenthesis h e r $ \ Capitalwidehat Delta{\ psi open} $ parenthesis has support A closing in parenthesis $K closing. $ \ parenthesisquad Thus parenrightbig , = L to the power of p to the power of prime open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis comma\ [ \mid .. weG .. can{ ..0 extend} ( .. G\varphi sub 0 .. by ..) continuity\mid .. to\ leq c \ parallel \varphi \ parallel { L p } \primean .. element{ ( ..\ ofOmega .. bracketleftbig\times L to\ theDelta power of(A)) p to the power of}\ prime] open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis bracketrightbig prime = L to the power of p open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis period .. Hence .. there .. exists .. u\noindent sub 0 in f o r a l l $ \varphi \ in K( \Omega ;C( \Delta ( A ) ) ) = K(L to the\ powerOmega of p open\times parenthesis\Delta Capital Omega( A times ) Capital ) Delta . $ open\quad parenthesisUsing A closing the density parenthesis of closing $K parenthesis ( \Omega such;C( that G sub\Delta 0 open parenthesis( A phi ) closing ) parenthesis ) $ = integral integral sub Capital Omega times Capital Delta open parenthesisin \quad A closing$ L ˆ parenthesis{ p ˆ{\prime u sub 0}} open( parenthesis\Omega x comma; s L closing ˆ{ p parenthesis ˆ{\prime phi}} open( parenthesis\Delta x comma(A) s closing parenthesis) ) = dxd betaL ˆ{ openp ˆ parenthesis{\prime s}} closing( parenthesis\Omega \times \Delta ( A ) ) , $ \quad we \quad can \quad extend \quad $ Gfor{ all0 phi}$ in\ Kquad openby parenthesis\quad c Capital o n t i n u Omega i t y \quad semicolonto C open parenthesis Capital Delta open parenthesis A closing parenthesisan \quad closingelement parenthesis\quad closingo f \quad parenthesis$ [ period L ˆ..{ Thusp ˆ{\ commaprime }} ( \Omega \times \Delta ( A))]F sub 0 open parenthesis\prime psi closing= L parenthesis ˆ{ p } =( integral\Omega integral\ subtimes Capital\Delta Omega times( Capital A ) Delta ) open . parenthesis $ \quad Hence \quad ther e \quad e x i s t s \quad A$ closingu { 0 parenthesis}\ in $ u sub 0 open parenthesis x comma s closing parenthesis psi-hatwide open parenthesis x comma s closing parenthesis dxd beta open parenthesis s closing parenthesis \ beginfor all{ a psi l i g in n K∗} open parenthesis Capital Omega semicolon A closing parenthesis and therefore for all psi in L to the power of p toL the ˆ{ powerp } of( prime\Omega open parenthesis\times Capital\Delta Omega semicolon( A A closing) ) parenthesis such comma that thanks\\ G to{ the0 } density( of\varphi )K = open\ parenthesisint \ int Capital{\ OmegaOmega semicolon\times A closing\Delta parenthesis(A) in L to the power} u of p{ to0 the} power( of x prime , open s parenthesis ) \varphi Capital( x Omega , semicolon s ) A dxd closing\ parenthesisbeta ( period s .. The ) theorem follows period square \endRemark{ a l i g ..n ∗} 3 period 1 0 period .. The above compactness theorem is the main reason for requiring a homogenization algebra to be separable period \noindent f o r a l l $ \varphi \ in K( \Omega ;C( \Delta ( A ) ) ) . $ \quad Thus ,

\ [F { 0 } ( \ psi ) = \ int \ int {\Omega \times \Delta (A) } u { 0 } ( x , s ) \widehat{\ psi } ( x , s ) dxd \beta ( s ) \ ]

\noindent f o r a l l $ \ psi \ in K( \Omega ; A ) $ and therefore for all $ \ psi \ in L ˆ{ p ˆ{\prime }} ( \Omega ; A ) , $ thanks to the density of

\noindent $ K ( \Omega ; A ) $ in $ L ˆ{ p ˆ{\prime }} ( \Omega ; A ) . $ \quad The theorem follows $ . \ square $

\noindent Remark \quad 3 . 1 0 . \quad The above compactness theorem is the main reason for requiring a homogenization algebra to be separable . 1 22 .. G period NGUETSENG comma N period SVANSTEDT \noindent3 period 21 period 22 \quad .. TheG strong . NGUETSENG Capital Sigma , N hyphen . SVANSTEDT convergence in L to the power of p open parenthesis Capital Omega closing3 . 2 parenthesis . \quad periodThe strong.. The concept $ \Sigma of strong Capital− $ convergence Sigma hyphen convergence in $ L ˆ{ p } ( \Omega ) . $ \quad The concept of strong 1 22 G . NGUETSENG , N . SVANSTEDT 3 . 2 . The strong Σ− convergence in $ \inSigma L to the− power$convergence of p open parenthesis Capital Omega closing parenthesis leans on the density of L to the power of p open Lp(Ω). The concept of strong Σ− convergence parenthesis Capital Omega semicolon C open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis in Lp(Ω) leans on the density of Lp(Ω; C(∆(A))) in Lp(Ω × ∆(A)). closing\noindent parenthesisin in $ L L to ˆ the{ p power} ( of p open\Omega parenthesis) $ Capital leans Omega on thetimes density Capital Delta of open $ L parenthesis ˆ{ p } A( closing\Omega parenthesis; C(closing parenthesis\Delta period( A ) ) )$in$Lˆ{ p } ( \Omega \times \Delta (A) Let1 ≤ p < ∞. )Let . 1 $ less or equal p less infinity period p DefinitionDefinition .. 3 period 1 1 period 3 . 1 .. 1 A . .. sequenceA .. sequence open parenthesis(uε) uε∈ subE, epsilon uε ∈ closingL (Ω) parenthesis, i s sub said epsilon in E comma \ [ Let 1 \ leq p < \ infty . \ ] p p u sub epsilonto in L be to the strongly power of p openΣ− parenthesisconvergent Capital in OmegaL (Ω) closingto some parenthesisu0 ∈ commaL (Ω ..× i∆( s ..A said)) toif .. be .. strongly .. Capital Sigmathe hyphen following condition is convergent .. in L to the power of p open parenthesis Capital Omega closing parenthesis .. to some u sub 0 in L to the power of p\noindent open parenthesisD e f i Capital n i t i o n Omega\quad times3 . Capital 1 1 . Delta\quad openA parenthesis\quad sequence A closing parenthesis\quad $ closing ( u parenthesis{\ varepsilon .. if the following} ) {\ varepsilon condition\ in E is} , u {\ varepsilon }\ in L ˆ{ p } ( \Omega ) , $ \quad i s \quad s a i d to \quad be fulfilled\quad : s t r o n g l y \quad $ \Sigma − $ fulfilled : open parenthesis SSC closing parenthesis braceex-braceex-braceleftmid-braceex-braceex-braceleftbt0 suchv sub openp (Ω; A) η>that k andu − ∈LpL(Ω×∆(A))≤ η Given 0 vb convergent(SSC)\quadbraceexin− $braceex L ˆ{−pbraceleftmid} ( \Omega− braceex)− $braceex\quad− braceleftbtto somesuch $ u( with{ 0 }\= G inv) , L ˆ{ p } ( 0≤suchα.that parenthesis bar with u sub epsilon minus to the power of Given sub hatwide-v sub v to the powerk u ofε− v epsilonε p (◦ to thethere powerissome ofEα =3 bv kL Ω)≤ηprovided 2 >ε sub\Omega bar sub\ Ltimes sub p to\ theDelta power of( G to A the ) power ) of $ eta\quad greaterif that the sub following open parenthesis condition to the power is of circ sub Capital Omega closing parenthesis toWe the express power of this v closing by writing parenthesisu sub→ u lessin orL equalp(Ω)− tostrong the powerΣ. of comma to the power of there sub \ begin { a l i g n ∗} ε 0 eta provided to the power ofLet bar us to theverify power the of unicity 0 and u of subu 0in minus Definition sub v-hatwide 3 . to 1 the 1 . power of v sub is some E alpha ni to f u l f i l l e d : \\ ( SSC ) braceex−braceex−braceleftmidp −braceex−braceex−braceleftbt such ˆ{ Given } { ( {\ parallel } the power ofProposition in L p L open parenthesis3 . 1 2 . CapitalIf Omega a s equence sub times Capital(uε)ε∈E Delta, uε ∈ toL the(Ω) power, is of s tronglyp sub openΣ parenthesis− A closing parenthesiswith { uconvergent to the{\ powervarepsilon of open parenthesis} − }}ˆ Capital{\eta Omega> semicolon{ that }} sub{\ closingwidehat parenthesis{v} to{ thev ˆ power{\ varepsilon of A closing parenthesis}}ˆ{ = } {\ parallel { L }}ˆ{ G } { p }}ˆ{\ parallel ˆ{ 0 } and { u { 0 } − } {\widehat{v}}ˆ{ v }} { ( ˆ{\ circ }ˆ{ v ) } {\Omega ) }ˆ{ , } {\ leq }ˆ{ the re } {\eta sub less or equalin L etap(Ω) dividedto sby ome 2 sub greateru ∈ Lp epsilon(Ω × ∆( 0A less)), orthen equal suchu alphais unique period . that provided }}ˆ{\ in { L p } 0L { ( \Omega }ˆ{ p 0} {\times \Delta }ˆ{ ( \Omega ; } { ( We expressProof this . by writingIn the u sub above epsilon notation right arrow , suppose u sub0 we in have L to theu power→ u1 and of p openu → parenthesisu2 in Lp(Ω) Capital− Omega closing A) }ˆ{ A) } { ) } {\ leq }} { i s some{ E }\alpha{\ε ni0 }}\ f rε a c {\0 eta }{ 2 } { > {\ varepsilon }} parenthesisstrong hyphen strongΣ. CapitalLet η Sigma > period0. The space K(Ω; A) being dense in Lp(Ω; A), 0 Let{\ usleq verifywe} thesuch may unicity{\ ofalpha u sub 0 in. Definition} that .. 3 period 1 1 period \end{ a l i g n ∗} Proposition 3 periodv 1∈ 2 K period(Ω; A) .. If a s equence openk ui parenthesis− v k Lp(Ω u sub× ∆( epsilonA)) closing≤ η i parenthesis= 1, 2. sub epsilon in E comma u choose i such that 0 bi 6 , Ac- sub epsilon in L to the power of p open parenthesis Capital Omega closing parenthesis comma .. is s trongly Capital Sigma hyphen \ centerlinecording{We express to this by writing $ u {\ varepsilon }\rightarrow u { 0 }$ in $ L ˆ{ p } convergent ε η Definition 3 . 1 1 , this yields some α > 0 such that k uε − v k Lp(Ω) ≤ (i = 1, 2) for ( in\ LOmega to the power) of− p$ open strong parenthesis $ \ CapitalSigma Omega. $ closing} parenthesis .. toi s ome u sub3 0 in L to the power of p open all E 3 ε ≤ α. It fo llows k vε − vε k ≤ 2η for E 3 ε ≤ α. Observing that parenthesis Capital Omega times Capital Delta open2 1 parenthesisLp(Ω) 3 A closing parenthesis closing parenthesis comma .. then u sub 0 \ centerline { Let us verify the unicityR of $ u { 0 }$ in Definition \quad 3 . 1 1 . } .. is unique period p k vε − vε k = ( | v (x, H (x)) − v (x, H (x)) |p χK(x)dx) , Proof period .. In the above2 notation1 Lp(Ω) commaΩ suppose2 ε we have2 u subε epsilon right arrow u sub1 0 to the power of 1 and u sub \noindent Proposition 3 . 1 2 . \quad If as equence $( u {\ varepsilon } ) {\ varepsilon epsilon rightwhere arrow uK subis a0 to compact the power set of in 2 inΩ Lcontaining to the power the of p supports open parenthesis of v1 and Capitalv2, Omegawe see closing that we parenthesis hyphen \ instrongE ..}can Capital, pass u Sigma to{\ the periodvarepsilon limit .. , Let as ..E eta}\3 ε greater→in0, in 0L period the ˆ{ precedingp ..} The( .. space inequality\Omega .. K open () use parenthesis Proposition, $ \quad Capitalis 3 Omega . s trongly semicolon $ A\Sigma −closing$ convergent parenthesis3 ) and .. obtain being .. dense .. in .. L to the power of p open parenthesis Capital Omega semicolon A closing parenthesis comma .. we .. may 2η \noindentchoose v subin i in $ K L open ˆ{ parenthesisp } ( Capital\Omega Omega) semicolon $ \quad A closingto s ome parenthesis $ u ..{ such0 }\ that ..in bar uL sub ˆ{ 0p to} the power( \Omega of i k vb2 − vb1 k Lp(Ω × ∆(A)) ≤ minus\times v-hatwide\Delta sub i bar( L p A open ) parenthesis ) , Capital $ \quad Omegathen times $ Capital u 3{ Delta0 }$ open\quad parenthesisi s unique A closing . parenthesis closing 2 1 2 1 parenthesisConsequently less or equal eta , divided by writing by 6 subu0 comma− u0 = iu =0 − 1vb comma2 + vb2 − 2vb period1 + vb1 − ..u According0, we get to \noindentDefinitionProof .. 3 period . \quad 1 1 commaIn the this aboveyields some notation alpha greater , suppose 0 such thatwe have bar u sub $ u epsilon{\ minusvarepsilon v sub i to}\ therightarrow power of epsilonu ˆ{ 1 bar} L{ p0 open}$ parenthesis and $ u Capital{\ varepsilon Omegaku closing2 − u1k}\ parenthesisLp(Ωrightarrow× ∆(A less)) ≤ orη. equalu eta ˆ{ divided2 } { by0 3} open$ in parenthesis $ L ˆ{ i =p 1} comma( 2\Omega ) − $ 0 0 closing parenthesis for2 1 Hence u = u , s ince η is arbitrary . Before we can present one funda- strongall E ni\ epsilonquad less$0 \ orSigma equal0 alpha. $ period\quad .. ItLet fo llows\quad .. bar$ v\ subeta 2 to> the power0 of. $epsilon\quad minusThe v sub\quad 1 to thespace power\quad of mental example of a strongly Σ− convergent epsilon$ K bar ( sub\Omega L p open; parenthesis A ) Capital $ \quad Omegabeing closing\quad parenthesisdense less\quad or equalin 2\ etaquad divided$ L by ˆ 3{ forp E} ni( epsilon\Omega less or ; A )sequence , $ ,\ wequad requirewe \quad a preliminarymay lemma . equal alphaLemma period .. Observing 3 . 13 . that We have bar v sub 2 to the power of epsilon minus v sub 1 to the power of epsilon bar sub L p open parenthesis Capital Omega closing parenthesis\noindent =choose parenleftbigg $ v to the{ i power}\ ofin integralK( sub Capital\Omega Omega bar; v A sub 2 ) open $ parenthesis\quad such x comma that H\quad sub epsilon$ \ openparallel l im k Φε k Lp(Ω) = kΦkLp(Ω × ∆(A)) (Φ ∈ Lp(Ω; A)). parenthesisu ˆ{ i } x{ closing0 } −parenthesis \widehatε closing→0 {v} parenthesis{ i }\ minusparallelb v sub 2 openL parenthesis p ( x\ commaOmega H sub\times epsilon open\Delta parenthesis(A x closing)) parenthesis\Proofleq .\ closingf r a cThe{\ parenthesiseta first}{ step bar6 i} to s{ tothe, recall power} i that of p = chi the K 1 lemma open , parenthesis is 2 true . with $ x closing\quadK(Ω; parenthesisAAccording) in place dx of parenrightbigg to hline from p to 1L commap(Ω; A), as i s straightforward by Proposition 3 . 3 and use of a routine argument \noindentwhere K( is seeD a e compact f the i n i t proof i o nset\ in ofquad Capital Theorem3 .Omega 1 1 3 containing , . this 9 ) . yields the Now supports some , fix of freely v $ sub\alpha 1Φ and∈ L vp sub(Ω;> A 2) comma0. $Let such weη see > that0 that. we $ \ parallel u can{\ passvarepsilonBy to the a density limit} comma − argument asv E ˆ{\ ni , epsilonvarepsilon we may right consider arrow} { 0 somecommai }\ψ inparallel∈ the K(Ω; precedingA) suchL inequality that p ( open\Omega parenthesis) use Proposition\ leq \ f r a c {\eta }{ 3 } ( i = 1 , 2 )$for 3 period 3 closing parenthesis and obtain R p η bar v-hatwide sub 2 minusk hatwide-vΦ − ψ k Lp sub(Ω; 1A bar) ≡ L p(Ω openk Φ(x parenthesis) − ψ(x) k p Capital∞dx) Omega1 ≤ times Capital Delta open parenthesis A 2 . closing\noindent parenthesisa l l closing $ E parenthesis\ ni \ varepsilon less or equal 2 eta\ dividedleq by\alpha 3 . $ \quad I t f o l l o w s \quad $ \ parallel v ˆConsequently{\ varepsilon comma} by{ writing2 } − u subv 0 ˆ to{\ thevarepsilon power of 2 minus} { u1 sub}\ 0 toparallel the power of{ 1L = u p sub 0 ( to the\Omega power of 2) minus}\ leq v-hatwide\ f r a c { 2 sub 2\eta plus hatwide-v}{ 3 }$ sub f o 2r minus $ E v-hatwide\ ni sub\ varepsilon 1 plus hatwide-v\ leq sub 1 minus\alpha u sub 0. to $ the\quad power ofObserving comma to the that power of 1 we get \ [ vextenddouble-vextenddouble\ parallel v ˆ{\ varepsilon u sub 0 to} the{ power2 } of − 2 minusv ˆ u{\ subvarepsilon 0 to the power} of{ 11 vextenddouble-vextenddouble}\ parallel { L p L p open ( parenthesis\Omega Capital) } = Omega ( ˆtimes{\ int Capital} {\ DeltaOmega open}\ parenthesismid Av closing{ 2 parenthesis} ( x closing , parenthesis H {\ varepsilon less or equal eta} period( x ))Hence u− sub 0v to{ the2 power} ( of 2 x = u sub , 0 H to the{\ powervarepsilon of comma} to the( power x of ) 1 s ince ) eta\mid is arbitraryˆ{ p }\ periodchi squareK( xBefore ) .. dx we .. ) can present\ r u l e { ..3em one}{ ..0.4 fundamental pt } ˆ{ p .. example} { 1 } .. of, a\ strongly] .. Capital Sigma hyphen convergent sequence comma we require a preliminary lemma period Lemma 3 period 13 period .. We have \noindentl epsilon imwhere right arrow $K$ 0 bar is Capital a compact Phi to the set power in of $ epsilon\Omega bar$ L p containing open parenthesis the Capital supports Omega of closing $ v parenthe-{ 1 }$ sisand = vextenddouble-vextenddouble-vextenddouble $ v { 2 } , $ we see that we hatwide-Phi vextenddouble-vextenddouble-vextenddouble L p open parenthesis Capitalcan pass Omega to times the Capital limit Delta , as open $ parenthesis E \ ni A closing\ varepsilon parenthesis closing\rightarrow parenthesis open0 parenthesis , $ in the Capital preceding Phi in L to inequality ( use Proposition the3 power . 3 ) of and p open obtain parenthesis Capital Omega semicolon A closing parenthesis closing parenthesis period Proof period .. The first step i s to recall that the lemma is true with K open parenthesis Capital Omega semicolon A closing parenthesis\ [ \ parallel in place of\widehat{v} { 2 } − \widehat{v} { 1 }\ parallel L p ( \Omega \times \DeltaL to the power(A)) of p open parenthesis\ leq Capital\ f r a Omega c { 2 semicolon\eta }{ A closing3 }\ ] parenthesis comma as i s straightforward by Proposition .. 3 period 3 and use of a routine argument open parenthesis see the proof of Theorem .. 3 period 9 closing parenthesis period .. Now comma fix freely Capital Phi in L to the\noindent power of pConsequently open parenthesis , Capital by writing Omega semicolon $ u ˆ{ A2 closing} { 0 parenthesis} − periodu ˆ{ 1 .. Let} { eta0 greater} = 0 period u ˆ{ ..2 By} { a 0 } − \widehatdensity{ argumentv} { 2 comma} + we may\widehat consider{v some} { psi2 in} K − open \ parenthesiswidehat{ Capitalv} { Omega1 } + semicolon\widehat A closing{v} parenthesis{ 1 } − such thatu ˆ{ 1 } { 0 ˆ{ , }}$ we get bar Capital Phi minus psi bar L p open parenthesis Capital Omega semicolon A closing parenthesis equiv parenleftbigg to the power\ [ \Arrowvert of integral subu Capital ˆ{ 2 Omega} { 0 bar} Capital − u Phi ˆ{ open1 } parenthesis{ 0 }\ xArrowvert closing parenthesisL minus p ( psi open\Omega parenthesis\times x closing\Delta parenthesis(A)) bar p infinity\ leq dx parenrightbigg\eta . \ hline] from p to 1 less or equal eta divided by 2 sub period

\noindent Hence $ u ˆ{ 2 } { 0 } = u ˆ{ 1 } { 0 ˆ{ , }}$ s i n c e $ \eta $ is arbitrary $ . \ square $ Before \quad we \quad can present \quad one \quad fundamental \quad example \quad of a strongly \quad $ \Sigma − $ convergent

\noindent sequence , we require a preliminary lemma .

\noindent Lemma 3 . 13 . \quad We have

\ [ l {\ varepsilon } im {\rightarrow 0 }\ parallel \Phi ˆ{\ varepsilon }\ parallel L p ( \Omega ) = \Arrowvert \widehat{\Phi}\Arrowvert L p ( \Omega \times \Delta (A))( \Phi \ in L ˆ{ p } ( \Omega ;A)). \ ]

\noindent Proof . \quad The first step i s to recall that the lemma is true with $K ( \Omega ; A )$ inplaceof $ L ˆ{ p } ( \Omega ; A ) , $ as i s straightforward by Proposition \quad 3 . 3 and use of a routine argument ( see the proof of Theorem \quad 3 . 9 ) . \quad Now , fix freely $ \Phi \ in L ˆ{ p } ( \Omega ; A ) . $ \quad Let $ \eta > 0 . $ \quad By a density argument , we may consider some $ \ psi \ in K( \Omega ; A )$ suchthat

\ [ \ parallel \Phi − \ psi \ parallel L p ( \Omega ;A) \equiv ( ˆ{\ int } {\Omega } \ parallel \Phi ( x ) − \ psi ( x ) \ parallel p {\ infty } dx ) \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } \ leq \ f r a c {\eta }{ 2 } { . }\ ] Capital Sigma hyphen CONVERGENCE .. 1 23 $ With\Sigma this in− mind$ comma CONVERGENCE we have on\quad the other1 23 hand WithLine this 1 vextendsingle-vextendsingle-vextendsingle-vextendsingle in mind , we have on the other hand bar Capital Phi to the power of epsilon bar sub L p open Σ− CONVERGENCE 1 23 With this in mind , we have on the other hand parenthesis Capital Omega closing parenthesis minus vextenddouble-vextenddouble-vextenddouble Phi-hatwide vextenddouble- \ [ \ begin { a l i g n e d }\arrowvert \ parallel \Phi ˆ{\ varepsilon }\ parallel { L p ( \Omega vextenddouble-vextenddoubleε L p open parenthesis Capital Omega timesε Capital Deltaε open parenthesis A closing parenthesis | k Φ kLp(Ω) −kΦbkLp(Ω × ∆(A))| ≤ | k Φ kLp(Ω) − k ψ k Lp(Ω)| closing) } − parenthesis \Arrowvert vextendsingle-vextendsingle-vextendsingle-vextendsingle\widehat{\Phi}\Arrowvert L less p or ( equal\ vextendsingle-vextendsingle-vextendsingleOmega \times \Delta (A )) \arrowvert \ leq \arrowvert \ parallel ε \Phi ˆ{\ varepsilon }\ parallel { L p bar Capital Phi to the power of epsilon bar sub L p open parenthesis+| k ψ kLp Capital(Ω) −kψb OmegakLp(Ω × closing∆(A)) parenthesis| minus bar psi to the power( \Omega of epsilon) bar} L − p open \ parallel parenthesis Capital\ psi ˆ Omega{\ varepsilon closing parenthesis}\ parallel vextendsingle-vextendsingle-vextendsingleL p ( \Omega ) Line\arrowvert \\ +|kψbkLp(Ω × ∆(A)) − kΦbkLp(Ω × ∆(A))|. 2 plus+ vextendsingle-vextendsingle-vextendsingle-vextendsingle\arrowvert \ parallel \ psi ˆ{\ varepsilon bar psi}\ toparallel the power of{ epsilonL p bar ( sub L\Omega p open parenthesis) } − Capital\Arrowvert OmegaIt follows closing\widehat parenthesis{\ psi minus}\ vextenddouble-vextenddouble-vextenddoubleArrowvert L p ( \Omega \ psi-hatwidetimes vextenddouble-vextenddouble-\Delta (A)) vextenddouble\arrowvert \\ L p open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing paren- + \arrowvert \Arrowvertε \widehat{\ psi }\Arrowvertε ε L p ( \Omega \times \Delta thesis vextendsingle-vextendsingle-vextendsingle-vextendsingle| k Φ kLp(Ω) −kΦbkLp(Ω × ∆( LineA))| 3plus ≤ k vextendsingle-vextendsingle-vextendsingle-vextendsingleΦ − ψ kLp(Ω) (A)) − \Arrowvert \widehat{\Phi}\Arrowvert L p ( \Omega \times \Delta vextenddouble-vextenddouble-vextenddouble psi-hatwide vextenddouble-vextenddouble-vextenddoubleε L p open parenthesis Cap- +| k ψ kLp(Ω) −kψbkLp(Ω × ∆(A))| ital(A)) Omega times Capital\arrowvert Delta open parenthesis. \end{ a Al i g closing n e d }\ parenthesis] closing parenthesis minus vextenddouble-vextenddouble- vextenddouble hatwide-Phi vextenddouble-vextenddouble-vextenddouble+kΦb − ψb Lk pLp open(Ω × parenthesis∆(A)). Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis vextendsingle-vextendsingle-vextendsingle-vextendsingle period But the first and third t erms on the r ight are majorized by k Φ − ψ kLp(Ω;A) . Hence \noindentIt follows I t f o l l o w s Line 1 vextendsingle-vextendsingle-vextendsingle-vextendsingle| k Φε k Lp(Ω) − kΦbkLp(Ω × ∆(A))| ≤ η + | k barψε k CapitalLp(Ω) − Phi kψbk toLp the(Ω × power∆(A)) of|. epsilon bar sub L p open parenthesis\ [ \ begin { Capitala l i g n e d Omega}\arrowvert closing parenthesis\ parallel minus vextenddouble-vextenddouble-vextenddouble\Phi ˆ{\ varepsilon }\ parallel Phi-hatwide{ L p vextenddouble- ( \Omega vextenddouble-vextenddouble) } −From \Arrowvert which the\ L lemmawidehat p open follows parenthesis{\Phi}\ in an CapitalArrowvert abvious Omega way times.L Capital p ( Delta\Omega open parenthesis\times A closing\Delta parenthesis(A closing)) parenthesis\arrowvert vextendsingle-vextendsingle-vextendsingle-vextendsingle\ leqWe are\ parallel now able to\Phi giveˆ the{\ claimedvarepsilon less example or} equal − . bar \ psi Capitalˆ{\ Phivarepsilon to the power}\ of epsilonparallel { L p ( \Omega ) }\\ p ε minus psi toExample the power of 3 epsilon . 14 . barLet sub Lu p∈ openL parenthesis(Ω; A). Then Capital , the Omega sequence closing(u parenthesis)ε>0 i s strongly Line 2 plus vextendsingle- vextendsingle-vextendsingle-vextendsingle+ \arrowvertΣ− \ parallel \ barpsi psiˆ{\ to thevarepsilon power of epsilon}\ parallel bar sub L p{ openL parenthesis p ( \ CapitalOmega Omega) } clos- − \Arrowvert \widehat{\ psip }\Arrowvert L p ( \Omega \timesp \Delta (A)) ing parenthesisconvergent minus vextenddouble-vextenddouble-vextenddouble in L (Ω) to ub. Indeed , for any psi-hatwidearbitrary vextenddouble-vextenddouble-vextenddoublev ∈ L (Ω; A), we have L p \arrowvert \\ ε ε open parenthesisk u Capital− v k OmegaLp(Ω) → times k ub Capital− vb Lp Delta(Ω × ∆( openA)) parenthesisas ε → 0 A. closingWe deduce parenthesis immediately closing parenthesis that vextendsingle- vextendsingle-vextendsingle-vextendsingle+ \Arrowvertthe \widehat{\Phi Line} 3 − plus vextenddouble-vextenddouble-vextenddouble \widehat{\ psi }\Arrowvert L hatwide-Phi p ( minus\Omega psi-hatwide\times \Delta (A)).ε \end{ a l i g n e d }\ ] vextenddouble-vextenddouble-vextenddoublesequence (u )ε>0 and the function L p openub satisfy parenthesis condition Capital ( Omega SSC ) times of Definition Capital Delta 3 . open 1 1 . parenthesis A closing parenthesis closingThe parenthesis remainder period of the present subsection is concerned with a series of results of But thepractical first and third interestt erms on the as r ight are regards majorized homogenization by .... bar Capital Phitheory minus . psi bar To sub L begin p open ,parenthesis Capital Omega\noindent semicolonthereBut A the closing i s first theparenthesisand following third period proposition t .... erms Hence on whose the r proof ight i s are an easy majorized verification by \ lefth f i l to l the$ \ parallel \Phi −vextendsingle-vextendsingle-vextendsingle-vextendsingle \ psi reader\ parallel . { L p ( \Omega bar;A) Capital Phi} to. the $ power\ h f i l of l epsilonHence bar L p open parenthesis p Capital OmegaProposition closing parenthesis 3 . 1minus 5 . vextenddouble-vextenddouble-vextenddoubleSuppose a s equence (uε)ε∈E, Phi-hatwide uε ∈ L vextenddouble-vextenddouble-(Ω), is s vextenddouble\ [ \arrowverttrongly L p open\ parallel parenthesis Capital\Phi ˆ Omega{\ varepsilon times Capital}\ Deltaparallel open parenthesisL p A closing ( \Omega parenthesisΣ− ) closing− paren- \Arrowvert \widehat{\Phi}\Arrowvertp L p ( \Omegap \times \Delta (A)) \arrowvert thesis vextendsingle-vextendsingle-vextendsingle-vextendsingleconvergent in L (Ω) to s ome u0 ∈ L less(Ω × or∆( equalA)). etaAssume plus vextendsingle-vextendsingle-vextendsingle- further that u0 ∈ \ leq \etap + \arrowvert \ parallelp \−psi1 ˆ{\ varepsilon }\ε parallel L p ( \Omega vextendsingleL (Ω; barC psi(∆( toA))) the. powerLet ofv epsilon0 ∈ L (Ω; barA L), p v0 open= G parenthesis◦ u0. Then Capitalk Omegauε − v0 closingk Lp(Ω) parenthesis→ 0 as minus vextenddouble- vextenddouble-vextenddouble) − \Arrowvert \widehat hatwide-psi{\ vextenddouble-vextenddouble-vextenddoublepsi }\Arrowvert L p ( \Omega L p open parenthesis\times Capital\Delta Omega(A times Capital)) Delta\arrowvert open parenthesis. \ A] closing parenthesis closing parenthesis vextendsingle-vextendsingle-vextendsingle-vextendsingle period E 3 ε → 0. From whichThe the lemma next proposition follows in an and abvious it s way corollary period square are likely to help us have a clear idea of the \noindentWe are nowFrom able whichto give the the claimed lemma example follows period in an abvious way $ . \ square $ somewhat abstract concept of strong Σ− convergence . Example .... 3 period 14 period .... Let u in L to the power of p open parenthesis Capital Omega semicolon A closing parenthesis Proposition 3 . 16 . Suppose a s equence (u ) , u ∈ Lp(Ω), is s period\ centerline .... Then{We comma are .... now the sequenceable to .... give open the parenthesis claimed u to example the power .ofε} epsilonε∈E closingε parenthesis sub epsilon greater 0 .... trongly Σ− i s .... strongly Capital Sigma hyphen convergent in Lp(Ω) to s ome u ∈ Lp(Ω × ∆(A)). Then \noindentconvergentExample .... in ....\ Lh to f i the l l power3 . 14 of p . open\ h f parenthesisi l0 l Let $Capital u \ Omegain closingL ˆ{ p parenthesis} ( \ ....Omega to .... u-hatwide; A sub ) period . $ ( i )u → u in Lp(Ω)− weak Σ; ....\ h fIndeed i l l Then comma , ....\ h ffor i ll ....the any sequence.... arbitrary\ ....εh f i v ll in0 L$ to ( the poweru ˆ{\ ofvarepsilon p open parenthesis} ) Capital{\ varepsilon Omega semicolon> A closing0 }$ parenthesis\ h f i l l i comma s \ h f ....i l l wes t .... r o nhave g l y $ \Sigma − $ (ii) k u k → k u k Lp(Ω × ∆(A)) asE 3 ε → 0. bar u to the power of epsilon minus vε toLp the(Ω) power of0 epsilon bar sub L p open parenthesis Capital Omega closing parenthesis right\noindent arrow barconvergent u-hatwide minus\ h f i hatwide-v l l in \ h L f p i l open l $ parenthesis L ˆ{ p } Capital( Omega\Omega times) Capital $ \ h Delta f i l l opento \ parenthesish f i l l $ A\widehat closing {u} { . }$ parenthesis\ h f i l l Indeed closing parenthesis , \ h f i l l ..f o as r epsilon\ h f i l rightl any arrow\ h f 0 i periodl l a r b .. i Wet r a deducer y \ h immediatelyf i l l $ v that\ in the L ˆ{ p } ( \Omega ;sequence A ) open , parenthesis $ \ h f i l ul towe the\ h power f i l l ofhave epsilon closing parenthesis sub epsilon greater 0 and the function u-hatwide satisfy condition open parenthesis SSC closing parenthesis of Definition 3 period 1 1 period \ hspaceThe remainder∗{\ f i l l of} the$ \ presentparallel subsectionu ˆ is{\ concernedvarepsilon with a} series − of resultsv ˆ{\ varepsilon }\ parallel { L p ( \Omegaof practical) }\ .. interestrightarrow .. as .. regards\ parallel .. homogenization\widehat theory{ periodu} − .. To \ ..widehat begin comma{v} .. thereL p.. i s ..( the\Omega \times \Deltafollowing( proposition A ) whose ) $ proof\quad i s an easyas verification $ \ varepsilon left to the\ readerrightarrow period 0 . $ \quad We deduce immediately that the Proposition .... 3 period 1 5 period .... Suppose .... a .... s equence .... open parenthesis u sub epsilon closing parenthesis sub epsilon\noindent in E commasequence u sub epsilon $ ( in uˆ L{\ to thevarepsilon power of p open} ) parenthesis{\ varepsilon Capital Omega> closing0 }$ parenthesis and the comma function .... is .... $ s\widehat{u} $ tronglysatisfy .... condition Capital Sigma ( hyphen SSC ) of Definition 3 . 1 1 . convergent in L to the power of p open parenthesis Capital Omega closing parenthesis .... to .... s ome u sub 0 in L to the powerThe remainder of p open parenthesis of the Capitalpresent Omega subsection times Capital is Deltaconcerned open parenthesis with a seriesA closing of parenthesis results closing parenthesis period ....of Assume practical further\ thatquad u subi n t 0 e in r e s t \quad as \quad regards \quad homogenization theory . \quad To \quad begin , \quad ther e \quad i s \quad the followingL to the power proposition of p open parenthesis whose proof Capital i Omega s an easy semicolon verification C open parenthesis left Capital to the Delta reader open . parenthesis A closing parenthesis closing parenthesis closing parenthesis period .... Let v sub 0 in L to the power of p open parenthesis Capital Omega semicolon\noindent A closingProposition parenthesis\ h comma f i l l 3 v sub . 1 0 5= G . to\ h the f i l power l Suppose of minus\ h 1 fcirc i l l ua sub\ h 0 fperiod i l l s .... equence Then bar\ uh sub f i ll epsilon$ ( minus u {\ varepsilon } v) sub{\ 0 tovarepsilon the power of epsilon\ in barE L} p open, parenthesis u {\ varepsilon Capital Omega}\ closingin parenthesisL ˆ{ p } right( arrow\Omega 0 .... as ) , $ \ h f i l l i s \ h f i l l s t r o n g l y \ h f i l l $ \ESigma ni epsilon− right$ arrow 0 period The next proposition and it s corollary are likely to help us have a clear idea of \noindentthe somewhatconvergent abstract concept in $ of L strong ˆ{ p Capital} ( Sigma\Omega hyphen convergence) $ \ h f i period l l to \ h f i l l s ome $ u { 0 }\ in L ˆProposition{ p } ( ....\ 3Omega period 16\ periodtimes .... Suppose\Delta .... a( .... As equence ) .... ) open . $ parenthesis\ h f i l l uAssume sub epsilon further closing parenthesis that $ u sub{ 0 } epsilon\ in $ in E comma u sub epsilon in L to the power of p open parenthesis Capital Omega closing parenthesis comma .... is .... s trongly .... Capital Sigma hyphen \noindentconvergent in$ L L to ˆ{ thep power} ( of p\ openOmega parenthesis;C( Capital Omega\Delta closing( parenthesis A ) .. to ) s ome ) u sub . $0 in\ Lh to f i the l l powerLet of $ v { 0 } p\ in open parenthesisL ˆ{ p } Capital( \ OmegaOmega times; Capital A Delta ) open , parenthesis v { 0 } A= closing G parenthesis ˆ{ − 1 closing}\ circ parenthesisu period{ 0 } .. Then. $ \ hopen f i l l parenthesisThen $ i\ closingparallel parenthesisu u{\ subvarepsilon epsilon right arrow} − u subv 0 ˆ ..{\ invarepsilon L to the power} of{ p open0 }\ parenthesisparallel CapitalL Omega p closing( \Omega parenthesis) hyphen\rightarrow weak Capital Sigma0 $ semicolon\ h f i l l as open parenthesis ii closing parenthesis bar u sub epsilon bar sub L p open parenthesis Capital Omega closing parenthesis right arrow\ begin bar{ a u l i sub g n ∗} 0 bar L p open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesisE \ ni as E\ varepsilon ni epsilon right arrow\rightarrow 0 period 0 . \end{ a l i g n ∗}

The next proposition and it s corollary are likely to help us have a clear idea of the somewhat abstract concept of strong $ \Sigma − $ convergence .

\noindent Proposition \ h f i l l 3 . 16 . \ h f i l l Suppose \ h f i l l a \ h f i l l s equence \ h f i l l $ ( u {\ varepsilon } ) {\ varepsilon \ in E } , u {\ varepsilon }\ in L ˆ{ p } ( \Omega ) , $ \ h f i l l i s \ h f i l l s t r o n g l y \ h f i l l $ \Sigma − $

\noindent convergent in $ L ˆ{ p } ( \Omega ) $ \quad to s ome $ u { 0 }\ in L ˆ{ p } ( \Omega \times \Delta ( A ) ) . $ \quad Then

\ centerline {( i $ ) u {\ varepsilon }\rightarrow u { 0 }$ \quad in $ L ˆ{ p } ( \Omega ) − $ weak $ \Sigma ; $ }

\ [ ( i i ) \ parallel u {\ varepsilon }\ parallel { L p ( \Omega ) }\rightarrow \ parallel u { 0 }\ parallel L p ( \Omega \times \Delta ( A ) ) as E \ ni \ varepsilon \rightarrow 0 . \ ] 1 24 .. G period NGUETSENG comma N period SVANSTEDT \noindentProof period1 24.. open\quad parenthesisG . NGUETSENG i closing parenthesis , N . SVANSTEDT : .. Let psi in L to the power of p to the power of prime open parenthesis Capital Omega semicolon A closing parenthesis period .. Fix a real c greater 0 with .. bar psi bar L p to the power of prime sub \noindent1 24Proof G . . NGUETSENG\quad ( i , N) . : SVANSTEDT\quad Let $ \ psi \ in L ˆ{ p ˆ{\prime }} ( \Omega ; open parenthesis Capital Omega semicolon A closing parenthesis0 less or equal c period .. Now comma fix Proof . ( i ) : Let ψ ∈ Lp (Ω; A). Fix a real c > 0 with k ψ k Afreely ) eta . greater$ \quad 0 ..Fix and choose a r e a v l in $ L c to the> power0 $ of p with open\ parenthesisquad $ Capital\ parallel Omega semicolon\ psi \ Aparallel closing parenthesisL p ˆ{\prime } { ( Lp0 ≤ c. Now , fix freely η > 0 and choose v ∈ Lp(Ω; A) such ..\Omega such that;A) ..(Ω; barA) u sub 0}\ minusleq v-hatwidec L . p $ open\quad parenthesisNow , Capital f i x Omega times Capital Delta open parenthesis A closing f r e e l y $ \etak u >− v0 Lp $(Ω ×\quad∆(A))and≤ chooseη $ v \ in L ˆ{ p } ( α \Omega ; A ) $ \quad such that \quad parenthesisthat closing parenthesis0 b less or equal eta divided4c . byBy 4 c sub hypothesis period .. By there i s some 0 such that $ \ parallel uε { 0 } −η \widehat{v} L p ( \Omega \times \Deltaε (A)) hypothesisk uε there− v ikLp s some(Ω) ≤ alpha2c subfor 0 ..E such3 ε ≤ thatα ..0. barOn u sub the epsilon other minus hand v , recallingto the power that of epsilonv → vb bar sub L p open parenthesis\ leq \ f rin Capital a c {\Lp(Ω)eta Omega− weak}{ closing4Σ( cCorollary parenthesis} { . } 3$ less .\ 4 orquad ) equal , weBy eta may divided by 2 c .. for E ni epsilon less or equal alpha sub 0 period .. hypothesis there i s some $ \alpha { 0 }$ \quad such that \quad $ \ parallel u {\ varepsilon } On consider some α1 > 0 such that − thev other ˆ{\ handvarepsilon comma recalling}\ parallel that v to the{ powerL p of epsilon ( \ rightOmega arrow v-hatwide) }\ leq in L\ tof r the a c {\ powereta of}{ p open2 parenthesis c }$ \quad f o r $ E \ ni \ varepsilon \ leq Z \alpha ZZ{ 0 } . $ \quad Onη Capital Omega closing parenthesis hyphen| weakvεψε Capitaldx − Sigma openvψdxdβ parenthesis| ≤ Corollary 3 period 4 closing parenthesis comma the other hand , recalling that $ v ˆ{\ varepsilonb b }\rightarrow \widehat{v} $ in $ L ˆ{ p } we may Ω Ω×∆(A) 4 ( consider\Omega some alpha) − sub$ 1 weakgreater 0 $ such\Sigma that ( $ Corollary 3 . 4 ) , wemay provided 0 < ε ≤ α1. Hence , by writing vextendsingle-vextendsingle-vextendsingle-vextendsingle integral sub Capital Omega v to the power of epsilon psi to the power of\noindent epsilon dxconsider minus integral some integralZ $ \ subalpha CapitalZZ{ 1 Omega} > times0 Capital $ZZ such Delta that open parenthesis A closing parenthesis hatwide-v u ψεdx − u ψdxdβ = (v − u )ψdxdβ psi-hatwide dxd beta vextendsingle-vextendsingle-vextendsingle-vextendsingleε 0 b lessb or0 equalb eta divided by 4 \ [ \arrowvert \ int {\Ω Omega } vΩ× ˆ∆({\A)varepsilon }\Ω×∆(psiA) ˆ{\ varepsilon } dx − \ int \ int {\Omega provided 0 less epsilon less or equal alpha sub 1 period .. Hence comma byZ writing \times \Delta (A) }\widehat{v}\widehat{\ psi } dxdε ε \beta \arrowvert \ leq \ f r a c {\eta }{ 4 }\ ] Line 1 integral sub Capital Omega u sub epsilon psi to the power of epsilon+ dx(uε minus− v )ψ integraldx integral sub Capital Omega times Ω Capital Delta open parenthesis A closing parenthesis u subZ 0 psi-hatwideZZ dxd beta = integral integral sub Capital Omega times Capital Delta open parenthesis A closing parenthesis open+ parenthesisvεψεdx hatwide-v− minusvψdxdβ, u sub 0 closing parenthesis psi-hatwide dxd \noindent provided $ 0 < \ varepsilon \ leq \alpha { 1b b} . $ \quad Hence , by writing beta Line 2 plus integral sub Capital Omega open parenthesisΩ u sub epsilon minusΩ×∆(A v) to the power of epsilon closing parenthesis psi to the powerone of epsilon quickly dx arrives Line 3 plusat integral sub Capital Omega v to the power of epsilon psi to the power of epsilon dx minus integral\ [ \ begin integral{ a l i g sub n e d Capital}\ int Omega{\Omega times Capital} u Delta{\ varepsilon open parenthesis}\ Apsi closingˆ{\ parenthesisvarepsilon v-hatwide} dx hatwide-psi− \ dxdint beta\ int {\Omega \times \Delta (A) } Zu { 0 }\ZZwidehat{\ psi } dxd \beta = \ int \ int {\Omega comma ε \times \Delta (A) } | ( uεψ\widehatdx − {v}u −0ψdxdβb u |{ ≤0 η} ) \widehat{\ psi } dxd \beta \\ one quickly arrives at Ω Ω×∆(A) +vextendsingle-vextendsingle-vextendsingle-vextendsingle\ int {\Omega } ( u {\ varepsilon integral} − subv Capital ˆ{\ varepsilon Omega u sub} epsilon) psi\ psi to theˆ{\ powervarepsilon of epsilon } dx \\ dx minus integralfor E integral3 ε ≤ α sub= min Capital(α0 Omega, α1), which times showsCapital ( Delta i ) . open parenthesis A closing parenthesis u sub 0 hatwide-psi dxd + \ int {\Omega } v ˆ{\ varepsilon }\ psi ˆp {\ varepsilon } dx − \ int \ int {\Omega beta vextendsingle-vextendsingle-vextendsingle-vextendsingle( i i ) : Let η > 0. Choose v ∈ lessL or(Ω; equalA) etasuch that k u0 − vb Lp(Ω × \times \Delta (A)η }\widehat{v}\widehat{\ psi } dxdε \beta , \end{ a l i g n e d }\ ] for E ni∆( epsilonA)) less≤ or6 equal. This alpha yields = min a open real parenthesisα0 > 0 such alpha that sub 0 commak uε − alphav k subLp(Ω) 1 closing≤ eta − parenthesisline3 comma which shows openprovided parenthesisE i closing3 ε ≤ α parenthesis0. Thus period , we have open parenthesis i i closing parenthesis : .. Let eta greater 0 period .. Choose v in L to the power of p open parenthesis Capital \noindent one quickly arrives at η Omega semicolon A closing parenthesis| k u ..0 k suchLp(Ω× that∆(A)) ..− bar k uv sub Lp 0(Ω minus× ∆(A v-hatwide))| ≤ L p open parenthesis Capital Omega times b 6 Capital Delta open parenthesis A closing parenthesis closing parenthesis less or equal eta divided by 6 sub period .. This \ [ yields\arrowvert aand real alpha\ subint 0 greater{\Omega 0 such} thatu ..{\ barvarepsilon u sub epsilon}\ minuspsi v toˆ{\ thevarepsilon power of epsilon} bardx L p− open \ parenthesisint \ int {\Omega Capital\times Omega\Delta closing parenthesis(A) less} or equalu { eta-line0 }\ 3widehat provided{\ E nipsi epsilon} dxd less or equal\beta alpha\ subarrowvert 0 period .. Thus\ leq comma\eta we \ ] have η | k u k Lp(Ω)− k vε k | ≤ (E 3 ε ≤ α ). vextendsingle-vextendsingle-vextendsingle bar u sub 0 bar sub L p open parenthesis Capitalε Omega times CapitalLp(Ω) Delta open3 0 \noindent f o r $ E \ ni \ varepsilon \ leq η \alpha = $ min $ ( \alpha { 0 } , \alpha { 1 } parenthesis AOn closing| k theother parenthesis− closinghand, parenthesisaccording minus| bar ≤ hatwide-vto Lemma L p open3.1 parenthesisα3,there Capitalissomethe Omegaα times> 0obvioussuchthatinequality Capital Delta ) ,$ whichshowsεv k ( i ) . × ∆(A)) for 0<ε≤ 1. Hence ,by 1 open parenthesis A closingLp(Ω) parenthesisvb closingkLp(Ω parenthesis vextendsingle-vextendsingle-vextendsingle2 less or equal eta divided by 6 and | k u k − k u k Lp(Ω × ∆(A))| ≤ | k u k Lp(Ω)− k vε k | \noindent ( i i ) : \quad Let $ \eta > 0 . $ \ε quadLp(Ω) Choose0 $ v \ in L ˆ{ p } ε ( \Omega Lp(Ω) Line 1 vextendsingle-vextendsingle-vextendsingle bar u sub epsilon bar L p open parenthesis Capital Omegaε closing parenthesis +| k v kLp(Ω) − k v Lp(Ω × ∆(A))| minus; A bar v ) to $ the\quad powersuch of epsilon that bar\ subquad L p open$ \ parallel parenthesis Capitalu { Omega0 } − closing \widehat parenthesis{v} vextendsingle-vextendsingle-L p ( \Omegab +| k v k − k u k Lp(Ω × ∆(A))| vextendsingle\times \Delta less or equal(A)) eta divided by 3 open\ leq parenthesis\ f r a c E{\ ni epsiloneta }{ less6 or} equal{ . alpha}$ \ subquad 0 closingThisb parenthesisLp(Ω×∆(A)) period Line0 2 yields a real $ \alpha { 0 } > 0 $ such that \quad $ \ parallel u {\ varepsilon } − On vextendsingle-vextendsingle-vextendsinglewe obtain readily bar the epsilon v bar sub L p open parenthesis Capital Omega closing parenthesis to thev ˆ power{\ varepsilon of other minus}\ handparallel hatwide-v barL sub L p p open ( parenthesis\Omega Capital) Omega\ leq subeta times−l i n to e the{ power3 }$ of comma provided sub Capital $ E \ ni \ varepsilon \ leq \alpha { 0 } . $ \quad Thus , we Delta open parenthesis A closing parenthesis| k u closingk parenthesis− k u k Lp to(Ω the× power∆(A)) of| according ≤ η vextendsingle-vextendsingle-vextendsingle lesshave or equal eta divided by 2 to sub for Lemmaε Lp(Ω) sub 0 less epsilon0 less or equal 3 period 1 alpha sub 1 sub period to the power of 3 to the powerfor of commaE 3 ε ≤ thereα = Hencemin is(α comma0, α1), thereby by to the proving power of( some i i the). alpha sub 1 greater 0 obvious such that inequality Line 3 \ [ \arrowvert \ parallel u { 0 }\ parallel { L p (2 \Omega \times \Delta (A vextendsingle-vextendsingle-vextendsingleCorollary 3 . 1 7 . Let bar u sub(uε) epsilonε∈E be bar a s sub equence L p open in parenthesisL (Ω). CapitalIn order Omega that closing this s parenthesis minus bar)) u sub} 0equence bar− L \pparallel open parenthesis\widehat Capital Omega{v} timesL Capitalp ( Delta\Omega open parenthesis\times A closing\Delta parenthesis(A)) closing parenthesis\arrowvert \ leq \ f r a c {\eta }{ 6 }\ ] 2 2 vextendsingle-vextendsingle-vextendsingles trongly Σ− converge in lessL (Ω) or equalt o vextendsingle-vextendsingle-vextendsingleu0 ∈ L (Ω × ∆(A)), i t is necessary bar and u sub suffi epsilon bar L p open parenthesis- Capital ci ent Omega that closingthe fol parenthesis lowing two minus conditions bar v to be the satisfied power of : epsilon bar sub L p open parenthesis Capital Omega 2 closing parenthesis vextendsingle-vextendsingle-vextendsingle( i )uε → u0 in LineL (Ω) 4− plusweak vextendsingle-vextendsingle-vextendsingleΣ; bar v to the power\noindent of epsilonand bar sub L p open parenthesis Capital Omega closing parenthesis minus bar hatwide-v L p open parenthesis Capital 2 Omega times Capital Delta open(ii parenthesis) k uε kL2 A(Ω) closing→ k u parenthesis0 k L (Ω × ∆( closingA)) parenthesisasE 3 ε → vextendsingle-vextendsingle-vextendsingle0. Line\ [ \ begin 5 plus{ vextendsingle-vextendsingle-vextendsinglea l i g n e d }\arrowvert \ parallel baru hatwide-v{\ varepsilon bar sub L p}\ openparallel parenthesis CapitalL p Omega ( times\Omega Capital Delta) − open \ parenthesisparallel A closingv ˆ{\ parenthesisvarepsilon closing}\ parenthesisparallel minus{ barL u sub p 0 bar( L\ pOmega open parenthesis) }\ Capitalarrowvert Omega times\ leq Capital\ f r a c {\ Deltaeta open}{ parenthesis3 } (E A closing\ ni parenthesis\ varepsilon closing parenthesis\ leq vextendsingle-vextendsingle-vextendsingle\alpha { 0 } ). \\ Onwe{\ obtainarrowvert readily }\ parallel the {\ varepsilon { v }\ parallel { L p ( \Omega ) }}ˆ{ other } − vextendsingle-vextendsingle-vextendsinglehand {\widehat{v}\ parallel bar u{ subL epsilon p bar( sub\Omega L p open}}ˆ parenthesis{ , } {\ Capitaltimes Omega}ˆ{ according closing parenthesis} {\Delta minus(A)) bar u sub 0}\ bar Larrowvert p open parenthesis\ leq Capital\ f r aOmega c {\eta times}{ Capital2 } to Delta{ openf o r parenthesis} Lemma A closing{ 0 parenthesis< \ varepsilon closing parenthesis\ leq } 3 vextendsingle-vextendsingle-vextendsingle . 1 {\alpha } { 1 }ˆ{ 3 } less{ or. equal}ˆ{ eta, } the re { Hence } i s { , by }ˆ{ some } thefor E\ nialpha epsilon{ less1 or} equal> alpha0 { =obvious min open} parenthesissuch alpha that sub{ 0i n comma e q u a l alpha i t y }\\ sub 1 closing parenthesis comma thereby proving\arrowvert open parenthesis\ parallel i i closing parenthesisu {\ varepsilon period square}\ parallel { L p ( \Omega ) } − \ parallel u Corollary{ 0 }\ 3parallel period 1 7 periodL .... p Let open ( parenthesis\Omega u\times sub epsilon\ closingDelta parenthesis(A)) sub epsilon in\ Earrowvert be a s equence\ inleq L to the\arrowvert power of 2 open\ parallel parenthesis Capitalu {\ Omegavarepsilon closing parenthesis}\ parallel period ....L In order p that ( this\Omega s equence ) − \ parallel v ˆs{\ tronglyvarepsilon Capital Sigma}\ hyphenparallel converge{ L in L p to the ( power\Omega of 2 open) parenthesis}\arrowvert Capital Omega\\ closing parenthesis .. t o u sub+ 0 in\ Larrowvert to the power of\ 2parallel open parenthesisv ˆ{\ Capitalvarepsilon Omega times}\ Capitalparallel Delta open{ L parenthesis p ( A closing\Omega parenthesis) } − closing \ parallel parenthesis\widehat{ commav} L .. i t p is necessary ( \Omega and suffi hyphen\times \Delta (A)) \arrowvert \\ +ci ent\arrowvert that the fol lowing\ parallel two conditions\widehat be satisfied{v :}\ parallel { L p ( \Omega \times \Delta (A))open parenthesis} i closing − \ parenthesisparallel u subu epsilon{ 0 right}\ arrowparallel u sub 0 ..L in L pto the ( power\Omega of 2 open parenthesis\times Capital\Delta Omega( A))closing parenthesis\arrowvert hyphen weak\end Capital{ a l iSigma g n e d semicolon}\ ] open parenthesis ii closing parenthesis bar u sub epsilon bar sub L to the power of 2 open parenthesis Capital Omega closing parenthesis right arrow bar u sub 0 bar L to the power of 2 open parenthesis Capital Omega times Capital Delta open parenthesis A\noindent closing parenthesiswe obtain closing readily parenthesis as E ni epsilon right arrow 0 period \ [ \arrowvert \ parallel u {\ varepsilon }\ parallel { L p ( \Omega ) } − \ parallel u { 0 }\ parallel L p ( \Omega \times \Delta (A)) \arrowvert \ leq \eta \ ]

\noindent f o r $ E \ ni \ varepsilon \ leq \alpha = $ min $ ( \alpha { 0 } , \alpha { 1 } ) ,$ thereby proving( i i $) . \ square $

\noindent Corollary 3 . 1 7 . \ h f i l l Let $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ be a s equence in $ L ˆ{ 2 } ( \Omega ) . $ \ h f i l l In order that this s equence

\noindent s t r o n g l y $ \Sigma − $ converge in $L ˆ{ 2 } ( \Omega ) $ \quad t o $ u { 0 } \ in L ˆ{ 2 } ( \Omega \times \Delta ( A ) ) , $ \quad i t is necessary and suffi − ci ent that the fol lowing two conditions be satisfied :

\ centerline {( i $ ) u {\ varepsilon }\rightarrow u { 0 }$ \quad in $ L ˆ{ 2 } ( \Omega ) − $ weak $ \Sigma ; $ }

\ [ ( i i ) \ parallel u {\ varepsilon }\ parallel { L ˆ{ 2 } ( \Omega ) }\rightarrow \ parallel u { 0 }\ parallel L ˆ{ 2 } ( \Omega \times \Delta ( A ) ) as E \ ni \ varepsilon \rightarrow 0 . \ ] Capital Sigma hyphen CONVERGENCE .. 1 25 $ Proof\Sigma period− .. In$ view CONVERGENCE .. of Proposition\quad .. 31 period 25 1 .. 6 comma .. we .. only .. have .. to .. show the .. sufficiency period .. SoProof comma . \quad In view \quad of Proposition \quad 3 . 1 \quad 6 , \quad we \quad only \quad have \quad to \quad show the \quad sufficiency . \quad So , Σ− CONVERGENCE 1 25 Proof . In view of Proposition 3 . 1 6 , we assumingonly open parenthesis have to i closing show parenthesis the sufficiency hyphen open . parenthesis So , ii closing parenthesis comma consider any arbitrary v in\noindent L to the powerassuming of 2 open ( parenthesis i ) − ( Capitalii ) , Omega consider semicolon any A arbitrary closing parenthesis $ v comma\ in andL use ˆ{ 2 } ( \Omega ; assuming ( i ) - ( ii ) , consider any arbitrary v ∈ L2(Ω; A), and use Abar ) u sub , epsilon $ and minus use v to the power of epsilon bar 2 L sub 2 open parenthesis Capital Omega closing parenthesis = bar u sub epsilon bar sub L to the power of 2 open parenthesis Capital Omega closing parenthesis to the power of 2 minus integral sub Capital \ [ \ parallel u {\ varepsilon } − vZ ˆ{\ varepsilon Z}\ parallel 2{ L } { 2 } ( \Omega Omega u sub epsilon to theε power of line-epsilon2 v dx minusline integral−epsilon sub Capital Omegaε hline subε 2 u sub epsilon v to the power of k u − v k 2L (Ω) =k u k 2 − u vdx − dx+ k v k 2 ) = \ parallelε u {\2 varepsilonε L (Ω)}\ parallelε ˆ{ 2 } { L ˆ{uεv2 } ( L\Omega(Ω) ) } − \ int {\Omega } epsilon dx plus bar v to the power of epsilon bar sub LΩ to the power of 2 openΩ parenthesis Capital Omega closing parenthesis to theu power{\ varepsilon of 2 }ˆ{ l i n e −e p s i l o n } v dx − \ int {\Omega }\ r u l e {3em}{0.4 pt } { u {\ varepsilon } v }ˆ{\ varepsilon } dx + \ parallel ε v ˆ{\ varepsilon }\ parallel2 ˆ{ 2 } { L ˆ{ 2 } ( \Omega to s ee thatto s when ee that E ni epsilonwhen rightE 3 ε arrow→ 0, 0 commak uε − v barkL u2(Ω) subt epsilon endsto minusk u v0 to− vb the L power(Ω × of∆( epsilonA)). Hence bar sub , L to the power of 2 open) }\ parenthesis] it Capitalfo llows Omega that closing condition parenthesis ( t SSC ends ) to .... of bar Definition u sub 0 minus 3 v-hatwide . 1 1 iL s to the satisfied power of . 2 open parenthesis Capital OmegaThis times Proves Capital the Delta corollary open parenthesis.  A closing parenthesis closing parenthesis period .... Hence comma it .. fo llows that .. conditionWe .. turn open now parenthesis to one SSC result closing of very parenthesis practical .. of interest Definition . .. 3 period 1 1 .. i s .. satisfied period \noindent to s ee that when $ E \ ni \ varepsilon \rightarrow p 01 1 , \ parallel u {\ varepsilon } .. This .. ProvesProposition the 3 . 18 . Suppose a real q ≥ 1 is such that 1 + q = r ≤ 1 and a − corollaryv ˆ{\s period equencevarepsilon square }\ parallel { L ˆ{ 2 } ( \Omega ) }$ t ends to \ h f i l l $ \ parallel u { 0 } − \widehat{v} L ˆ{ 2 } ( \Omegap \times \Deltap ( A ) ) . $ \ h f i l l Hence , We turn( nowuε)ε∈ toE oneis s result trongly of veryΣ− practicalconvergent interest in periodL (Ω) to s ome u0 ∈ L (Ω × ∆(A)), and a Propositions equence 3 period 18(v period) ....is Supposeweakly aΣ real− convergent q greater equal in 1 isL suchq(Ω) thatto hline s ome fromv p∈ toL 1q plus(Ω × 1∆( dividedA)). by q = 1 divided \noindent i t \quad foε ε∈ llowsE that \quad c o n d i t i o n \quad ( SSC ) \quad0 of Definition \quad 3 . 1 1 \quad i s \quad s a t i s f i e d . \quad This \quad Proves the by r less orThen equal 1 andu v a→ s equenceu v in Lr(Ω)− weak Σ as E 3 ε → 0. corollary $ . ε ε\ square0 0 $ open parenthesisProof . u sub epsilonWe closing may assume parenthesis without sub epsilon loss in E .... of is generality s trongly Capital that SigmaE hyphenis a convergent in L to q the power offundamental p open parenthesis s e - Capital quence Omega . closing The result parenthesis is that .... to s(v omeε)ε∈E u subis bounded 0 in L to the in powerL (Ω) of ( p open parenthesis Capital\ centerline OmegaProposition{ timesWe turn Capital now 3 Delta . to5 ) open one. parenthesis Thisresult of A closing very practical parenthesis closing interest parenthesis . } comma .... and a s equence open parenthesis v sub epsilonr0 closing parenthesis1 1 sub epsilon in E .... is weakly Capital Sigma hyphen convergent in being so , fix freely ψ ∈ L (Ω; A)( r0 = 1 − r ) and let c > 0 with L\noindent to the powerProposition of q open parenthesis 3 . 18 Capital . \ h Omega f i l l Suppose closing parenthesis a real .... $ to q s ome\geq v sub 0 in1 L $ to isthe suchpower of that q open $ parenthesis\ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } +Capital\ f r aOmega c { 1 times}{ q Capital} = Delta\ f r a open c { 1 parenthesis}{ r }\ A closingleq parenthesis1 $ and closing a s equence parenthesis period Then u sub epsilon v sub epsilon right arrow u sub 0 v subc 0≥ ..max in L{k tov0 thekLq power(Ω×∆(A of))k rψ openkLr parenthesis0(Ω;A), k ψ CapitalkLr 0(Ω; OmegaA) sup k closingvε kLq(Ω)}. \noindent $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ \ h f i l l is s tronglyε∈E $ \Sigma parenthesis hyphen weak Capital Sigma as E ni epsilon right arrow 0 period convergence the η revextenddouble−g strong − η − $ convergent in, $ L ˆ{ p } ( \Omega ) $ \ h f i l l to s ome $ u { 0 }\ in L ˆ{ p } ( Proof periodOn(u ..) We∈E other .. mayintroduce assumehand without, flet∈ ..Lp loss(> Ω; .. 0 of.A generality)Havingsuchthat that .. E .. isk ..ard au fundamental0−tofbthekLp s(Ω e× hyphen∆(A))Σ≤ andkeepofin ε ε 6c , \Omegaquence period\times .. The\ resultDelta is that( .. A open )parenthesis ) , v $ sub\ epsilonh f i l l closingand a parenthesis sub epsilon in E is bounded in L to the power of q open parenthesis Capital Omega closing parenthesisα > open0 parenthesis Propositionk u − f ε ..k Lp 3 period(Ω) ≤ 5η closing parenthesis \noindentminds equence that this $ infers ( the v {\ existencevarepsilon of some} )0 {\suchvarepsilon that \ εin E }$ \ h3 fc i l l i s weakly period .. This q0 $ \Sigma for− $E convergent3 ε ≤ α in0. Finally $ L ˆ{ ,q } noting( \Omega that fψ) $∈ \Lh f(Ω; i l l Ato), sand ome using $ v { 0 }\ in being sothe comma weak fix freelyΣ− psiconvergence in L to the power of (v of) r to, thewe power may consider of prime open some parenthesisα > 0 such Capital that Omega semicolon A closing Lparenthesis ˆ{ q } parenleftbig( \Omega 1 divided\times by r to the\Delta powerε ofε∈ primeE( A = 1 minus ) )1 divided . $ by r1 parenrightbig .. and let c greater 0 with Z ZZ Line 1 c greater equal maximum braceleftbiggε ε bar v sub 0 bar sub L q open parenthesisη Capital Omega times Capital Delta open parenthesis\noindent AThen closing $parenthesis u {\ varepsilon closing| parenthesisvεf ψ} dxv− bar{\ psivarepsilon bar subv0fb Lψdxdβb r prime}\| sub≤rightarrow open parenthesisu Capital{ 0 } Omegav { semicolon0 }$ \ Aquad in Ω Ω×∆(A) 2 closing$ L ˆ{ parenthesisr } ( comma\Omega bar psi bar) sub− L$ r prime weak sub $ open\Sigma parenthesis$ as Capital $ E Omega\ ni semicolon\ varepsilon A closing parenthesis\rightarrow supremum 0 . $ epsilon in Efor bar all v subE 3 epsilonε ≤ α1. barHence sub L q , open by writing parenthesis Capital Omega closing parenthesis bracerightbigg period Line 2 On open parenthesis sub u sub epsilonZ closing parenthesisZZ sub epsilon sub inZ E to the power of comma to the power of the other sub \noindent Proof . \quad We \quadε may assume without \quad l o sε s \quadε of generality that \quad $ E $ introduce hand comma f let in L tou theεvεψ powerdx − of p to the poweru0v0ψdxdβb of eta= open( parenthesisuε − f )vεψ greaterdx Capital Omega semicolon 0 period A\quad closingi s parenthesis\quad a Having fundamental suchΩ that s to e the− powerΩ×∆( ofA) re vextenddouble-gΩ vextenddouble-vextenddouble ard u sub 0 minus to quence . \quad The result is that \quad $ (ZZ v {\ varepsilon } ) {\ varepsilon \ in E }$ f-hatwide the vextenddouble-vextenddouble-vextenddouble sub L p open parenthesisfb Capital Omega sub times Capital Delta open is bounded in $ L ˆ{ q } ( \Omega ) (+ $ Proposition( −u0)v\0quadψdxdβb 3 . 5 ) . \quad This parenthesis A closing parenthesis closing parenthesis to the power of strongΩ×∆(A) Capital Sigma less or equal to the power of hyphen eta divided by 6 c sub comma to the power of convergenceZ sub and keep ofZZ in \noindent being so , fix freely $ \ psi \ inε ε L ˆ{ r ˆ{\prime }} ( \Omega ;A)( mind that this infers the existence of some alpha+ sub 0vε greaterf ψ dx 0− such that ....v bar0fbψdxdβ,b u sub epsilon minus f to the power of epsilon bar\ f r aL c p{ open1 }{ parenthesisr ˆ{\prime Capital}} Omega= closing 1 parenthesis− \ fΩ r a c { less1 or}{ equalr }Ω eta×∆() dividedA $) \quad by 3 cand l e t $ c > 0 $ with for .. E nione epsilon easily less arrives or equal at alpha sub 0 period .. Finally comma .. noting .. that .. f psi in L to the power of q to the power of\ [ prime\ begin open{ a l parenthesis i g n e d } c Capital\geq Omega\max semicolon\{\ A closingparallel parenthesisv comma{ 0 }\ .. andparallel .. using .. the{ ..L weak q ( \Omega \times \Delta (A))Z }\ parallelZZ \ psi \ parallel { L r }\prime { ( \Omega Capital Sigma hyphen convergence of| openu v parenthesisψεdx − v sub epsilonu v ψdxdβ closing| parenthesis ≤ η sub epsilon in E comma we may consider ;A) } , \ parallel \ psiε ε \ parallel { L0 0 b r }\prime { ( \Omega ;A) }\sup {\ varepsilon some alpha sub 1 greater 0 such that Ω Ω×∆(A) \ invextendsingle-vextendsingle-vextendsingle-vextendsingleE }\ parallel v {\ varepsilon }\ integralparallel sub Capital{ L Omega q ( v sub\Omega epsilon f to) the}\} power of. epsilon\\ psi On{ ( } provided{ u {\Evarepsilon3 ε ≤ α = min} ()α0,{\ α1).varepsilonThis shows}} theˆ{ propositionthe } {\ in. EThis ˆ{ proposition, }} other { introduce } to the powerhas of epsilon one useful dx minus corollary integral . integral sub Capital Omega times Capital Delta open parenthesis A closing parenthesis v subhand 0 hatwide-f , { f psi-hatwide} l e t dxd{\ betain vextendsingle-vextendsingle-vextendsingle-vextendsingleL ˆ{ p }}ˆ{\eta } ( >{\Omega } ;less 0 or . equal{ A eta} divided) Having by 2 { such thatfor all}ˆ{ E nire epsilon vextenddouble less or equal alpha−g }\ sub 1Arrowvert period .. Henceard comma{ byu writing{ 0 } − } to {\widehat{ f }} the {\Arrowvert } { L pLine ( 1 integral\Omega sub}ˆ Capital{ strong Omega} {\ u subtimes epsilon v\ subDelta epsilon psi(A)) to the power of epsilon}\Sigma dx minus{\ integralleq } integralˆ{ − } sub \ f Capital r a c {\eta }{ 6 Omegac }ˆ{ timesconvergence Capital Delta} { open, } parenthesis{ and A keep closing} parenthesiso f { in u}\ subend 0{ va sub l i g n 0 e psi-hatwide d }\ ] dxd beta = integral sub Capital Omega open parenthesis u sub epsilon minus f to the power of epsilon closing parenthesis v sub epsilon psi to the power of epsilon dx Line 2 plus integral integral sub Capital Omega times Capital Delta open parenthesis A closing parenthesis open parenthesis to the\noindent power of f-hatwidemind that minus this u sub infers 0 closing the parenthesis existence v sub of 0 hatwide-psi some $ \ dxdalpha beta Line{ 0 3} plus> integral0 $ sub such Capital that Omega\ h f v i l l sub$ \ epsilonparallel f to theu power{\ ofvarepsilon epsilon psi to} the − powerf of ˆ{\ epsilonvarepsilon dx minus integral}\ parallel integral sub CapitalL p Omega ( times\Omega Capital) Delta\ leq open\ f r a cparenthesis{\eta }{ A closing3 c parenthesis}$ v sub 0 f-hatwide hatwide-psi dxd beta comma one easily arrives at \noindentvextendsingle-vextendsingle-vextendsingle-vextendsinglef o r \quad $ E \ ni \ varepsilon integral\ leq sub Capital\alpha Omega{ u0 sub} epsilon. $ v\quad sub epsilonF i n a psi l l y to the , \ powerquad noting \quad that \quad of$ epsilon f \ psi dx minus\ in integralL integral ˆ{ q ˆ sub{\ Capitalprime Omega}} ( times\Omega Capital Delta; open A parenthesis ) , $ A\quad closingand parenthesis\quad uusing sub 0 v\ subquad the \quad weak 0 hatwide-psi$ \Sigma dxd− beta$ vextendsingle-vextendsingle-vextendsingle-vextendsingle convergence of $ ( v {\ varepsilon } less) or{\ equalvarepsilon eta \ in E } , $ we may consider some $ \providedalpha E{ ni1 epsilon} > less0 or $ equal such alpha that = min open parenthesis alpha sub 0 comma alpha sub 1 closing parenthesis period .. This shows the proposition period square \ [ This\arrowvert proposition has\ int one useful{\Omega corollary} periodv {\ varepsilon } f ˆ{\ varepsilon }\ psi ˆ{\ varepsilon } dx − \ int \ int {\Omega \times \Delta (A) } v { 0 }\widehat{ f }\widehat{\ psi } dxd \beta \arrowvert \ leq \ f r a c {\eta }{ 2 }\ ]

\noindent f o r a l l $ E \ ni \ varepsilon \ leq \alpha { 1 } . $ \quad Hence , by writing

\ [ \ begin { a l i g n e d }\ int {\Omega } u {\ varepsilon } v {\ varepsilon }\ psi ˆ{\ varepsilon } dx − \ int \ int {\Omega \times \Delta (A) } u { 0 } v { 0 }\widehat{\ psi } dxd \beta = \ int {\Omega } ( u {\ varepsilon } − f ˆ{\ varepsilon } ) v {\ varepsilon } \ psi ˆ{\ varepsilon } dx \\ + \ int \ int {\Omega \times \Delta (A) } ( ˆ{\widehat{ f } } − u { 0 } ) v { 0 }\widehat{\ psi } dxd \beta \\ + \ int {\Omega } v {\ varepsilon } f ˆ{\ varepsilon }\ psi ˆ{\ varepsilon } dx − \ int \ int {\Omega \times \Delta (A) } v { 0 }\widehat{ f }\widehat{\ psi } dxd \beta , \end{ a l i g n e d }\ ]

\noindent one easily arrives at

\ [ \arrowvert \ int {\Omega } u {\ varepsilon } v {\ varepsilon }\ psi ˆ{\ varepsilon } dx − \ int \ int {\Omega \times \Delta (A) } u { 0 } v { 0 }\widehat{\ psi } dxd \beta \arrowvert \ leq \eta \ ]

\noindent provided $ E \ ni \ varepsilon \ leq \alpha = $ min $ ( \alpha { 0 } , \alpha { 1 } ) . $ \quad This shows the proposition $ . \ square $ This proposition has one useful corollary . 1 26 .. G period NGUETSENG comma N period SVANSTEDT \noindentCorollary ..1 3 26 period\quad 19 periodG . NGUETSENG .. Let E .. be , .. N a . fundamental SVANSTEDT s equence period .. Let open parenthesis u sub epsilon closing parenthesisC o r o l l a r sub y \ epsilonquad 3 in E... 19 be . ..\ aquad s equenceLet $ E $ \quad be \quad a fundamental s equence . \quad Let 1 26 G . NGUETSENG , N . SVANSTEDT Corollary 3 . 19 . Let E be a $ (in .... u L to{\ thevarepsilon power of p open} parenthesis) {\ varepsilon Capital Omega\ in closingE parenthesis}$ \quad .... withbe \ ....quad 1 lessa p s less equence infinity comma .... and fundamental s equence . Let (uε)ε∈E be a s equence .... open parenthesis v sub epsilon closing parenthesis sub epsilon in E .... be .... a .... s equence0 .... in .... L to the power of p in Lp(Ω) with 1 < p < ∞, and (v ) be a s equence in Lp (Ω) ∩ L∞(Ω) to\noindent the power ofin prime\ h f openi l l parenthesis$ L ˆ{ p Capital} ( Omega\Omega closingε ε)∈ parenthesisE $ \ h f i cap l l Lwith to the\ h power f i l l of$ infinity 1 < openp parenthesis< \ Capitalinfty Omega, $ \ closingh f i l l parenthesisand \ h f i l l $ ( v {\ varepsilon } ) {\ varepsilon \ in E }$ \ h f i l l be \ h f i l l a \ h f i l l s equence \ h f i l l in \ h f i l l $ L ˆ{ p ˆ{\prime }} ( \Omega ) \cap L ˆ{\ infty } ( \Omega ) $ Equation: open parenthesis sub p to the power of prime to the power of 1 underbar =1 1 minus hlinep from p to 1 closing suchthat :( 0 = 1 − ) parenthesis .. such that : p 1 \ begin { a l i g n ∗} open parenthesis i closing parenthesis u sub epsilon right arrowp u sub 0 .. in L to the power of p open parenthesis Capital Omega \ tag ∗{$ ( ˆ{ 1 {\underline {\}}}( i )uε {→pu0 ˆ{\inprimeL (Ω)}}− weak= 1Σ; − \ r u l e {3em}{0.4 pt } ˆ{ p } { 1 } closing parenthesis hyphen weak Capital Sigma semicolon 0 ( ii )v → v in Lp (Ω)− s trong Σ; ) $open} such parenthesis that ii closing : parenthesis v subε epsilon0 right arrow v sub 0 .. in L to the power of p to the power of prime open ( iii )(v ) is bounded in L∞(Ω). parenthesis\end{ a l i g Capitaln ∗} Omega closing parenthesis hyphenε ε∈E s trong Capital Sigma semicolon Then u v → u v in Lp(Ω)− weak Σ. open parenthesis iii closing parenthesis openε parenthesisε 0 0 v sub epsilon closing parenthesis sub epsilon in E is bounded in L to Proof . According to Proposition 3 . 1 8 , we have u v → u v in the\ centerline power of infinity{( i open $ ) parenthesis u {\ Capitalvarepsilon Omega closing}\rightarrow parenthesis periodu { 0 }$ ε ε\quad in0 0 $ L ˆ{ p } ( \Omega L1(Ω)− weak Σ. Thus , as E 3 ε → 0, ) Then− $ u sub weak epsilon $ v\Sigma sub epsilon; right $ arrow} u sub 0 v sub 0 .. in L to the power of p open parenthesis Capital Omega closing parenthesis hyphen weak Capital SigmaZ period ZZ ε \ centerlineProof period{( .. According i i $ ) to.. v Proposition{\uεvεvarepsilonψ dx → .. 3 period}\ 1 8urightarrow comma0v0ψdxdβb .. we(ψ have∈v K u(Ω; sub{ A0)) epsilon.}$ \ vquad sub epsilonin $ right L ˆ arrow{ p ˆ u{\ subprime 0 v }} Ω Ω×∆(A) sub( 0\ ..Omega in L to the) power− $ of 1 s open trong parenthesis $ \Sigma Capital Omega; $ closing} parenthesis hyphen weak .. Capital Sigma period p Thus commaOn the as E other ni epsilon hand right , arrow observe 0 comma that the s equence (uεvε)ε∈E i s bounded in L (Ω). \ centerlineintegralHence sub{ Capital( , i ithanks i Omega $ ) to u Theorem sub ( epsilon v {\ v 3 subvarepsilon . 9 epsilon , we can psi} toextract the) power{\E0varepsilonfrom of epsilonE such dx right that\ in arrow theE sequence integral}$ is integral bounded sub Capital in $L ˆ{\ infty } ( \Omega ) . $ } p p Omega times(uε Capitalvε)ε∈E0 Deltaweakly openΣ parenthesis− converges A closing in L (Ω) parenthesisto some u subz0 0∈ vL sub(Ω 0× psi-hatwide∆(A)). Thus dxd beta , as open parenthesis psi in K open parenthesis Capital Omega semicolon A closing parenthesis closing parenthesis period \ centerlineOn the other{Then .... hand $ u comma{\ varepsilon .... observe that} ....v the{\ svarepsilon equence .... open}\ parenthesisrightarrow u sub epsilonu { 0 v} subv epsilon{ 0 closing}$ \quad in 0 parenthesis$ L ˆ{ p sub} epsilon( \Omega in E .... i s) bounded− $ in weak L to the $ power\Sigma of p open. $ parenthesisE} 3 ε → Capital0, Omega closing parenthesis period Z ZZ Hence comma thanks to Theorem .. 3 periodε 9 comma we can extract E to the power of prime from E such that the sequence \noindent Proof . \quad Accordinguεvεψ dx to→ \quad Propositionz0ψdxdβb (ψ\quad∈ K(Ω;3A)) .. 1 8 , \quad we have $ u {\ varepsilon } open parenthesis u sub epsilon vΩ sub epsilon closingΩ× parenthesis∆(A) sub epsilon in E to the power of prime weakly Capital Sigma hyphenv {\ convergesvarepsilon in L to}\ therightarrow power of p openu parenthesis{ 0 } Capitalv { Omega0 }$ closing\quad parenthesisin $ L to ˆ{ some1 } z sub( 0 in\Omega L to the power) − of $ pweak open\ parenthesisquadFrom$ all\ CapitalSigma that Omegawe. deduce $ times Capital Delta open parenthesis A closing parenthesis closing parenthesis period .. Thus Thus , as $ E \ ni \ varepsilon \rightarrow 0 , $ comma as ZZ E to the power of prime ni epsilon right arrow 0 comma(z integral− u v )ϕdxdβ sub Capital= 0 Omega u sub epsilon v sub epsilon psi to the power \ [ \ int {\Omega } u {\ varepsilon } v 0 {\0 varepsilon0 }\ psi ˆ{\ varepsilon } dx \rightarrow of epsilon dx right arrow integral integral sub CapitalΩ×∆(A Omega) times Capital Delta open parenthesis A closing parenthesis z sub 0 psi-hatwide\ int \ int dxd beta{\ openOmega parenthesis\times psi in K\Delta open parenthesis(A) Capital} Omegau { semicolon0 } v A closing{ 0 }\ parenthesiswidehat closing{\ psi parenthesis} dxd for all ϕ ∈ K(Ω×∆(A)) ( s ee Remark 4 . 2 ) . Hence z = u v almost everywhere period\beta ( \ psi \ in K( \Omega ;A)). 0 \ ] 0 0 in Ω × ∆(A). The corollary follows thereby . From all that we deduce  We conclude the present subsection by showing that strong Σ− convergence gen - integraleralizes integral sub usual Capital strong Omega convergence times Capital . Delta Specifically open parenthesis , we have A closing parenthesis open parenthesis z sub 0 minus u\noindent sub 0 v subOn 0 closing the otherparenthesis\ h f phi i l l dxdhand beta , =\ 0h f i l l observe that \ h f i l l the s equence \ h f i l l $ ( u {\ varepsilon } Proposition 3 . 20 . Suppose a s equence (u ) is s trongly convergent in v for{\ allvarepsilon phi in K open} parenthesis) {\ varepsilon Capital Omega times\ in CapitalE }$ Delta\ h openε f iε∈ l lE parenthesisi s bounded A closing in parenthesis $ L ˆ{ p closing} ( parenthesis\Omega Lp(Ω) open) parenthesis . $ s ee Remark .. 4 period 2 closing parenthesis period .. Hence z sub 0 = u sub 0 v sub 0 almost everywhere to some u ∈ Lp(Ω). Then u → u in Lp(Ω)− strong Σ. in Capital Omega times0 Capital Delta openε parenthesis0 A closing parenthesis period .. The corollary follows thereby period Proof . Let us begin by observing that the function u ∈ Lp(Ω) may as well square\noindent Hence , thanks to Theorem \quad 3 . 9 , we can extract0 $Eˆ{\prime }$ from $ E $ be viewed as a function in Lp(Ω; A)( resp .Lp(Ω × ∆(A))) depending on the sole suchWe that conclude the the sequence present subsection by showing that strong Capital Sigma hyphen convergence gen hyphen variable x ∈ Ω. Having made this point , let v ∈ Lp(Ω; A). By applying Lemma 3 . $eralizes ( u usual{\ strongvarepsilon convergence} periodv {\ .. Specificallyvarepsilon comma} ) we have{\ varepsilon \ in E ˆ{\prime }}$ weakly 1 3 with Φ = u − v, we see that i f η > 0 is freely fixed , then $ \PropositionSigma − 3 period$ converges 20 period .... in0 Suppose $ L ˆ a{ sp equence} ( open\Omega parenthesis) u $ sub to epsilon some closing $ z parenthesis{ 0 }\ subin epsilonL ˆ in{ Ep ....} some α > 0 is( s trongly\Omega convergent\times in L to\ theDelta power of( p open A parenthesis ) ) Capital . $ \ Omegaquad closingThus, parenthesis as exists such that k u − vε k Lp(Ω) ≤ k u − v k + η and k u − u k Lp(Ω) ≤ η to some u sub 0 in L to the power0 of p open parenthesis0 Capitalb Lp(Ω Omega×∆(A)) closing4 parenthesisε 0 period .. Then4 u sub epsilon right for E 3 ε ≤ α. Hence k u − vε k Lp(Ω) ≤ η + k u − v k . We deduce that arrow\ begin u{ suba l i 0 g .. n ∗} in L to the power of p openε parenthesis Capital2 Omega0 closingb Lp parenthesis(Ω×∆(A)) hyphen strong Capital Sigma period (u ) u , EProof ˆ{\prime periodcondition ..}\ Letni ( us SSC begin\ )varepsilon by of observing Definition that\rightarrow the 3 . function 1 1 u is sub0 satisfied 0 in , L\\\ to by theint powerε ε{\∈E ofandOmega p open0} thereby parenthesisu {\ varepsilon Capital Omega} . closingv {\ parenthesisvarepsilonproving .. may the}\ aspropositionpsi well beˆ{\ varepsilon } dx \rightarrow \ int \ int {\Omega \times \Delta Σ− (A)viewed as a} functionz {4 in0 . L}\ toThe thewidehat power vague{\ ofpsi p open} convergence parenthesisdxd \beta Capital of Omega( Radon\ psi semicolon measures\ in A closingK( parenthesis\Omega open parenthesis;A resp)). period L toLet the the power basic of p open notation parenthesis and hypotheses Capital Omega be times as in Capital the preceding Delta open parenthesissection A ( closing see in parenthesis closing parenthesis\end{ a l i gparticular closingn ∗} parenthesis the beginning .. depending of on Section the sole 3 ) . On the other hand , the space of all Z M(Z). variablecomplex x in Capital Radon Omega measures period .. Having on a lo made cally this compact point comma space let vwill in L be to the denoted power byof p open parenthesis Capital M(Z) K(Z)( Omega\noindent semicolonThusFrom A , closing all that parenthesisi s nothing we deduce period else .. than By applying the topological Lemma dual of provided with the σ− 3 periodusual 1 3 .. with inductive .. Capital l imit Phi = topology u sub 0 minus ) . v comma Also , .. the we .. notion see that of .. a i f etacompact greater 0 .. lo is cally .. freely fixed comma .. then\ [ \ ..int somecompact ..\ int alpha greater{\ spaceOmega 0 \times \Delta (A) } ( z { 0 } − u { 0 } v { 0 } ) exists\varphi such thatdxd .... bar u\beta sub 0 minus= v to 0 the\ ] power of epsilon bar L p open parenthesis Capital Omega closing parenthesis less or equal bar u sub 0 minus v-hatwide bar sub L p open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis plus eta divided by 4 and .... bar u sub epsilon minus u sub 0 bar L p open parenthesis Capital Omega\noindent closingf oparenthesis r a l l $ less\varphi or equal eta\ dividedin K( by 4 \Omega \times \Delta ( A ) ) ($ seeRemark \quad 4 . 2 ) . \quad Hence $ zfor{ E0 ni} epsilon= less u or{ equal0 } alphav { period0 }$ .... almost Hence .... everywhere bar u sub epsilon minus v to the power of epsilon bar L p open parenthesisin $ \Omega Capital Omega\times closing\ parenthesisDelta ( less orA equal ) eta . divided $ \quad by 2 plusThe bar corollary u sub 0 minus follows v-hatwide thereby bar sub $ L p. open\ square $ parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis period .... We deduce that\noindent We conclude the present subsection by showing that strong $ \Sigma − $ convergence gen − condition open parenthesis SSC closing parenthesis of Definition .. 3 period 1 1 .. is satisfied by open parenthesis u sub epsilon closing\noindent parenthesiseralizes sub epsilon usual in E strong and u sub convergence 0 comma thereby . \quad provingSpecifically , we have the proposition period square \noindent4 period ..Proposition The vague Capital 3 . Sigma 20 . hyphen\ h f i l convergence l Suppose of a Radon s equence measures $ ( u {\ varepsilon } ) {\ varepsilon \ inLet theE } basic$ \ notationh f i l l andis s hypotheses trongly be convergent as in the preceding in $ section L ˆ{ ..p open} parenthesis( \Omega see in ) $ particular the beginning of Section 3 closing parenthesis period .. On the other hand comma the space of all complex \noindentRadon measuresto some on a $lo ucally{ compact0 }\ spacein ZL will ˆ{ bep denoted} ( by\Omega M open parenthesis) . $ Z\ closingquad Then parenthesis $ u period{\ varepsilon .. Thus } comma\rightarrow u { 0 }$ \quad in $ L ˆ{ p } ( \Omega ) − $ strong $ \Sigma . $ M open parenthesis Z closing parenthesis i s nothing else than the topological dual of K open parenthesis Z closing parenthesis open\noindent parenthesisProof provided . \quad with theLet usual us begin by observing that the function $ u { 0 }\ in L ˆ{ p } ( inductive\Omega l imit) topology $ \quad closingmay parenthesis as well period be .. Also comma the notion of a sigma hyphen compact lo cally compact space viewed as a function in $ L ˆ{ p } ( \Omega ; A ) ($resp$. Lˆ{ p } ( \Omega \times \Delta ( A ) ) ) $ \quad depending on the sole v a r i a b l e $ x \ in \Omega . $ \quad Having made this point , let $ v \ in L ˆ{ p } ( \Omega ; A ) . $ \quad By applying Lemma 3 . 1 3 \quad with \quad $ \Phi = u { 0 } − v , $ \quad we \quad see that \quad i f $ \eta > 0 $ \quad i s \quad freely fixed , \quad then \quad some \quad $ \alpha > 0 $

\noindent exists such that \ h f i l l $ \ parallel u { 0 } − v ˆ{\ varepsilon }\ parallel L p ( \Omega ) \ leq \ parallel u { 0 } − \widehat{v}\ parallel { L p ( \Omega \times \Delta (A)) } + \ f r a c {\eta }{ 4 }$ and \ h f i l l $ \ parallel u {\ varepsilon } − u { 0 }\ parallel L p ( \Omega ) \ leq \ f r a c {\eta }{ 4 }$

\noindent f o r $ E \ ni \ varepsilon \ leq \alpha . $ \ h f i l l Hence \ h f i l l $ \ parallel u {\ varepsilon } − v ˆ{\ varepsilon }\ parallel L p ( \Omega ) \ leq \ f r a c {\eta }{ 2 } + \ parallel u { 0 } − \widehat{v}\ parallel { L p ( \Omega \times \Delta (A)) } . $ \ h f i l l We deduce that

\noindent condition ( SSC ) of Definition \quad 3 . 1 1 \quad is satisfied by $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ and $ u { 0 } , $ thereby proving the proposition $ . \ square $

\ centerline {4 . \quad The vague $ \Sigma − $ convergence of Radon measures }

Let the basic notation and hypotheses be as in the preceding section \quad ( see in particular the beginning of Section 3 ) . \quad On the other hand , the space of all complex Radon measures on a lo cally compact space $ Z $ will be denoted by $M ( Z ) . $ \quad Thus , $M ( Z ) $ i s nothing else than the topological dual of $K ( Z ) ( $ provided with the usual inductive l imit topology ) . \quad Also , the notion of a $ \sigma − $ compact lo cally compact space Σ− CONVERGENCE 1 27 i s worth recalling . By this i s meant any lo cally compact space which can be expressed as the union of a countable family of compact subspaces . Definition 4 . 1 . A s equence (µε)ε∈E of Radon measures on Ω is said to be vaguely Σ− convergent to some µ0 ∈ M(Ω × ∆(A)) if as E 3 ε → 0, Z ZZ ψ(x, Hε(x))dµε(x) → ψb(x, s)dµ0(x, s) Ω Ω×∆(A) for all ψ ∈ K(Ω; A). We express this by writing µε → µ0 in M(Ω)− vague Σ. Remark 4 . 2 . It is easy to verify that the transformation ψ → ψb = G ◦ ψ i s a topological i somorphism of K(Ω; A) onto K(Ω × ∆(A)) ≡ K(Ω; C(∆(A))), each of the two spaces being endowed with the appropriate inductive l imit topology . Consequently , for fixed ε ∈ E, it i s easily seen that to each µ ∈ M(Ω) there is attached a unique Tε(µ) ∈ M(Ω × ∆(A)) such that Z hTε(µ), ψbi = ψ(x, Hε(x))dµ(x) Ω for all ψ ∈ K(Ω; A), where the brackets denote the duality pair- ing between M(Ω×∆(A)) and K(Ω×∆(A)). This yields a transformation µ → Tε(µ) that maps l inearly M(Ω) into M(Ω×∆(A)). Thus , to say that a sequence (µε)ε∈E in M(Ω) i s vaguely Σ− convergent amounts to saying that as E 3 ε → 0, the sequence of Radon measures Tε(µε)(ε ∈ E) on Ω × ∆(A) i s convergent in the weak ∗ topology on M(Ω × ∆(A)). The usefulness of the following lemma will come to l ight in a short while . Lemma 4 . 3 . Let Z be a locally compact space , and let P ⊂ M(Z). The fol lowing two ass ertions are equivalent : ( i )P is bounded in the weak ∗ topology on M(Z), i . e ., supµ∈P | µ(ϕ) | < +∞

foreachϕ ∈ K(Z).

( ii )P is locally bounded in norm , i . e ., supµ∈P | µ | (K) < +∞ for each compact

setK ⊂ Z. Proof . According to [ 4 , p . 60 , Proposition 1 5 ] , assertion ( i ) i s equivalent to the

following :

( ii i ) For any compact s et H ⊂ Z, there exists a constant cH ≥ 0 such that

k ϕ k ∞forallϕ ∈ KH (Z) = {f ∈ K(Z): Suppf ⊂ H}. supµ∈P | µ(ϕ) |≤ cH

Thus , the problem reduces to proving the equivalence ( ii ) ⇔ ( ii i ) . The ( i i ) ⇒ ( i ii ) part being evident , we need only to concentrate on the proof of ( i ii ) ⇒ ( ii ) . So , assume ( ii i ) , and fix freely a compact s et K ⊂ Z. Let U be a relatively compact

open neighbourhood of K, and put H = U( closure of U). Then , in vue of ( i ii ) , we have | µ | (f) ≤ cH k f k ∞ for any µ ∈ P and for all f ∈ KH (Z) Capital Sigma hyphen CONVERGENCE .. 1 27 \ hspacei s ....∗{\ worthf i .... l l } recalling$ \Sigma period ....− $ By thisCONVERGENCE .... i s .... meant\quad .... any1 27 .... lo cally .... compact .... space .... which .... can be expressed as the union of a countable family of compact subspaces period with f ≥ 0, where c is a nonnegative constant . With this in mind , let µ ∈ P. \noindentDefinitioni 4 period s \ h f 1 i lperiod l worth A s equence\ hH f i l l openr e c parenthesisa l l i n g . mu\ h f epsilon i l l By closing t h i s parenthesis\ h f i l l i sub s epsilon\ h f i l l inmeant E of Radon\ h f imeasures l l any \ h f i l l l o c a l l y \ h f i l l compact \ h f i l l space \ h f i l l which \ h f i l l can be Considering that χU( the characteristic function of U) i s lower semicontinuous on Z, on Capital Omegawe have is said to be vaguely \noindentCapital Sigmaexpressed hyphen convergent as the union to some of mu a 0in countable M open parenthesis family Capital of compact Omega subspacestimes Capital . Delta open parenthesis A Definition 4 . 1 . A s equence $ ( \mu \ varepsilon ) {\ varepsilon \ in E }$ of Radon measures on closing parenthesis closing parenthesis if as E ni| epsilonµ | (U) right = sup arrow| 0µ comma| (f), $ \integralOmega sub$ Capital is said Omega to psi be open vaguely parenthesis x comma H sub epsilon open parenthesis x closing parenthesis closing parenthesis f ∈ K (Z), f ≤ χ d mu epsilon open parenthesis x closing parenthesis right arrow+ integralU integral sub Capital Omega times Capital Delta open parenthesis\noindent A closing$ \Sigma parenthesis− $ psi-hatwide convergent open parenthesis to some x $ comma\mu s closing0 \ in parenthesisM( d mu\ 0Omega open parenthesis\times x comma\Delta s closing( A parenthesis ) )$ ifas$E \ ni \ varepsilon \rightarrow 0 , $ for all psi in K open parenthesis Capital Omega semicolon A closing parenthesis period .. We express this by writing mu epsilon right\ [ \ arrowint mu{\ 0Omega in M open}\ parenthesispsi ( Capital x Omega , H closing{\ varepsilon parenthesis hyphen} ( vague x Capital ) Sigma ) d period\mu \ varepsilon (Remark x ) .. 4 period\rightarrow 2 period .. It\ ..int is .. easy\ int to verify{\Omega that .. the\ transformationtimes \Delta psi right(A) arrow psi-hatwide}\widehat = G circ{\ psipsi .. i} s ..( a x , s ) d \mu 0 ( x , s ) \ ] topological i somorphism of K open parenthesis Capital Omega semicolon A closing parenthesis onto K open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis equiv K open parenthesis Capital Omega semicolon\noindent C openf o r parenthesis a l l $ \ psi Capital\ Deltain openK( parenthesis\Omega A closing; parenthesis A ) closing . $ parenthesis\quad We closing express parenthesis this commaby writing each$ \mu \ varepsilon \rightarrow \mu 0 $ in $ M ( \Omega ) − $ vague $ \Sigma . $of the two spaces being endowed with the appropriate inductive l imit topology period Consequently comma for fixed epsilon in E comma it i s easily seen that to each mu in M open parenthesis Capital Omega closing\noindent parenthesisRemark there\quad is 4 . 2 . \quad I t \quad i s \quad easy to verify that \quad the transformation $ \attachedpsi \ arightarrow unique T sub epsilon\widehat open parenthesis{\ psi } mu= closing G parenthesis\ circ \ inpsi M open$ \ parenthesisquad i s Capital\quad Omegaa times Capital Deltatopological open parenthesis i somorphism A closing parenthesis of $ K closing ( parenthesis\Omega such; that A )$ onto $K ( \Omega \times \DeltaangbracketleftBig(A)) T sub epsilon\equiv open parenthesisK( mu closing\Omega parenthesis;C( comma psi-hatwide\Delta angbracketrightBig( A ) ) = integral ) , sub $ Capitaleach Omega psi open parenthesis x comma H sub epsilon open parenthesis x closing parenthesis closing parenthesis d mu open parenthesisof the two x closing spaces parenthesis being endowed with the appropriate inductive l imit topology . Consequentlyfor .. all .. psi in , K for open fixed parenthesis $ \ Capitalvarepsilon Omega semicolon\ in AE closing , $ parenthesis it i s comma easily .. where seen .. that the .. tobrackets each .. denote $ \mu ..\ in the ..M( duality .. pairing\Omega .. between) $ there i s M open parenthesis Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis .. and .. K open\noindent parenthesisattached Capital Omega a unique times $Capital T {\ Deltavarepsilon open parenthesis} ( A closing\mu parenthesis) \ in closingM( parenthesis\Omega period\ ..times This .. yields\Delta .. a ..( transformation A ) .. )$ mu right suchthat arrow T sub epsilon open parenthesis mu closing parenthesis that .. maps .. l inearly M open parenthesis Capital Omega closing parenthesis .. into .. M open parenthesis Capital Omega times\ [ \ langle Capital DeltaT open{\ varepsilon parenthesis A} closing( parenthesis\mu ), closing parenthesis\widehat{\ periodpsi }\ .. Thusrangle comma ..= to ..\ int say that{\ ..Omega a .. } sequence\ psi ( x , H {\ varepsilon } ( x ) ) d \mu ( x ) \ ] open parenthesis mu epsilon closing parenthesis sub epsilon in E in M open parenthesis Capital Omega closing parenthesis i s vaguely Capital Sigma hyphen convergent amounts to saying that as E ni epsilon right arrow 0 comma \noindentthe sequencef o r of\ Radonquad measuresa l l \quad T sub$ epsilon\ psi open\ in parenthesisK( mu epsilon\Omega closing; parenthesis A ) open , $ parenthesis\quad where epsilon in\quad E the \quad brackets \quad denote \quad the \quad d u a l i t y \quad p a i r i n g \quad between closing$ M parenthesis ( \Omega .. on Capital\times Omega times\Delta Capital( Delta A open ) parenthesis ) $ \quad A closingand parenthesis\quad $ .. Ki s convergent ( \Omega in \times \Deltathe weak( * topology A ) on M ) open . parenthesis $ \quad CapitalThis Omega\quad timesy i e l d Capital s \quad Deltaa open\quad parenthesistransformation A closing parenthesis\quad $ closing\mu parenthesis\rightarrow period T {\ varepsilon } ( \mu ) $ thatThe usefulness\quad maps of the\quad followingl inearly lemma will come$M to (l ight\ inOmega a short while) $ period\quad i n t o \quad $ M ( \Omega \times \DeltaLemma 4( period A 3 period ) ) .. Let . Z $ be a\quad locallyThus compact , space\quad commato \quad and letsay P subset that M\ openquad parenthesisa \quad Zsequence closing parenthesis period$ ( .. The\mu fol lowing\ varepsilon ) {\ varepsilon \ in E }$ in $ M ( \Omega ) $ i s vaguely $ \twoSigma ass ertions− $ are convergent equivalent : amounts to saying that as $ E \ ni \ varepsilon \rightarrow 0 , $open parenthesis i closing parenthesis P .. is bounded in the weak * .. topology on M open parenthesis Z closing parenthesis comma .. i period e period comma supremum sub mu in P bar mu open parenthesis phi closing parenthesis bar less plus infinity \noindentfor each phithe in K sequence open parenthesis of Radon Z closing measures parenthesis $ Tperiod{\ varepsilon } ( \mu \ varepsilon )( \ varepsilonopen parenthesis\ in ii closingE parenthesis ) $ \quad P is locallyon $ bounded\Omega in norm\times comma ..\ iDelta period e period( A comma ) $supremum\quad subi smu convergent in P in barthe mu weak bar open $ parenthesis∗ $ topology K closing on parenthesis $M ( less plus\Omega infinity for\times each compact\Delta ( A ) ) . $ s et K subset Z period \ centerlineProof period{The .... According usefulness to .... of open the square following bracket 4 lemma comma ....will p period come 60 to comma l ight .... in Proposition a short .... while 1 5 closing . } square bracket comma .... assertion .... open parenthesis i closing parenthesis .... i s .... equivalent .... to the \noindentfollowing :Lemma 4 . 3 . \quad Let $ Z $ be a locally compact space , and let $ P \subset Mopen ( parenthesis Z ) ii . i closing $ \quad parenthesisThe fol .. For lowing .. any compact .. s et .. H subset Z comma .. there exists .. a constant .. c sub Htwo greater ass equal ertions 0 .. such are that equivalent : Equation: supremum sub mu in P bar mu open parenthesis phi closing parenthesis bar less or equal c sub H .. bar phi bar infinity\ hspace for∗{\ allf phi i l l in}( K i sub $ H ) open P parenthesis $ \quad Z closingis bounded parenthesis in = the open weak brace $ f in∗ K$ open\quad parenthesistopology Z closing on parenthesis $M ( : ZSuppf ) subset , H $ closing\quad bracei . period e $ . , \sup {\mu \ in P }\mid \mu ( \varphi ) \mid < Thus+ comma\ infty the$ problem reduces to proving the equivalence .. open parenthesis ii closing parenthesis Leftrightarrow open parenthesis ii i closing parenthesis period .. The .. open parenthesis i i closing parenthesis double stroke right arrow open parenthesis i\ iibegin closing{ a parenthesis l i g n ∗} f opart r being each evident\varphi comma .. we\ in needK(Z). only to concentrate on the proof of open parenthesis i ii closing parenthesis double stroke right\end arrow{ a l i g openn ∗} parenthesis ii closing parenthesis period .. So comma assume open parenthesis ii i closing parenthesis comma and fix freely a compact s et K subset Z period .. Let U be a relatively compact\ hspace ∗{\ f i l l }( ii $ ) P$ is locally bounded in norm , \quad i . e $ . , \sup {\mu \ inhlineP }\mid \mu \mid (K) < + \ infty $ for each compact open neighbourhood of K comma and put H = U open parenthesis closure of U closing parenthesis period .. Then comma in vue\ begin of open{ a l parenthesisi g n ∗} i ii closing parenthesis comma swe have et .. K bar mu\subset bar open parenthesisZ. f closing parenthesis less or equal c sub H bar f bar infinity .. for any mu in P and for all\end f in{ a K l isub g n ∗} H open parenthesis Z closing parenthesis .. with f greater equal 0 comma where c sub H is a nonnegative constant period .. With this in mind comma let mu in P period .. Considering \noindentthat chi UProof open parenthesis . \ h f i l ltheAccording characteristic to function\ h f i l of l U[ closing 4 , \ parenthesish f i l l p . i s 60 lower , \ semicontinuoush f i l l Proposition on Z comma\ h wef i l havel 1 5 ] , \ h f i l l a s s e r t i o n \ h f i l l ( i ) \ h f i l l i s \ h f i l l e q u i v a l e n t \ h f i l l to the Line 1 bar mu bar open parenthesis U closing parenthesis = supremum bar mu bar open parenthesis f closing parenthesis comma Line\ begin 2 f{ ina K l i g sub n ∗} plus open parenthesis Z closing parenthesis comma f less or equal chi sub U f o l l o w i n g : \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }( i i i ) \quad For \quad any compact \quad s et \quad $ H \subset Z , $ \quad there exists \quad a constant \quad $ c { H }\geq 0 $ \quad such that

\ begin { a l i g n ∗} \ tag ∗{$ \sup {\mu \ in P }\mid \mu ( \varphi ) \mid \ leq c { H }$}\ parallel \varphi \ parallel \ infty f o r a l l \varphi \ in K { H } ( Z ) = \{ f \ in K ( Z ) : Suppf \subset H \} . \end{ a l i g n ∗}

\noindent Thus , the problem reduces to proving the equivalence \quad ( i i $ ) \Leftrightarrow ( $ i i i ) . \quad The \quad ( i i $ ) \Rightarrow ( $ i i i ) part being evident , \quad we need only to concentrate on the proof of ( i ii $ ) \Rightarrow ( $ i i ) . \quad So , assume ( ii i ) , and fix freely a compact s et $K \subset Z . $ \quad Let $ U $ be a relatively compact

\ [ \ r u l e {3em}{0.4 pt }\ ]

\noindent openneighbourhood of $K ,$ andput $H = U ($ closure of $U ) . $ \quad Then , in vue of ( i ii ) , we have \quad $ \mid \mu \mid ( f ) \ leq c { H }\ parallel f \ parallel \ infty $ \quad f o r any $ \mu \ in P$ and for all $ f \ in K { H } ( Z ) $ \quad with $ f \geq 0 , $ where $ c { H }$ is a nonnegative constant . \quad With this in mind , let $ \mu \ in P . $ \quad Considering that $ \ chi U ( $ the characteristic function of $ U ) $ i s lower semicontinuous on $ Z , $ we have

\ [ \ begin { a l i g n e d }\mid \mu \mid ( U ) = \sup \mid \mu \mid ( f ) , \\ f \ in K { + } ( Z ) , f \ leq \ chi { U }\end{ a l i g n e d }\ ] 1 28 .. G period NGUETSENG comma N period SVANSTEDT \noindentwhere ....1 K 28sub\ plusquad openG parenthesis . NGUETSENG Z closing , N parenthesis . SVANSTEDT .... i s .... the .... set .... of all .... phi in K open parenthesis Z closing parenthesis .... with .... phi greater equal 0 period .... But .... each .... f .... such that 1 28 G . NGUETSENG , N . SVANSTEDT \noindentf in K sub pluswhere open\ h parenthesis f i l l $ K Z closing{ + parenthesis} ( Z .... and) $ f less\ h or f i equall l i chi s \ Uh belongs f i l l the to K\ subh f Hi l lopens e tparenthesis\ h f i l l Zo fclosing a l l \ h f i l l where K (Z) i s the set of all ϕ ∈ K(Z) with ϕ ≥ 0. But each f such that parenthesis$ \varphi .... and\ in satisfies+ K .... ( bar Z f bar )sub $ infinity\ h f i less l l orwith equal\ h 1 f period i l l $ ....\ Thereforevarphi \geq 0 . $ \ h f i l l But \ h f i l l each \ h f i l l f ∈ K (Z) and f ≤ χU belongs to K (Z) and satisfies k f k ≤ 1. Therefore $ fsupremum $ \ h f isub l l musuch+ in P that bar mu bar open parenthesis U closingH parenthesis less or equal∞ c sub H and hence supremum sub mu sup | µ | (U) ≤ c and hence sup | µ | (K) ≤ c , which shows ( i i ). in P bar mu barµ∈P open parenthesisH K closing parenthesisµ∈P less or equalH c sub H comma which shows open parenthesis i i closing parenthesis\noindent periodOur$ f square goal\ in nowK is to{ establish+ } ( a ZΣ− compactness ) $ \ h f i l l resultand s $ imilar f to\ leq Theorem\ chi 3U$ . 9 . belongs to $K { H } (Our Z goal )Specifically now $ is\ h to f iestablish l l ,and assuming a satisfies Capital thatSigma\ hE hyphen fi l s l a compactnessfundamental$ \ parallel result s equence sf imilar\ parallel ,to we Theorem want to{\ .. 3 show periodinfty that 9}\ periodleq 1 . $ \ hSpecifically f i l l Thereforefrom comma any s .. equence assuming that(µε)ε E∈E i sin a fundamentalM(Ω) which s equence is bounded comma in we the want weak to show∗ topology that on from anyM s(Ω) equence, one open can parenthesis extract mu a subsequenceepsilon closing that parenthesis i s subvaguely epsilonΣ in− convergent E in M open . parenthesis Ac- Capital Omega closing\noindent parenthesistually$ \sup ,which this is{\ will boundedmu arise\ in asin the a consequence weakP }\ * topologymid of on\ amu more\ generalmid (U) result , viz . \ leq c { H }$ and hence $ \Msup open{\Theorem parenthesismu \ in Capital 4P . Omega}\ 4 . mid closingLet\ parenthesismu Z \midbe comma a(K) .. one metrizable can extract\ leq ..σ a−c subsequencecompact{ H } that,local $ .. which i s vaguely shows Capital ( i i $ ) . ly\ square compact $ space . Let (µn) be an ordinary s equence of Sigma hyphen convergent period .. Actually comma n∈N this willRadon arise as a consequence measures of a moreon generalZ. Assume result comma that viz period this OurTheorem goals now .. equence 4 period is to is 4 establishperiod bounded .. Let in a.. Z $ .. the\ beSigma .. weak a .. metrizable−∗ $topology compactness .. sigma on hyphenM result(Z compact). Then s .. imilar local one ly .. to compact can Theorem .. space\quad period3 . 9 . ..Specifically Let extract , a \quad assuming that $ E $ i s a fundamental s equence , we want to show that from anysubsequence s equence( $µk ( ) \mufrom \ varepsilon(µn) such) that{\ varepsilonµk → µ in\ inM(ZE)−}weak$ in∗ $ M ( \Omega open parenthesis mu n closingn n parenthesis∈N sub nn in∈N N .. be .. an .. ordinaryn .. s equence .. of Radon .. measures .. on .. Z period) $ which.. Assumewhen is .. bounded that .. this in the weak $ ∗ $ topology on s equence is .... bounded in .... the weak * .... topology .... on M open parenthesis Z closing parenthesis period .... Then .... one ....\noindent can .... extract$ M a ( \Omega ) , $ \quad one can extract \quad a subsequence that \quad i s vaguely n → +∞. $ \subsequenceSigma − ....$ open convergent parenthesis mu. \ kquad sub nActually closing parenthesis , sub n in N from .... open parenthesis mu n closing parenthesis this will arise as a consequence of a more general result , viz . sub n in N ....Proof such . .... thatWe mu achieve k sub n rightthis arrowin two mu steps .... in . MStep open parenthesis 1 . Z closing Let U parenthesisbe a relatively hyphen weak * .... when n right arrow plus infinity period compact open set in Z. The aim here is to verify that any subsequence (µtn)n∈ \noindentProof periodTheorem .. We achieve\quad this4 in . two 4 . steps\quad periodLet \quad $ Z $ \quad be \quad a \quad mN e t r i z a b l e \quad extracted from (µn)n∈ contains a subsequence $ \Stepsigma .. 1 period− $ .. compactLet U be a\ relativelyquadN l o compact c a l l y open\quad setcompact in Z period\ ..quad The aimspace here . is\ toquad verifyLet $that ( .. any\mu .. subsequencen ) { ..n open\ parenthesisin N }$ mu\ tquad sub nbe closing\quad parenthesisan \quad sub nordinary in N .. extracted\quad froms equence .. open parenthesis\quad o f Radon \quad measures \quad on \quad mu$ Z n closing . $ parenthesis\quad Assume sub n in\ Nquad .. containsthat ..\quad a subsequence0t h i s (µtn)n∈Nsuchthat parenleftbig mu t sub n to the power of prime parenrightbig sub n in N such that \noindent s equence i s \ h f i l l bounded in 0 \ h f i l0 l the weak $ ∗ $ \ h f i l l topology \ h f i l l on $ M mu t sub n to the power of prime bar sub U rightµt arrown |U → nuν toin theM power(U)− ofweak prime in∗ Mas openn → parenthesis+∞. (4. U1) closing parenthesis hyphen( Z weak ) ..To * . .. this $ as n\ end righth f i ,l larrow letThenK plus=\ Uh infinity fand i l l putone periodB\ h0 openfor f i l l theparenthesiscan closed\ h f 4 uniti l period l e ball x t 1 r aclosing in c tM a ( parenthesisK)( strong dual To this endof C comma(K)). let KProvided = overbar Uwith and the put relativeB to the power weak of∗ primetopology for the on closedM(K unit),B ball0 i s in a M metriz open parenthesis - K closing parenthesis\noindentable opensubsequence compactparenthesis space strong\ h f i ( ldual l see$ , (e . g\mu . ,k [ 1{ n 7} , p) .{ 426n ] )\ .in HavingN }$ frommade\ thish f i l l $ ( \mu n ) { n \ in N }$ \ h f i l l such \ h f i l l that $ \mu k { n }\rightarrow \mu $ \ h f i l l in of C openpoint parenthesis , let K closing(µtn)n∈ parenthesisbe any closingarbitrary parenthesis subsequence period .... extracted Provided fromwith the relative(µn)n∈ weak. For * topology on M open $ M ( Z ) − $ weakN $ ∗ $ \ h f i l l when N parenthesiseach K closing integer parenthesisn ≥ comma0, put B toν then power= µt ofn | primeK . iThen s a metrizνn hyphen∈ M(K) and further able compactsupn k spaceνn k open= parenthesis supn | νn | ( seeK) comma = sup en period| µtn | g(K period) < +∞ comma( use .. openLemma square 4bracket . 3 ) 1 , where.. 7 comma p period 426 \ begin { a l i g n ∗} closing squaren runs bracket through closingN parenthesis. Thus , period we may .. Having made this point comma let .. open parenthesis mu t sub n closing n \rightarrow + \ infty . 0 parenthesisassume sub n in withoutN lo ss of generality that the sequence (νn)n∈ is contained in B . Hence \end{ a l i g n ∗} N be any arbitrarywe can subsequenceextract a subsequence extracted from( ..νrn open)n∈N parenthesisfrom (νn) mun∈N nsuch closing that parenthesis as n → sub+∞ n, in νrn N→ periodν in .. For each integer 0 n greater equalM(K 0) comma− weak ∗, whence νrn |U → ν |U = ν in M(U)− weak ∗. Therefore , ( 4 . 1 \noindent Proof . \quad We achieve0 this in two steps . put .. nu) sub fo llowsn = mu by t sub lett n ing bar subtn K= periodtrn (n ..∈ ThenN) and .. nu noting sub n in that M openνn parenthesis|U = µtn |U K. closingStep parenthesis 2 . .. and .. further Step \quad 1 . \quad Let $U$ be a relatively compact open set in $ Z . $ \quad The aim here is to verify .. supremumLet sub n bar(Ui) nui∈ subbe n a bar sequence = supremum of open sub n sets barin nu subZ such n bar that open parenthesisUi ⊂ KUi+1 closing, Ui compact parenthesis = that \quad any \quadN subsequence \quad $ ( \mu t { n } ) { n \ in N }$ \quad extracted from \quad supremumand sub∪ ni∈ barUi mu= Z. t subBy n bar suitably open parenthesis applying Kthe closing result parenthesis of Step 1 less we plus are infinity readily open led parenthesis to two use Lemma .. 4 $ ( \mu nN ) { n \ in N }$ \quad c o n t a i n s \quad a subsequence period 3 closingsequences parenthesis(ν commai)i∈N where(νi ∈ M n runs(Ui)) throughand N period(µtn (i))( .. Thusi, n) ∈ commaN × N wein mayM(Z) framed as assume withoutfollows lo:( ss ofµ generalitytn (0))n ∈ N thatis the a subsequence sequence .. open extracted parenthesis from nu sub(µn n)n closing∈ in suchparenthesis a way sub that n in N is contained in \ begin { a l i g n ∗} N B to the powerµtn (0) of prime|U0 → ν period0 in M(U0)− weak ∗ as n → +∞; for i ≥ 1, (µtn (i))n ∈ N i s a subsequence ( \mu t ˆ{\prime } { n } ) { n \ in N } such that Hence weextracted can extract from a subsequence(µtn (i − open1))n parenthesis∈ N in such nu sub a way r sub that n closingµtn (i parenthesis) |Ui → νi in subM n( inUi N)− fromweak open parenthesis nu sub\end n{ closinga l i g∗ n as parenthesis∗} n → + sub∞. n inHence N such , that by the as n usual right arrow diagonal plus infinity process comma , it i s immediate that the nu sub rsequence sub n right(µk arrow) nuwith in M openk = parenthesisn(n) i s a subsequence K closing parenthesis extracted hyphen from weak(µn *) commaso that whence for nu sub r sub n bar \ hspace ∗{\ f i l l } $ \mun n∈N t ˆ{\primen t } { n }\mid { U }\rightarrow n∈N\nu ˆ{\prime }$ in $ M sub U righteach arrowi nu∈ bar, µk sub| U→ =ν nuin to theM(U power)− weak of prime∗ as in Mn open→ +∞ parenthesis. Furthermore U closing , parenthesis it is clear hyphen that weak * period .. (U) − $N weakn U\iquadi $ ∗ $ i \quad as $ n \rightarrow + \ infty . ( 4 . 1 Therefore commaν = ν | (i ∈ ), hence a ( unique ) Radon measure µ on Z such that µ | = ν for ) $ i i+1 Ui N Ui i open parenthesisany i ∈ 4N period( this 1i closings a classical parenthesis property fo llows ) by . lett S ing ince t sub each n toϕ the∈ K power(Z) ofl prime ies in =K t( subUi) r sub n open parenthesis n in N closingfor parenthesis some suitable and noting index thati, we nu subdeduce n bar that sub Uµk = mu→ µ t subin M n bar(Z) sub− weak U period∗ as n → +∞, \ hspace ∗{\ f i l l }To this end , let $K = \ overlinen {\}{ U }$ and put $ B ˆ{\prime }$ for the closed unit ball in Step .. 2thereby period .. proving Let .. open the theorem parenthesis. U sub i closing parenthesis sub i in N be a sequence of open sets in Z such that overbar$M U ( sub K i subset ) U ($ sub i plus strongdual 1 comma overbar U sub i compact and cup sub i in N U sub i = Z period .. By suitably applying the result of Step 1 we are readily \noindentled to twoo sequences f $ C .. open ( parenthesis K ) nu) sub . i $ closing\ h f parenthesis i l l Provided sub i in with N open the parenthesis relative nu weaksub i in M$ ∗ open$ parenthesis topology on U$ M sub i (closing K parenthesis ) , closing B ˆ{\ parenthesisprime } ..$ and i s.. a open metriz parenthesis− mu sub t sub n open parenthesis i closing parenthesis closing parenthesis open parenthesis i comma n closing parenthesis in N times N in M open parenthesis Z closing parenthesis .. framed\noindent able compact space ( see , e . g . , \quad [ 1 \quad 7 , p . 426 ] ) . \quad Having made this point , let \quad $ (as follows\mu : opent { parenthesisn } ) mu{ subn t sub\ in n openN } parenthesis$ 0 closing parenthesis closing parenthesis n in N is a subsequence extractedbe any from arbitrary open parenthesis subsequence mu n closing extracted parenthesis from sub\ nquad in N in$ such ( a way\mu thatn ) { n \ in N } . $ \quad For each integer $ nmu sub\geq t sub n0 open , parenthesis $ 0 closing parenthesis bar sub U sub 0 right arrow nu sub 0 in M open parenthesis U sub 0 closingput \ parenthesisquad $ \ hyphennu { weakn } * as= n right\mu arrowt plus{ n infinity}\mid semicolon{ K for} i greater. $ \ equalquad 1Then comma\ openquad parenthesis$ \nu mu{ n sub} t sub\ in n openM parenthesis ( K i )closing $ \ parenthesisquad and closing\quad parenthesisf u r t h e r n\quad in N i s a$ subsequence\sup { n }\ parallel \nu { n }\ parallel = extracted\sup from{ n ..}\ openmid parenthesis\nu mu{ n sub}\ t submid n open( parenthesis K ) i minus = $ 1 closing parenthesis closing parenthesis n in N in such$ a\sup way that{ n mu}\ submid t sub n open\mu parenthesist { n i}\ closingmid parenthesis(K) bar sub U< sub i+ right\ arrowinfty nu sub( i$ in M use open Lemma parenthesis\quad 4 . 3 ) , where U$n$ sub i closing runs parenthesis through hyphen $N weak . $ * as\quad Thus , we may n right arrow plus infinity period .. Hence comma by the usual diagonal process comma it i s immediate that the sequence \noindentopen parenthesisassume mu without k sub n closing lo ss parenthesis of generality sub n in that N with the k sub sequence n = n t\ toquad the power$ ( of open\nu parenthesis{ n } ) n closing{ n parenthesis\ in N } i$ s a issubsequence contained extracted in from$Bˆ open{\prime parenthesis} mu. $ n closing parenthesis sub n in N so that for each Hencei in N we comma can mu extract k sub n bar a subsequence sub U sub i right $ arrow ( nu\nu sub i{ inr M{ openn parenthesis}} ) { Un sub\ iin closingN parenthesis}$ from hyphen $ ( weak\nu { n } *) as{ n rightn arrow\ in plusN } infinity$ such period that .. Furthermore as $n comma\rightarrow it is clear that+ \ infty , $ $nu\ subnu i ={ nur sub{ n i plus}}\ 1 barrightarrow sub U sub i open\nu parenthesis$ in i $ in M N closing ( K parenthesis ) − comma$ weak hence $a open∗ parenthesis, $ whence unique $ \nu { r { n }} closing\mid parenthesis{ U }\rightarrow Radon measure mu\nu on Z such\mid that{ muU bar} sub= U\ subnu iˆ ={\ nuprime sub i }$ in $ M ( U ) − $ weak $ ∗for any. $i in N\quad open parenthesisTherefore this , i s a classical property closing parenthesis period .. S ince each phi in K open parenthesis Z closing( 4 . parenthesis 1 ) fo ..llows l ies in by K open lett parenthesis ing $ t U ˆ sub{\ iprime closing parenthesis} { n } = t { r { n }} ( n \ in N ) $ andfor noting some suitable that index $ \ inu comma{ n we}\ deducemid that{ muU k} sub= n right\mu arrowt mu{ inn M}\ openmid parenthesis{ U } Z closing. $ parenthesis hyphen weakStep * as\quad n right2 arrow . \quad plus infinityLet \quad comma $ ( U { i } ) { i \ in N }$ be a sequence of open sets in $Z$thereby such proving that the theorem $\ overline period{\}{ squareU } { i }\subset U { i + 1 } , \ overline {\}{ U } { i }$ compact and $ \cup { i \ in N } U { i } = Z . $ \quad By suitably applying the result of Step 1 we are readily led to two sequences \quad $ ( \nu { i } ) { i \ in N } ( \nu { i }\ in M( U { i } ) ) $ \quad and \quad $ ( \mu { t { n }} ( i ) ) ( i , n ) \ in N \times N $ in $ M ( Z ) $ \quad framed as follows $ : ( \mu { t { n }} ( 0 ) ) n \ in N $ is a subsequence extracted from $ ( \mu n ) { n \ in N }$ in such a way that $ \mu { t { n }} ( 0 ) \mid { U { 0 }}\rightarrow \nu { 0 }$ in $ M ( U { 0 } ) − $ weak $ ∗ $ as $ n \rightarrow + \ infty ; $ f o r $ i \geq 1 , ( \mu { t { n }} ( i ) ) n \ in N $ i s a subsequence extracted from \quad $ ( \mu { t { n }} ( i − 1 ) ) n \ in N $ in such a way that $ \mu { t { n }} ( i ) \mid { U { i }}\rightarrow \nu { i }$ in $ M ( U { i } ) − $ weak $ ∗ $ as $ n \rightarrow + \ infty . $ \quad Hence , by the usual diagonal process , it i s immediate that the sequence $ ( \mu k { n } ) { n \ in N }$ with $ k { n } = n { t }ˆ{ ( n ) }$ i s a subsequence extracted from $ ( \mu n ) { n \ in N }$ so that for each $ i \ in N, \mu k { n }\mid { U { i }}\rightarrow \nu { i }$ in $ M ( U { i } ) − $ weak $ ∗ $ as $ n \rightarrow + \ infty . $ \quad Furthermore , it is clear that $ \nu { i } = \nu { i + 1 }\mid { U { i }} ( i \ in N ) , $ hence a ( unique ) Radon measure $ \mu $ on $Z$ such that $ \mu \mid { U { i }} = \nu { i }$ f o r any $ i \ in N ( $ this i s a classical property ) . \quad S i n c e each $ \varphi \ in K ( Z ) $ \quad l i e s in $ K ( U { i } ) $

\noindent for some suitable index $ i , $ we deduce that $ \mu k { n }\rightarrow \mu $ in $ M ( Z ) − $ weak $ ∗ $ as $ n \rightarrow + \ infty , $

\noindent thereby proving the theorem $ . \ square $ Capital Sigma hyphen CONVERGENCE .. 1 29 \ hspaceThis leads∗{\ f to i l the l } Capital$ \Sigma Sigma hyphen− $ CONVERGENCE compactness result\quad for measures1 29 comma as claimed period Corollary 4 period 5 period .. We assume that E is a fundamental s equence period .. Then comma from any Σ− CONVERGENCE 1 29 \ centerlines equence ..{ openThis parenthesis leads to mu the epsilon $ \ closingSigma parenthesis− $ compactness sub epsilon in E result .. in M open for parenthesis measures Capital , as claimed Omega closing . } This leads to the Σ− compactness result for measures , as claimed . parenthesis .. which is .. bounded in .. the weak * .. t opology .. on M open parenthesis Capital Omega closing parenthesis comma Corollary 4 . 5 . We assume that E is a fundamental s equence . Then , \noindentone can extractCorollary a subsequence 4 . 5 that . \ isquad vaguelyWe Capital assume Sigma that hyphen $ E convergent $ is a period fundamental s equence . \quad Then , from any from any s equence (µε) in M(Ω) which is bounded in the weak ∗ sProof equence period\quad .. Let us$ observe ( \mu that Capital\ varepsilonε∈E Omega times) Capital{\ varepsilon Delta open parenthesis\ in AE closing}$ \ parenthesisquad in is $ a M metrizable ( \Omega t opology on M(Ω), sigma) $ hyphen\quad compactwhich loi s cally\quad compactbounded in \quad the weak $ ∗ $ \quad t opology \quad on $ M ( one can extract a subsequence that is vaguely Σ− convergent . \Omegaspace period) .. , Hence $ comma considering Remark .. 4 period 2 .. and Theorem .. 4 period 4 comma the corollary i f proved i f Proof . Let us observe that Ω × ∆(A) is a metrizable σ− compact lo cally compact we can showspace that . the Hence sequence , considering open parenthesis Remark T sub epsilon 4 . 2open and parenthesis Theorem mu epsilon 4 . closing 4 , the parenthesis corollary closing parenthesis sub\noindent epsilon inone E is bounded can extract in the weak a subsequence * topology on that is vaguely $ \Sigma − $ convergent . i f proved i f we can show that the sequence (Tε(µε))ε∈E is bounded in the weak ∗ M opentopology parenthesis on Capital Omega times Capital Delta open parenthesis A closing parenthesis closing parenthesis period .... This\noindent i s straightforwardProof . period\quad ....Let If psi us in observe K open parenthesis that $ Capital\Omega Omega\times semicolon\ ADelta closing parenthesis( A comma )$ is and a i f metrizable K i M(Ω × ∆(A)). This i s straightforward . If ψ ∈ K(Ω; A), and i f K i s a compact s$ a\ compactsigma − $ compact lo cally compact spaceset .... . set in\quad Capital inHenceΩ Omegacontaining , containing considering the the support support Remark .... of\quadψ, of psithen4 comma . 2 |h\ ....quadTε(µε then),andψb ....i| Theorem vextendsingle-vextendsingle-vextendsingle ≤ c supx\∈quadΩ kψb(x4)k∞ . 4 , the corollary i f proved i f angbracketleftBigwe can showfor all Tthat subε ∈ epsilonE, thewhere sequence open parenthesisc = sup $r∈ (E mu| µr Tepsilon| ({\K) closingvarepsiloni s parenthesisfinite , according} comma( \ hatwide-psimu to Lemma\ varepsilon angbracketrightBig 4 . 3 . )) vextendsingle-{\ varepsilon vextendsingle-vextendsingle\ in E }The$ is corollary bounded less is provedin or theequal. weak c supremum $ ∗ $ sub topology x in Capital on Omega vextenddouble-vextenddouble-vextenddouble psi- We will end with a few remarks . hatwide open parenthesis x closing parenthesis vextenddouble-vextenddouble-vextenddoublep infinity \noindentfor all epsilon( 1$ )M Suppose in E comma( a\Omega sequence where c =\(times supremumuε)ε∈E in \ subLDelta(Ω)(1 r in E≤ barp( < mu∞ A) ris bar such ) open ) that parenthesis . , as $ E\ h3 K fε i closing→ l l 0This, parenthesis i s straightforward .. i s finite . \ h f i l l I f comma$ \ psi according\ in toK( Lemma .. 4\ZOmega period 3 period; A .. The )ZZ ,$ andif $K$ isacompact corollary is proved period squareuε(x)ψ(x, Hε(x))dx → u0(x, s)ψb(x, s)dxdβ(s) \noindentWe will ends e with t \ h a f few i l l remarksinΩ $ period\Omega $ containingΩ×∆( theA) support \ h f i l l o f $ \ psi , $ \ h f i l l then \ h f i l l $ \arrowvert \ langle T {\ varepsilon } p ( \mu \ varepsilon ), \widehat{\ psi }\rangle open parenthesisfor all ψ 1 closing∈ K parenthesis(Ω; A), where Supposeu0 a sequence∈ L (Ω open× ∆( parenthesisA)). It u is sub clear epsilon that closing the sequence parenthesis sub epsilon in E \arrowvert \ leq c \sup { x \ in \Omegap }\Arrowvert \widehat{\ psi } ( x ) \Arrowvert in L to the( poweruε)ε∈E ofis p not open weakly parenthesisΣ− Capitalconvergent Omega in closingL (Ω) parenthesis. However open , parenthesis each function 1 less oruε equalbeing p less infinity closing parenthesis\ infty $ viewed is such that as acomma Radon as E measure ni epsilon on rightΩ, arrowthe above 0 comma sequence i s vaguely Σ− convergent : integralMore sub Capital precisely Omega u , sub epsilonwehave open parenthesisuεdx x→ closingu0(dx parenthesis⊗ dβ) in psi openM(Ω) parenthesis− vague x commaΣ. H sub epsilon open\noindent parenthesisWef o r x deduce closinga l l $ parenthesis that\ varepsilon the closing vague parenthesisΣ\−inconvergenceE dx right ,$ is arrow where a natural integral $c generalization integral = sub\sup Capital of{ weak Omegar \Σin− timesE Capital}\mid Delta open\mu parenthesisr convergence\mid A closing( parenthesis K ) $ u\quad sub 0 openi s parenthesis finite , x according comma s closing to Lemma parenthesis\quad psi-hatwide4 . 3 . open\quad parenthesisThe x commacorollary s closing is parenthesis proved dxd $ .beta\ opensquare parenthesis $ s closing parenthesis for all psi in K open parenthesis Capital Omega semicolon A closing parenthesis comma .. where u sub 0 in L to the power of p p\ centerline open parenthesis{We Capital will endOmega with times a Capital few remarks Delta openin .L parenthesis}(Ω). A closing parenthesis closing parenthesis period .. It .. is clear that the sequence \noindent ( 1 ) Suppose a sequence $ ( u {\ varepsilon } ) {\ varepsilon \ in E }$ in open parenthesis( 2 ) uSuppose sub epsilon a sequence closing parenthesis(µε)ε∈E subis vaguely epsilon inΣ E− isconvergent not weakly Capital in M(Ω) Sigmato some hyphenµ convergent0 ∈ in L to the $ L ˆ{ p } ( \Omega ) ( 1 \ leq p < \ infty ) $ is such that , as $E \ ni power of p openM(Ω parenthesis× ∆(A)). CapitalThen , Omega as E closing3 ε → 0 parenthesis, we have periodµε → µe ..0 Howeverin M(Ω) comma− weak .. each∗, where function u sub epsilon being \ varepsilonviewed as a Radon\rightarrow measure on Capital0 Omega , $ comma the above sequence i s vaguely Capital Sigma hyphen convergent : More .. precisely comma .. we .. have ..ZZ u sub epsilon dx right arrow u sub 0 open parenthesis dx oslash d beta closing \ [ \ int {\Omega } u {\ varepsilon } ( x ) \ psi ( x , H {\ varepsilon } ( parenthesis .. in M open parenthesis Capitalµe0(ϕ) Omega = closingϕ parenthesis(x)dµ0(x, s) hyphen, ϕ ∈ K(Ω) vague. .. Capital Sigma period .. We .. deduce xthat ) the ) vague dx Capital\rightarrow Sigma hyphen convergence\ int Ω\×int is∆( aA) natural{\Omega generalization\times of weak\Delta Capital Sigma(A) hyphen convergence} u { 0 } ( x , s ) \widehat{\ psi } ( x , s ) dxd \beta ( s ) \ ] in L to the power of p open parenthesis5 . Application Capital Omega of closingΣ− parenthesisconvergence period open parenthesis5 . 1 . 2 closingPreliminaries parenthesis .. . SupposeIn the a sequence present open section parenthesis we are mu concerned epsilon closing with parenthesis showing sub epsilon in E is vaguely Capital Sigma hyphen convergent in M open parenthesis Capital Omega closing parenthesis to some mu 0 in \noindenthowf o rΣ a− l lconvergence $ \ psi arises\ in inK( the homogenization\Omega ; of A partial ) differential , $ \quad equationswhere $. u { 0 }\ in M openTo parenthesis i llustrate Capital this Omega , we find times it more Capital convenient Delta open to parenthesis fo cus attention A closing on parenthesis the rather closing s imple parenthesis period .. LThen ˆ{ commap } ( as E ni\Omega epsilon right\times arrow 0 comma\Delta we have( mu A epsilon ) right ) arrow . $ mu-tildewide\quad I t \ 0quad in M openis clear parenthesis that Capital the sequence $ ( ucase{\ varepsilon of an ellipt ic} l) inear{\ varepsilon differential operator\ in E } of$ order is not two weakly , in divergence $ \Sigma − $ convergent in Omega closingform parenthesis . Specifically hyphen , weak let * comma where $ Lmu-tildewide ˆ{ p } ( 0 open\Omega parenthesis) phi closing. $ \ parenthesisquad However = integral , \quad integraleach sub Capital function Omega $ times u {\ Capitalvarepsilon Delta open} paren-$ being thesis A closing parenthesis phi open parenthesis x closing parenthesis d mu 0 open parenthesis x comma s closing parenthesis comma\noindent phi inviewed K open parenthesis as a Radon Capitali,j=1 measure Omega on closing $ \ parenthesisOmega period, $ the above sequence i s vaguely $ \Sigma − $ convergent : X ∂ ε ∂uε 1 1,2 5 period .. Application of Capital− Sigma hyphen(a ij convergence) = finΩ, uε ∈ H0 (Ω) = W0 (Ω), (5.1) ∂xi ∂xj More5 period\quad 1 periodp r e c .. i sPreliminaries e l y , \quadN periodwe ..\quad In thehave present\quad section we$ uare concerned{\ varepsilon with showing} dx how \rightarrow u { 0 } (Capital dx Sigma\otimes hyphend convergence\beta arises) in$ the\quad homogenizationin $ M of partial ( \Omega differential) equations− $ period vague .. To\quad $ \Sigma where ε > 0, Ω i s a fixed bounded open set in N , f ∈ H−1(Ω) = W −1,2(Ω), . $i llustrate\quad thisWe comma\quad wededuce find it more convenient to fo cus attentionRx on the rather s imple aεij(x) = a ( x )(x ∈ Ω) with a ∈ L∞( N ), a = a and the classical ellipt ic - ity thatcase.. the of an vague ellipt ic $ij l inear\εSigma .. differential− $ operatorconvergenceij .. ofR ordery isji two a natural commaij, .. generalizationin divergence form period of weak $ \Sigma − $ convergenceSpecificallycondition comma let: there is a constant α > 0 such that Equation: open parenthesis 5 period 1 closing parenthesisN .. minus sum from i comma j = 1 to N partialdiff divided by partialdiff \ begin { a l i g n ∗} x sub i parenleftbigg a to the power of epsilonX ij partialdiff u sub epsilon2 divided byN partialdiff x sub j parenrightbigg = f in Capital Omegain comma L ˆ{ p u sub} epsilon( \Omega in H sub 0).Re to theaij power(y)ξj of 1 openξi ≥ α parenthesis| ξ | (ξ Capital∈ C ) Omega closing parenthesis = W sub 0 to the power\end{ ofa l 1 i g comma n ∗} 2 open parenthesis Capital Omega closing parenthesis comma i , j = 1 for almost all y ∈ RN . For each real number ε > 0, (5.1) uniquely determines where epsilon greater 0 comma Capital Omega i s a fixed bounded open set in R sub x to the power of N comma f in H to the \noindentuε,( 2 ) \quad Suppose a sequence $ ( \mu \ varepsilon ) {\ varepsilon \ in E }$ power of minus 1 open parenthesis Capital Omega closing parenthesis = W to the1 power of minus 1 comma 2 open parenthesis so that we have in hand a generalized sequence (uε)ε>0 in H (Ω). Capitali s vaguely Omega closing $ \Sigma parenthesis− $ comma convergent in $M ( \Omega 0 ) $ to some $ \mu 0 \ in $ $a Mto the ( power\Omega of epsilon ij\times open parenthesis\Delta x closing( parenthesis A ) = ) a sub . ij $ parenleftbig\quad Then x divided , as by epsilon $ E parenrightbig\ ni \ varepsilon open parenthesis\rightarrow x in Capital0 Omega , $ we closing have parenthesis $ \mu with\ varepsilon a sub ij in L to the\rightarrow power of infinity\ parenleftbigwidetilde R{\ submu} y to0 the $ power in of $ M N( to the\Omega power of) parenrightbig− $ weak comma $ a∗ sub ji, = $ overbar where a sub ij comma and the classical ellipt ic hyphen ity condition : .. there is a constant alpha greater 0 such that \ beginN { a l i g n ∗} \ widetildeRe sum a sub{\mu ij open} parenthesis0 ( \ yvarphi closing parenthesis) = xi sub\ int j to the\ int power{\ ofOmega hline xi sub\times i greater equal\Delta alpha bar(A) xi bar to } the\varphi power of 2( parenleftbig x ) xi din C to\mu the power0 of ( N parenrightbig x , s ) , \varphi \ in K( \Omega ) . i comma j = 1 \endfor{ almosta l i g n ∗} all y in R to the power of N period .. For each real number epsilon greater 0 comma open parenthesis 5 period 1 closing parenthesis uniquely determines u sub epsilon comma \ centerlineso that we have{5 in. \ handquad a generalizedApplication sequence of open $ \ parenthesisSigma − u sub$ epsilonconvergence closing parenthesis} sub epsilon greater 0 in H sub 0 to the power of 1 open parenthesis Capital Omega closing parenthesis period \noindent 5 . 1 . \quad Preliminaries . \quad In the present section we are concerned with showing how $ \Sigma − $ convergence arises in the homogenization of partial differential equations . \quad To i llustrate this , we find it more convenient to fo cus attention on the rather s imple case \quad of an ellipt ic l inear \quad differential operator \quad of order two , \quad in divergence form . Specifically , let

\ begin { a l i g n ∗} − \sum ˆ{ i , j = 1 } { N }\ f r a c {\ partial }{\ partial x { i }} ( a ˆ{\ varepsilon }{ i j }\ f r a c {\ partial u {\ varepsilon }}{\ partial x { j }} ) = f in \Omega , u {\ varepsilon }\ in H ˆ{ 1 } { 0 } ( \Omega ) = W ˆ{ 1 , 2 } { 0 } ( \Omega ), \ tag ∗{$ ( 5 . 1 ) $} \end{ a l i g n ∗}

\noindent where $ \ varepsilon > 0 , \Omega $ i s a fixed bounded open set in $Rˆ{ N } { x } , f \ in H ˆ{ − 1 } ( \Omega ) = W ˆ{ − 1 , 2 } ( \Omega ) , $ $ a ˆ{\ varepsilon }{ i j } ( x ) = a { i j } ( \ f r a c { x }{\ varepsilon } ) ( x \ in \Omega ) $ with $ a { i j }\ in L ˆ{\ infty } ( R ˆ{ N } { y }ˆ{ ) } , a { j i } = \ overline {\}{ a } { i j , }$ and the classical ellipt ic − ity condition : \quad there is a constant $ \alpha > 0 $ such that

\ centerline {N }

\ [ Re \sum a { i j } ( y ) \ xi { j }ˆ{\ r u l e {3em}{0.4 pt }}\ xi { i }\geq \alpha \mid \ xi \mid ˆ{ 2 } ( \ xi \ in C ˆ{ N } ) \ ]

\noindent i , j $ = 1 $ for almost all $ y \ in R ˆ{ N } . $ \quad For each real number $ \ varepsilon > 0 , ( 5 . 1 )$ uniquelydetermines $u {\ varepsilon } , $

\noindent so that we have in hand a generalized sequence $ ( u {\ varepsilon } ) {\ varepsilon > 0 }$ in $ H ˆ{ 1 } { 0 } ( \Omega ) . $ 1 30 .. G period NGUETSENG comma N period SVANSTEDT \noindentThe purpose1 30of homogenization\quad G . NGUETSENG in the present , case N . is SVANSTEDT to investigate the l imit behaviour comma as epsilon right arrow 0 comma of u sub epsilon provided the coefficients a sub ij satisfy a suitable hypothesis The purpose1 30 of G homogenization . NGUETSENG , N . in SVANSTEDT the present case is to investigate the l imit with .. respectThe .. purpose to .. the of .. homogenization so hyphen called .. in lo the cal present.. variable case .. y is = to open investigate parenthesis the y 1 l commaimit be- period period period commabehaviour y N closing , as parenthesis $ \ varepsilon period .. It ..\ isrightarrow .. common .. in 0 , $ o f $ u {\ varepsilon }$ provided the coefficients haviour , as ε → 0, of u provided the coefficients a satisfy a suitable hypothesis with $ ahomogenization{ i j }$ satisfyto .. require a the suitable ..ε a sub ij hypothesis quoteright s .. to .. satisfyij the periodicity hypothesis comma .. which respect to the so - called lo cal variable y = (y1, ..., yN). It is withmeans\quad that ther e s functions p e c t \quad a subto ij open\quad parenthesisthe \quad 1 lessso or equal− c a il l comma e d \quad j lessl or o equal c a l \ Nquad closingv a parenthesis r i a b l e \quad are periodic$ y =(y1,...,yN).$common in homogenization to require the aij\quad’ s toI t \quad satisfyi s the\quadperiodicitycommon \quad in comma sayhypothesis with period 1, in each which homogenizationcoordinate comma to ....\ iquad periodr e e period q u i r e comma the \ ....quad for ....$ every a { ....i j k} in$ Z to ’ the s \ powerquad ofto N\ commaquad ....satisfy one .... the has .... periodicity a sub hypothesis , \quad which means that the functions a (1 ≤ i, j ≤ N) are periodic , say with period 1 in each ij open parenthesis y plus k closing parenthesisij = a sub ij open parenthesis y closing parenthesis open parenthesis 1 less or equal i coordinate , i . e . , for every k ∈ N , one has comma\noindent j less ormeans equal thatN closing the parenthesis functions $ a { i j } ( 1 \ leq Z i , j \ leq N ) $ are periodic , say with period 1 in each a (y + k) = a (y) (1 ≤ i, j ≤ N) almost ..ij everywhere .. in yij in R to the power of N period Capital Sigma hyphen convergence comma .. which .. coincides .. in \noindentalmostcoordinate everywhere , \ h f i l l ini .y e∈ . , R\Nh. f iΣ l− l convergencef o r \ h f i l l ,every which\ h f i l l coincides$ k \ in in Z ˆ{ N } , $ the present the present setting with well - known two - scale convergence , has proved to be an \ hsetting f i l l one with\ wellh f i hyphen l l has known\ h f i twol l hyphen$ a { scalei j convergence} ( y comma + has k proved ) to = be an a efficient{ i j } tool( y ) ( 1 \ leq iefficient , j tool\ leq for studyingN ) the $ periodic homogenization of l inear as well as nonlinear for studyingboundary the periodic value homogenization problems of and l inear init as ial well boundary as nonlinear value boundary problems , including ( value problems .. and init ial boundary value problems comma .. including .. open parenthesis 5 period 1 closing parenthesis \noindent5almost . 1 ) .\quad We refereverywhere for example\quad toin [ 1 $ , y 2\ 6in , 2R 3 ˆ ]{ N (} see. in particular\Sigma − the$ convergence , \quad which \quad c o i n c i d e s \quad in the present period .. Wereferences refer in [ 2 3 ] ) . settingfor example with to open well square− known bracket two 1 comma− scale .. 2 6 convergence comma 2 .. 3 closing , has square proved bracket to .. be open an parenthesis efficient see toolin particular the for studyingHowever the periodic , the periodicity homogenization hypothesis of i ls only inear one as thing well among as nonlinear many other boundary hy - references inpotheses open square under bracket which 2 .. we 3 closing can consider square bracket the homogenization closing parenthesis of period say ( 5 . 1 ) . There is valueHowever problems comma the\quad periodicityand hypothesisinit ial i boundary s only one thing value among problems many other , \ hyquad hypheni n c l u d i n g \quad ( 5 . 1 ) . \quad We r e f e r for exampleno doubt to [ that 1 , in\quad a great2 number6 , 2 \quad of physical3 ] \ squad ituations( see the in periodicity particular hypothesis the references i s in [ 2 \quad 3 ] ) . pothesesinappropriate under which we can and consider should the homogenization be therefore of say substituted open parenthesis by a 5 realistic period 1 closing hypothesis parenthesis period .. There is . Howeverno doubt , that the in periodicity a great number hypothesis of physical s ituations i s only the oneperiodicity thing hypothesis among many other hy − We claim that Σ− convergence theory allows to tackle homogenization problems pothesesi s .. inappropriate under which .. and .. we should can be consider .. therefore the .. substituted homogenization by a realistic of .. say hypothesis ( 5 . period 1 ) . \quad There i s no doubtbeyond that inthe a classical great numberperiodic of s ett physical ing . s Before ituations we can the concentrate periodicity on the hypothesis proof of We claimthis that assertion .. Capital as Sigma regards hyphen convergence ( 5 . 1 ) , lettheory us allowsexhibit to a tackle few homogenizationconcrete examples problems of nonpe ibeyond s \quad the classicalinappropriate periodic s\ ettquad ing periodand \ ..quad Beforeshould we can beconcentrate\quad ont h ethe r e fproof o r e of\quad substituted by a realistic \quad hypothesis . - riodicity hypotheses on a under which it is possible to successfully this assertion as regards .. open parenthesis 5 periodij 1 closing parenthesis comma let us exhibit a few concrete examples of \noindentstudyWe claim the that \quad $ \Sigma − $ convergence theory allows to tackle homogenization problems nonpe hyphenhomogenization of ( 5 . 1 ) . beyondriodicity the .. hypotheses classical .. on periodic a sub ij .. under s ett which ing it . ..\ isquad possibleBefore to .. successfully we can concentrate study the on the proof of  −1 1  thishomogenization assertionExample of as open 5 regards . parenthesis 1 . Let\quad 5 periodY 0 =( 5 1 closing . 1 ) parenthesis , letN − us1 periodwith exhibitN ≥ a2, fewand let concreteL2 (Y 0) examplesbe the of nonpe − 2, 2 per r iExample o d i c i t y 5 period\quad 1hypotheses period .... Let Y\quad to theon power $ of a prime{ i = j Row}$ 1\ minusquad 1under underbar which 1 underbar it \quad Row 2is 2 comma possible 2 . N minus to \quad successfully study the usual 1 with N greater equal 2 comma and0 let L sub per to the power of2 2 openN−1 parenthesis) Y to the power of prime closing parenthesis Y − L ( 0 ( be\noindent the usualHilberthomogenization space of ofperiodic ( 5 . 1 functions ) . in loc Ry see section 1 ) . We may Hilbert ....replace space the .... of periodicity Y to the power hypothesis of prime onhyphenaij periodic(1 ≤ i, j ....≤ functionsN) by .... in L sub loc to the power of 2 parenleftbig R sub\noindent y to the powerExample of prime 5 . to 1 the . power\ h f i l of l NLet minus $ 1 Y to ˆ the{\ powerprime of} parenrightbig= \ l e f t open(\ begin parenthesis{ array see}{ ....cc } section − 1 .... 1{\ closingunderline {\}} & parenthesis1 {\underline period ....{\}}\\ We may 2 , 2 \end{ array }\ right )N − 1 $ with $ N \geq 2 , $ a ∈ B ( ; L2 (Y 0)) (1 ≤ i, j ≤ N), (5.2) andreplace l e t the $ Lperiodicity ˆ{ 2 } hypothesis{ per } on( a subij Y ij ˆ open∞{\Rprime parenthesisper } ) 1 less$ orbe equal the i usual comma j less or equal N closing parenthesis by Equation: open parenthesis2 5 period0 2 closing parenthesis .. a sub ij in B sub infinity parenleftbig R semicolon L sub per to the \noindentwhereH i l b eB r∞ t (R\;hL fper i l( lY space)) denotes\ h f thei l l spaceo f $ of Y all ˆ{\ continuousprime } functions − $ p e ryN i o d→ i c u\(yNh f) i lof l fR u n c t i o n s \ h f i l l in power of 2 open parenthesis2 0 Y to the power of prime closing parenthesis2 0 parenrightbig open parenthesis 1 less or equal i comma j less$ L or ˆ{ equal2 into} N{ closingLl oper c ( parenthesisY} ) such( R that comma ˆ{ Nu(yN−) converges1 } { iny ˆL{\perprime(Y ) as }}| yNˆ{ |→) ∞} . ( $ see \ h f i l l s e c t i o n \ h f i l l 1 ) . \ h f i l l We may where BExample sub infinity 5 parenleftbig . 2 . More R semicolon generally L sub , instead per to ofthe ( power 5 . 2 of ) we 2 open may parenthesis consider Y the to element the power of prime closing \noindent replace2 the0 periodicity hypothesis+ on2 $ a0 { i j } ( 1 \ leq i− , j \ leq parenthesisa parenrightbigij ∈ C(R; Lper denotes(Y )) thewith spaceaij of(., all yN continuous) → ijz in functionsLper(Y y) as N rightyN arrow→ +∞ uand open parenthesisaij(., yN) → y Nzij closing parenthesis N ) $ by 2 0 of R intoin L subL perper( toY ) theas poweryN of 2 open→ parenthesis −∞, 1 Y≤ to thei, j power≤ ofN, primewhere closing parenthesisaij(., yN) suchstands that u open parenthesis 0 0 N−1 y N closingfor parenthesis the function converges iny L= sub (y1, per ..., yN to the− 1) power→ aij( ofy , 2 yN open)( parenthesisfor fixed YyN to∈ theR) powerof R of primeinto closingC, parenthesis as bar\ begin y N{ bara land iright g n ∗}ij arrow+, z− infinityare two period functions in L2 (Y 0) that are in general different . a { i j }\ inz ijB {\ infty } ( Rper ; L ˆ{ 2 } { per } ( Y ˆ{\prime } ) ) ( 1 ExampleExample 5 period 2 period 5 . More 3 . generally( Almost comma insteadperiodicity of open hypothesisparenthesis 5 ) period . Let 2 closing(L2 parenthesis, l∞)( N ) we may consider \ leq i , j \ leq N), \ tag ∗{$ ( 5 . 2 ) $} R the elementbe a sub the ij in so - called amalgam of L2 and l∞ on RN [19], i . e ., (L2, l∞)(RN ) \endC{ parenleftbiga l i gi n s∗} the R space semicolon of all L sub per to the power of 2 open parenthesis Y to the power of prime closing parenthesis parenrightbig .. with a sub ij open parenthesis period comma y N closing parenthesis right arrow ij z to the power of plus in L sub per to the power of\noindent 2 open parenthesiswhere Y $ to B the{\ powerinfty of prime} ( closing R parenthesis ; L ˆ{ as2 y N} { rightper arrow} plus( infinity Y ˆ{\ andprime a sub} ij open) parenthesis) $ denotes the space of all continuous functions $ y N \rightarrow u ( y N ) $ 2 N period comma y N closing parenthesis right arrow suchthat u ∈ Lloc(R ) of $R$ into $Lˆ{ 2 } { per } ( Y ˆ{\prime } )$ suchthat $u ( y N )$ z sub ij to the power of minus .. in .. L sub per to theZ power of 2 open parenthesis1 Y to the power of prime closing parenthesis converges in $ L ˆ{ 2 } { per } ( Y ˆ{\prime } 2 ) $ as $ \mid y N \mid \rightarrow .. as .. y N right arrow minus infinity commak u k 12,∞ less= or sup equal( i comma| u(y) | j lessdy) or< equal∞ N comma .. where .. a sub ij open parenthesis N 2 period\ infty comma.y $ N closing parenthesis .. stands .. fork∈ ..Z thek+Y function y to the power of1 prime1 = open parenthesis y 1 comma period period period comma y N minus 1 closing parenthesis with Y = (− )N. This i s a Banach space under the norm k · k2,∞ . We right\noindent arrow aExample sub ij open 5 parenthesis .2 , 2 . More y to generallythe power of prime , instead comma of y N ( closing 5 . 2 parenthesis ) we may open consider parenthesis the for element fixed y N in $ R a { i j } define L2 ( N ) to be the space of all functions u ∈ (L2, l∞)( N ) such that the closing\ in $ parenthesis ofAP R toR the power of N minus 1 into C comma and R set {τ u : h ∈ N } ( with τ u(y) = u(y − h) for y ∈ N ) has a compact closure in $ij Cz to the ( power Rh of ; plus L commaR ˆ{ 2 z sub} { ijper toh the} power( of Y minus ˆ{\ areprime two functions}R ) in ) L $ sub\quad per towith the power $ of a 2{ openi j parenthesis} (. (L2, l∞)( N ). Such functions are t ermed almost periodic in the sense of Stepanoff Y, to the y power N of ) primeR\rightarrow closing parenthesisi j that{ arez } inˆ{ general+ }$ different in $ period L ˆ{ 2 } { per } ( Y ˆ{\prime } ) $ asExample $ y .. N 5 period\rightarrow 3 period .. open+ parenthesis\ infty $ Almost and .. periodicity$ a { i j ..} hypothesis( . closing , parenthesis y N period ) \ ..rightarrow Let .. open $ parenthesis$ z ˆ{ L− to } the{ poweri j }$ of 2\quad commain l to\ thequad power$ of L infinity ˆ{ 2 closing} { per parenthesis} ( parenleftbig Y ˆ{\prime R to} the power) $ of\quad N parenrightbigas \quad ..$ be y the N .. so\ hyphenrightarrow − \ infty , 1 \ leq i , j \ leq N , $ \quad where \quad $ acalled{ amalgami j } ( of L . to the , power y of N 2 .. and ) $ l to\ thequad powerstands of infinity\quad .. onf R o r to\ thequad powerthe of N open square bracket 1 9 closing squarefunction bracket $ comma y ˆ{\ .. iprime period} e period=(y1,...,yN comma open parenthesis L to the power of 2 comma l− to the1 power ) of\ infinityrightarrow closing parenthesisa { i j } parenleftbig( y ˆ R{\ toprime the power} of, N parenrightbig y N ) .. i ($s the space forfixed of all $y N \ in R ) $ o f $ R ˆ{ N − Equation:1 }$ intou in L sub $C loc to ,$ the power and of 2 parenleftbig R to the power of N parenrightbig .. such that bar u bar sub 2 comma infinity$ i j = supremum{ z }ˆ{ k+ in} Z to, the z power ˆ{ − of} N{ parenleftbiggi j }$ are integral two sub functions k plus Y bar in u open $ L parenthesis ˆ{ 2 } { yper closing} parenthesis( Y ˆ{\ barprime } to) the$ power that of are 2 dy in parenrightbigg general different 1 divided by . 2 less infinity with Y = parenleftbig minus 1 divided by 2 sub comma 1 divided by 2 parenrightbig N sub period .. This i s a Banach space under\noindent the normExample .. bar times\quad bar sub5 . 2 3 comma . \quad infinity(period Almost .. We\quad defineperiodicity \quad hypothesis ) . \quad Let \quad $ (L sub L AP ˆ{ to2 the} power, of l 2 ˆparenleftbig{\ infty R} to the) power ( of R N ˆ parenrightbig{ N } ) $ .. to\quad be thebe space the of all\quad functionsso − u in open parenthesis Lcalled to the power amalgam of 2 comma of $l to L the ˆ{ power2 }$ of infinity\quad closingand parenthesis $ l ˆ{\ infty parenleftbig}$ R\quad to the poweron $ of R N ˆ parenrightbig{ N } [ .. 1 such 9that the] set , $ \quad i . e $ . , ( L ˆ{ 2 } , l ˆ{\ infty } ) ( R ˆ{ N } ) $ \quad i s the space of all braceleftbig tau sub h u : h in R to the power of N bracerightbig open parenthesis with tau sub h u open parenthesis y closing parenthesis\ begin { a l = i g u n open∗} parenthesis y minus h closing parenthesis .. for y in R to the power of N closing parenthesis .. has a compact closure\ tag ∗{ in$ u \ in L ˆ{ 2 } { l o c } ( R ˆ{ N } ) $} such that \\\ parallel u \ parallel { 2 , open\ infty parenthesis} = L to\sup the power{ k of 2\ commain Z l toˆ{ theN }} power( of infinity\ int closing{ k parenthesis + Y }\ parenleftbigmid u R to ( the powery ) of N\mid ˆ{ 2 } parenrightbigdy ) \ f r period a c { 1 ..}{ Such2 functions} < are\ infty t ermed almost periodic in the sense of Stepanoff \end{ a l i g n ∗}

\noindent with $ Y = ( − \ f r a c { 1 }{ 2 } { , }\ f r a c { 1 }{ 2 } )N { . }$ \quad This i s a Banach space under the norm \quad $ \ parallel \cdot \ parallel { 2 , \ infty } . $ \quad We d e f i n e $ L ˆ{ 2 } { AP } ( R ˆ{ N } ) $ \quad to be the space of all functions $ u \ in ( L ˆ{ 2 } , l ˆ{\ infty } ) ( R ˆ{ N } ) $ \quad such that the set $ \{\tau { h } u : h \ in R ˆ{ N }\} ( $ with $ \tau { h } u ( y ) = u ( y − h ) $ \quad f o r $ y \ in R ˆ{ N } ) $ \quad has a compact closure in $ ( L ˆ{ 2 } , l ˆ{\ infty } ) ( R ˆ{ N } ) . $ \quad Such functions are t ermed almost periodic in the sense of Stepanoff Capital Sigma hyphen CONVERGENCE .. 1 31 $ open\Sigma square bracket− $ CONVERGENCE 1 .. 9 closing square\quad bracket1 31 period .. This being so comma we may as well replace the periodicity hypothesis by[ 1 \quad 9 ] . \quad This being so , we may as well replace the periodicity hypothesis by Σ− CONVERGENCE 1 31 [ 1 9 ] . This being so , we may as well replace the Equation:periodicity open parenthesis hypothesis 5 period by 3 closing parenthesis .. a sub ij in L sub AP to the power of 2 parenleftbig R to the power of\ begin N parenrightbig{ a l i g n ∗} open parenthesis 1 less or equal i comma j less or equal N closing parenthesis period a Example{ i j }\ 5 periodin 4 periodL ˆ{ ..2 Let} { L subAP infinity} ( comma R ˆ{ perN to} the) power ( of 1 2 open\ leq parenthesisi Y , closing j parenthesis\ leq N). denote the \ tag ∗{$ ( 5 . 3 ) $} closure in open parenthesis L to the power of 2 comma2 l toN the power of infinity closing parenthesis parenleftbig R to the power of aij ∈ LAP (R ) (1 ≤ i, j ≤ N). (5.3) N\end parenrightbig{ a l i g n ∗} of the space of all finite sums sum sub finite phi i to the power2 of u i open parenthesis phi i in B sub2 ∞ infinityN parenleftbig R sub y to the power \noindentExampleExample 5 5 . . 4 4 . . \quadLetLetL∞,per $( LY ) ˆdenote{ 2 } {\ theinfty closure in,(L per, l )(}R )(of the Y space ) $ denote the closure in of N to the power of parenrightbig comma u sub i in C sub per open parenthesis Y closing parenthesis closingN parenthesis comma $ ( L ˆ{ 2 } , l ˆ{\ infty } ) ( R ˆ{ N } ) $ of the space of−1 1  where Y =of Row all 1 minus finite 1 sums underbarP 1 underbarϕiui Row(ϕi ∈ 2 B 2 comma( N ), u 2 .∈ to C the(Y power)), where of N subY = comma all finite sums $ \sum {finitef i n i t e }\varphi∞ Ry i ˆ{peru } i ( \varphi2, 2 i \ in B {\ infty } C sub per open parenthesis Y closing parenthesis defined in section 1 and B sub infinity parenleftbig R sub, y to the power of N ( R ˆ{ N } { y }ˆ{ ) } , u { i }\Nin) C { per } ( Y ) ) ,$where$Y = \ l e f t (\ begin { array }{ cc } − to the powerCper of( parenrightbigY ) defined in .. sectiondefined in 1 Example and B∞ 2(R ..y perioddefined 1 0 period in Example .. We may 2 as . 1 0 . We may 1 well{\ considerunderlineas well the{\}} homogenizationconsider& the 1 homogenization of{\ openunderline parenthesis{\}}\\ of 5 ( period 5 . 1 1 ) closing2 under , parenthesis the 2 hypothesis\end under{ array the}\ hypothesisright )ˆ{ N } { , }$ $Equation: C { per open} parenthesis( Y 5 ) period $ defined 4 closing in parenthesis section .. 1and a sub ij $B in L sub{\ infinityinfty comma} ( per R to ˆ{ theN power} { y of}ˆ 2{ open) }$ parenthesis\quad defined Y closing in parenthesis Example open 2 \quad parenthesis. 1 0 1 less . \ orquad equalWe i comma may as j less or equal N closing parenthesis well consider the homogenization of (2 5 . 1 ) under the hypothesis in place of the periodicity hypothesis periodaij ∈ L∞,per(Y ) (1 ≤ i, j ≤ N) (5.4) Example 5 period 5 period .. More generally comma in place of open parenthesis 5 period 4 closing parenthesis we may consider \ beginEquation:{ a lin i g open placen ∗} parenthesis of the periodicity 5 period 5 closing hypothesis parenthesis . Example .. a sub ij in 5 L . sub 5 infinity . commaMore generally AP to the power , in of 2 parenleftbig Ra to the{ i power j place}\ ofin ofN parenrightbig ( 5L . ˆ4{ )2 we} open may{\ parenthesisinfty consider 1, less per or equal} ( i comma Y jless ) or ( equal 1 N closing\ leq parenthesisi , comma j \ leq N ) \wheretag ∗{ L$ sub ( infinity 5 comma . 4 AP )to $the} power of 2 parenleftbig R to the power of N parenrightbig .. denotes the closure in open parenthesis\end{ a l i g L n ∗} to the power of 2 comma l to the power of infinity closing parenthesis parenleftbig R to the power of N parenrightbig a ∈ L2 ( N ) (1 ≤ i, j ≤ N), (5.5) .. of all finite sums of the ij ∞,AP R \noindentform sum subin placefinite phi of i to the the periodicity power of u i open hypothesis parenthesis phi. i in B sub infinity parenleftbig R sub y to the power of N where L2 ( N ) denotes the closure in (L2, l∞)( N ) of all finite sums of the form parenrightbigExample 5 comma . 5 .∞ u,AP\ subquadR i inMore AP parenleftbig generally R to , the in power place of N of parenrightbigR ( 5 . 4 ) closing we may parenthesis consider open parenthesis s ee section P u N N 2 for the definitionfinite ϕi i(ϕi ∈ B∞(Ry ), ui ∈ AP (R )) ( s ee section 2 for the definition \ beginof AP{ a parenleftbig l i g n ∗} R to the power of N parenrightbig closing parenthesis period a Remark{ i j 5}\ periodin 6 periodL ˆ{ ..2 Hypothesis} {\ infty open parenthesis, AP 5} period( 5 R closing ˆ{ N parenthesis} ) ( generalizes 1 \ openleq parenthesisi , 5 periodj \ leq 3 ofAP ( N )). N),closing parenthesis\ tag ∗{ and$ open ( parenthesis 5 . 5 period ) $ 4} closing parenthesisR comma as well period \end{ a l i g n ∗} ExampleRemark .. 5 period5 7 . period 6 . .. WeHypothesis may as well ( 5 consider . 5 ) generalizes the homogenization ( 5 . 3 of ) andopen ( parenthesis 5 . 4 ) , as 5 period well . 1 closing parenthesis .. under theExample 5 . 7 . We may as well consider the homogenization of ( 5 . 1 ) \noindentfollowing hypothesiswhere $ comma L ˆ{ where2 } {\ 1 lessinfty or equal i, comma AP j} less( or equal R ˆ{ NN : } ) $ \quad denotes the closure in under the following hypothesis , where 1 ≤ i, j ≤ N : $ (a sub L ij ˆ is{ constant2 } , on each l ˆ cell{\ kinfty plus Y} parenleftbig) ( k R in ˆZ{ toN the} power) $ of\ Nquad parenrightbigof all .. finite with Y assums above of comma the a is constant on each cell k + Y (k ∈ N ) with Y as above , formand further $ \sum comma{ asf iij nbar i t ke bar}\ rightvarphi arrow infinityi ˆ{ commau } integraliZ ( sub\ kvarphi plus Y a subi ij open\ in parenthesisB {\ yinfty closing parenthesis} ( R ˆ{ N } { y } and further , as | k |→ ∞, R a (y)dy t ends to a finite l imit in . dy) t ends , to u a finite{ i l}\ imitin in C periodAP ( R ˆ{ Nk+}Y ij) ) ( $ s ee section 2 forC the definition The study ofThe the study homogenization of the homogenization problem for open problem parenthesis for 5 (period 5 . 1 1 ) closing under parenthesis any of the under hypotheses any of the hypotheses \ beginstated{ a in lstated i the g n ∗} preceding in the precedingexamples reduces examples to an reduces abstract settingto an abstract that we will setting now that we will now look o flook AP intointo period ( . R ˆ{ N } )). \end5 period{ a l i g5 n2 .∗} period 2 . ..The The .. abstract abstract .. homogenization homogenization .. problem .. for .. problem open parenthesis for 5 period ( 5 1 . closing 1 ) . parenthesis period .. The mainThe purpose main purpose of the present subsection i s to investigate the limit behaviour , \noindentof the presentasRemarkε → .. subsection0 5, .of 6u i .ε s( to\quad the investigate soHypothesis lut ion the limit of ( 5(behaviour . 5 . 1 5 ) comma ) ) under generalizes .. as the epsilon abstract ( right 5 . hypothesis arrow 3 ) 0 and comma ( 5 .. . of 4 u sub ) , epsilon as well . open parenthesis the so lut ion of open parenthesis 5 period .. 1 closing parenthesis closing parenthesis under the abstract hypothesis\noindent Example \quad 5 . 7 . \quad We may as well consider the homogenization of ( 5 . 1 ) \quad under the a ∈ X2 ( N ) (1 ≤ i, j ≤ N), (5.6) followingEquation: open hypothesis parenthesis , 5 where period 6 $closing 1 ij parenthesis\ leqA Ry i .. a , sub ij j in X\ subleq A toN the power : $ of 2 parenleftbig R sub y to the power of N to the power of parenrightbig open parenthesisN 1 less or equal i comma j less∞ or equal N closing parenthesis comma \ centerlinewhere{ $A a i s{ ani j H} -$ algebra is constantoneach on R with the property cell $k that +A Yis ( dense k \ inin A Z( ˆ{ N } ) $ where A i s an H hyphen algebra on R to the power of N .. with1 the,2 property that .. A to the power of infinity .. is .. dense in \quad withsee $Y$Section 2as ) .above We , also} require A to be W − proper in the following sense : A open parenthesis see (P )1D(∆(A)) is dense in H1(∆(A)) = W 1,2(∆(A)). Section 2 closing parenthesis period .. We also require A to beN W to the power of 1 comma 2 hyphen proper in the following \ centerline(P){and further2 Given , as an open$ \mid set kΩ ⊂\midRx , \rightarrowa fundamental\ infty sequence , E and\ int a { k + Y } sense : 1 0 a { i j }sequence( y(vε )ε∈E which dy$ i tendstoafinite s bounded in H (Ω), a limitin subsequence $CE can .$ be extracted} from open parenthesisE P closing parenthesis 1 D open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis is dense in H to the power of0 1 open parenthesis Capital Delta open parenthesis1 A closing parenthesis closing parenthesis The studysuch of that the homogenization as E 3 problemε → for0, (vε 5→ . 1v )0 underin anyH of(Ω) the− weak hypotheses and = W to the∂v powerε of 1∂v comma0 2 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis period stated in the→ preceding+ ∂jv1 examples reduces to an abstract setting that we will nowin open parenthesis∂xj P∂x closingj parenthesis 2 .. Given an open set Capital Omega subset R sub x to the power of N comma .. a fundamentallook i n t o sequence . E and a sequence open parenthesis v sub epsilon2 closing parenthesis sub epsilon in E which2 i s1 bounded in H to the power of 1 open parenthesis L (Ω) − weakΣ(1 ≤ j ≤ N), wherev1 ∈ L (Ω; H#(∆(A))). Capital\noindent Omega5 closing . 2 . parenthesis\quad The comma\quad a subsequencea b s t r a c t E\ toquad the powerhomogenization of prime can be\quad extractedproblem from E\quad f o r \quad ( 5 . 1 ) . \quad The main purpose ofsuch the that present .... as ....\quad E to thesubsection power of prime i s ni to epsilon investigate right arrow the0 comma limit v sub behaviour epsilon right , arrow\quad v subas 0 $ ....\ varepsilon in .... H to \rightarrowThe aim0 now , $is to\quad show thato f $ the u homogenization{\ varepsilon of} ($ 5 . 1 ) under ( 5 . 6 ) i s possible the power ofprovided 1 open parenthesis the H - algebra Capital OmegaA has closingthe preceding parenthesis properties hyphen weak . .... To and this .... end partialdiff , let v sub epsilon divided by partialdiff( the so x sub lut j right ion arrow of ( partialdiff 5 . \quad v sub1 0 ) divided ) under by partialdiff the abstract x sub j plus hypothesis partialdiff sub j v sub 1 .... in L to the power of 2 open parenthesis Capital1 Omega1 closing2 parenthesis1 hyphen weak Capital Sigma open parenthesis 1 less or F0 = H0 (Ω) × L (Ω; H#(∆(A))) equal\ begin j less{ a l or i g nequal∗} N closing parenthesis comma where v sub 1 in L to the power of 2 parenleftbig Capital Omega semicolon H suba hash{ i to j the}\ powerin ofX 1 open ˆ{ 2 parenthesis} { A } Capital( R Delta ˆ{ openN } parenthesis{ y }ˆ{ ) A} closing( parenthesis 1 \ leq closingi parenthesis , j parenrightbig\ leq N period), \ tag ∗{$ ( 5 . 6 ) $} \endThe{ a aim l i g n now∗} is to show that the homogenization of open parenthesis 5 period 1 closing parenthesis under open parenthesis 5 period 6 closing parenthesis i s possible \noindentprovided thewhere H hyphen $A$ algebra i A s has anH the− precedingalgebra properties on $ period R ˆ{ N .. To}$ this\quad end commawith let the property that \quad $ AF subˆ{\ 0infty to the power}$ of\quad 1 = Hi sub s \ 0quad to thedense power of in 1 open $A parenthesis ($ see Capital Omega closing parenthesis times L to the power ofSec 2 parenleftbig tion 2 ) Capital . \quad OmegaWe semicolon also require H sub hash $A$ to the power to be of 1 $Wˆ open parenthesis{ 1 , Capital 2 } Delta − $ open proper parenthesis in the A closing following sense : parenthesis closing parenthesis parenrightbig \ centerline {( P $ ) 1 D ( \Delta ( A ) )$ isdensein $Hˆ{ 1 } ( \Delta ( A ) ) = W ˆ{ 1 , 2 } ( \Delta ( A ) ) . $ }

( P ) 2 \quad Given an open set $ \Omega \subset R ˆ{ N } { x } , $ \quad a fundamental sequence $ E $ and a sequence $ ( v {\ varepsilon } ) {\ varepsilon \ in E }$ which i s bounded in $Hˆ{ 1 } ( \Omega ) , $ a subsequence $Eˆ{\prime }$ can be extracted from $ E $

\noindent such that \ h f i l l as \ h f i l l $ E ˆ{\prime }\ ni \ varepsilon \rightarrow 0 , v {\ varepsilon }\rightarrow v { 0 }$ \ h f i l l in \ h f i l l $ H ˆ{ 1 } ( \Omega ) − $ weak \ h f i l l and \ h f i l l $\ f r a c {\ partial v {\ varepsilon }}{\ partial x { j }}\rightarrow \ f r a c {\ partial v { 0 }}{\ partial x { j }} + \ partial { j } v { 1 }$ \ h f i l l in

\ begin { a l i g n ∗} L ˆ{ 2 } ( \Omega ) − weak \Sigma ( 1 \ leq j \ leq N ) , where v { 1 } \ in L ˆ{ 2 } ( \Omega ; H ˆ{ 1 } {\# } ( \Delta (A))). \end{ a l i g n ∗}

\noindent The aim now is to show that the homogenization of ( 5 . 1 ) under ( 5 . 6 ) i s possible provided the H − algebra $ A $ has the preceding properties . \quad To this end , let

\ [ F ˆ{ 1 } { 0 } = H ˆ{ 1 } { 0 } ( \Omega ) \times L ˆ{ 2 } ( \Omega ; H ˆ{ 1 } {\# } ( \Delta (A))) \ ] 1 32 .. G period NGUETSENG comma N period SVANSTEDT \noindentwith the norm1 32 \quad G . NGUETSENG , N . SVANSTEDT withbar v the bar subnorm F sub 0 1 = open parenthesis bar v sub 0 bar H to the power of 1 from 0 to 2 open parenthesis Capital Omega 1 32 G . NGUETSENG , N . SVANSTEDT with the norm closing parenthesis plus bar v sub 1 bar 2 L sub 2 open parenthesis Capital Omega semicolon H sub hash to the power of 1 open \ [ \ parallel v \ parallel { F { 0 }} 1 = ( \ parallel v { 0 }\ parallel H ˆ{ 1 }ˆ{ 0 } { 2 } parenthesis Capital Delta open parenthesis A closing10 parenthesis closing1 parenthesis1 closing parenthesis1 closing parenthesis 1 divided ( \Omega )k +v k \1parallel = (k v k Hv (Ω)+{ 1 }\k v kparallel2L (Ω; H (∆(2{A))))L } ,{v =2 (}v , v( ) ∈ \Omega, ; H ˆ{ 1 } {\# } by 2 comma v = open parenthesisF0 v sub0 0 comma2 v sub 11 closing2 parenthesis# in F2 sub 0 to0 the1 powerF0 of comma to the power of 1 ( which\Delta .... makes(A)))) .... it .... a Hilbert .... space\ ....f r a open c { 1 parenthesis}{ 2 } bar, times v bar = sub ( H sub v 0{ 1 sub0 } open, parenthesis v { 1 Capital} ) \ in F ˆwhich{ 1 } makes{ 0 ˆ{ it, a}}\ Hilbert] space (k · k 1 stands for the usual gradient norm ) . Omega closing parenthesis .... stands .... for .... the .... usualH0 (Ω) gradient .... norm closing parenthesis period By combiningBy combining property open property parenthesis ( P ) P1 closingwith ( parenthesis parts ( 2 1 ) with and open ( 3 ) parenthesis of ) Remark parts 2 .... . 2 open 6 , parenthesis it 2 closing parenthesisfollows and open parenthesis 3 closing parenthesis of closing parenthesis Remark 2 .... period 2 .... 6 comma it follows \noindentreadily thatreadilywhich that\ h f i l l makes \ h f i l l i t \ h f i l l a H i l b e r t \ h f i l l space \ h f i l l $ ( \ parallel \cdot \ parallel { H { 0 }} 1 { ( ∞\Omega ) }$ \ h f i l l stands \ h f i l l f o r \ h1 f i l l the \ h f i l l usual gradient \ h f i l l norm ) . F sub 0 to the power of infinity = D openF0 parenthesis= D(Ω) × Capital[D(Ω) ⊗ OmegaJ(D(∆( closingA))/C)] parenthesisis dense times in openF0, (5 square.7) bracket D open parenthesiswhere CapitalD Omega(∆(A)) closing/C denotes parenthesis the space oslash of J openϕ ∈ D parenthesis(∆(A)) such D thatopen parenthesis Capital Delta open parenthesis A \noindent By combining property ( P ) 1 with ( parts \ h f i l l ( 2 ) and ( 3 ) of ) Remark 2 \ h f i l l . 2 \ h f i l l 6 , it follows closing parenthesis closing parenthesis slash C closingZ parenthesis closing square bracket .. is dense in F sub 0 to the power of comma to the power of 1 open parenthesis 5 period 7 closingϕ(s) parenthesisdβ(s) = 0. \noindentwhere D openreadily parenthesis that Capital Delta open parenthesis∆(A) A closing parenthesis closing parenthesis slash C denotes the space of phi in D open parenthesis Capital Delta open parenthesis A closing parenthesis closing1 1 parenthesis such that We also need the s esquilinear form baΩ(., .) on F0 × F0 given by \ hspaceintegral∗{\ subf i Capital l l } $ Delta F ˆ{\ openinfty parenthesis} { 0 A} closing= parenthesis D ( phi\Omega open parenthesis) \times s closing parenthesis[D( d beta\Omega open paren-) \otimes J(D( \Deltai,j=1 ( A ) ) / C ) ] $ \quad is dense in $Fˆ{ 1 } { 0 ˆ{ , }} thesis s closing parenthesis = 0 period ZZ ∂u ∂v ( 5 . 7 ) $ X 0 0 We also need the s esquilinearbaΩ( formu, v) hatwide-a = sub Capitalbaij( Omega+ ∂ openju1) parenthesis( + ∂iv1) perioddxdβ comma period closing parenthesis on Ω×∆(A) ∂xj ∂xi F sub 0 to the power of 1 times F sub 0 toN the power of 1 given by \noindenthatwide-a subwhere Capital $ DOmega ( open\Delta parenthesis( u comma A ) v1 closing ) parenthesis / C$ = denotesthespaceof sum from i comma∞ j = 1 to N $ integral\varphi integral\ in for u = (u0, u1) and v = (v0, v1) in F0, where baij = G(aij) ∈ L (∆(A)) ( D(sub Capital\seeDelta Omega part times (( v Capital ) A of Corollary ) Delta )$ open suchthat parenthesis 2 . 2 0 ) A . closing There parenthesis is no difficulty a-hatwide in sub verifying ij parenleftbigg that partialdiff u sub 0 divided by partialdiff x sub j plus partialdiff sub j u sub 1 parenrightbigg overbar parenleftbigg partialdiff v sub 0 divided by the sesquilinear form baΩ(., .) is Hermitian , continuous and coercive ( use corollary 2 partialdiff\ [ \ int x{\ subDelta i plus partialdiff(A) sub i v sub}\ 1Rvarphi parenrightbigg( dxd s beta )1 d \beta ( s ) = 0 . \ ] . 2 0 , and note also that ∆(A) ∂ivdβ = 0 for v ∈ H#(∆(A)), as is straightforward by for u = openProposition parenthesis 2 u . sub 2 0 comma4 and u sub use 1 of closing Remark parenthesis 2 . .. 2 and v6 = ) open. parenthesisConsequently v sub , 0 i comma f l v sub 1 closing parenthesis .. in F sub 0 to the power of comma to the power of1 1 .. where hatwide-a sub ij = G open parenthesis a sub ij closing \ centerlinedenotes{We also the continuous need the antilinear s esquilinear form on formF0 $ given\widehat by l{(av} ){\ =Omegahf, v}0i (for .v , . ) $ parenthesis in L to the power of1 infinity open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis on $ F ˆ={ 1 (v}0, v{1)0 }\∈ Ftimes0, thenF the ˆ{ variational1 } { 0 }$ given by } open parenthesisproblem see part open parenthesis v closing parenthesis of Corollary .. 2 period 2 0 closing parenthesis period .. There is no difficulty in verifying\ [ \widehat that the{a} sesquilinear{\Omega } ( u , v ) = \sum ˆ{ i , j = 1 } { N }\ int \ int {\Omega \times \Delta (A) }\widehat{a} {) i j } 1 ( \ f r a c {\ partial u { 0 }}{\ partial x { j }} form hatwide-a sub Capital Omega open{ parenthesisau = ((u0, periodu1 (∈ comma)F0 :all periodv ∈ 1 closing parenthesis is Hermitian(5.8) comma continuous bΩ ,uv) =l v for F0 +and coercive\ partial open parenthesis{ j } u use{ corollary1 } ) 2 period\ overline 2 .. 0 comma{\}{ and( }\ notef r a c {\ partial v { 0 }}{\ partial x { i }} + also\ partial thathas integral one{ sub andi } Capital onlyv one Delta{ 1 solution} open) parenthesis . dxd A\beta closing\ parenthesis] partialdiff sub i vd beta = 0 for v in H sub hash to the power of 1 open parenthesisWe are now Capital in a Deltaposition open to parenthesis prove the A following closing parenthesis homogenization closing parenthesis theorem comma . as is straightforward by PropositionTheorem 5 . 8 . Under the preceding hypotheses , let u = (u0, u1) be uniquely \noindent2 periodde 2f .. o- r 4 u .. and $ = use of ( Remark u { ..0 2} period, 2 u .. 6{ closing1 } parenthesis) $ \quad periodand .. v Consequently $ = ( comma v { i0 f l} denotes, the v { 1 } ) $ \quad in $ F ˆ{ 1 } { 0 ˆ{ , }}$ \quad where $ \widehat{a} { i j } = G ( a { i j } continuous antilinearfined by ( 5 . 8 ) , and for each real ε > 0, let u be the ) \ in L ˆ{\ infty } ( \Delta ( A ) ) ( $ se e ε form onunique F sub 0 to the s olution power of 1 .. of given ( 5 by . 1 l open ) . parenthesis Then , v asclosingε → parenthesis0, = angbracketleft f comma overbar v sub 0part right angbracket ( v ) of .. Corollary for v .. = open\quad parenthesis2 . 2 v 0 sub ) 0 . comma\quad vThere sub 1 closing is no parenthesis difficulty in F sub in 0 verifying to the power that of comma the to sesquilinear theform power $ of\ 1widehat .. then the{a} variational{\Omega } ( . , . ) $ is Hermitian , continuous and coercive ( use corollary 2 . 2 \quad 0 , and note a l s o that $ \ int {\Delta (A) }\ partial { 1i } vd \beta = 0 $ f o r $ v \ in problem uε → u0 inH (Ω) − weak, (5.9) H ˆ{ 1 } {\# } ( \Delta ( A ) ) , $ as is0 straightforward by Proposition Equation: open parenthesis 5∂u period 8∂u closing parenthesis .. braceleftbigg hatwide-a sub Capital Omega to the power of u = 2 . 2 \quad 4 \quad and useε of Remark0 \quad2 2 . 2 \quad 6 ) . \quad Consequently , i f $ l $ open parenthesis open parenthesis comma→ u v+ sub∂ju closing1 inL parenthesis(Ω) − weak toΣ the (1 ≤ poweri, j ≤ ofN u). sub 0 comma u =(5. to10) the power of 1 sub ∂xj ∂xj ldenotes to the power the of continuous closing parenthesis antilinear open parenthesis in v closing parenthesis to the power of F sub for to the power of 0 to the form on $ F ˆ{ 1 } { 0 }$ \quad givenby $l ($ v $) = \ langle f , \ overline {\}{ v } { 0 } power of 1 toProof the power . ofFor : all fixed v in Fε sub> 0, 0we to thehave power of 1 \ranglehas one$ and\ onlyquad onef o solution r v \quad period $ = ( v { 0 } , v { 1 } ) \ in F ˆ{ 1 } { 0 ˆ{ , }}$ \quadWe arethen now the in a position variational to prove the following homogenization theorem period i,j=1 Z Theorem .... 5 period 8 period .... UnderX theε preceding∂uε hypotheses∂v comma .... let u = open parenthesis u sub 0 comma u sub 1 closing\noindent parenthesisproblem .... be uniquely de hyphen a ij dx = hf, vi (5.11) Ω ∂xj ∂xi fined by .. open parenthesis 5 periodN 8 closing parenthesis comma .. and for each .. real epsilon greater 0 comma .. let u sub \ begin { a l i g n ∗} epsilon .. be .. the .. unique1 .. s olution .. of open parenthesis 5 period 1 closing parenthesis period \{\widehatfor all {va}∈ ˆH{0 (Ω)u .} {\ByOmega taking} in={ particular( } ( {v =, uε{andu } makingv }ˆ{ useu of{ the0 } properties, } { ) } u { = }ˆ{ 1 }ˆ{ ) } { l } Then comma .. as epsilon right arrow 0 comma 1 ( \ in of{ thev } matrix) ˆ{ F(a}ijˆ){1≤i,j0≤ ˆN{, we1 }} see{ thatf o r the}ˆ{ sequence: } a l l(uε) vε>0 i\ in s boundedF ˆ{ 1 in } H{0 (Ω)0 }\. tag ∗{$ ( 5 Equation: open parenthesis 5 period 9 closing parenthesis .. u sub epsilon right arrow u sub 0 in H sub1,2 0 to the power of 1 open . 8 )Consequently $} , given an arbitrary fundamental sequence E, appeal to the W − parenthesis Capital Omega closing parenthesis hyphen weak comma Equation: open parenthesis 5 period0 1 0 closing parenthesis .. \end{ a l i gproperness n ∗} of A( see in particular property ( P ) 2 ) yields a subsequence E from E partialdiff u sub epsilon divided by partialdiff1 x sub j right arrow0 partialdiff u sub 0 divided by partialdiff x sub j plus partialdiff sub and some u = (u0, u1) ∈ F0 such that , as E 3 ε → 0, we have ( 5 . 9 ) - ( 5 . 1 0 ) . j u sub 1 inThus L to the , power of 2 open parenthesis Capital Omega closing parenthesis hyphen weak Capital Sigma open parenthesis 1\noindent less or equalhas i comma one j and less or only equal one N closing solution parenthesis . period Proof period .. For fixed epsilon greater 0 comma we have \ centerlineEquation: open{We parenthesis are now in5 period a position 1 1 closing to parenthesis prove the .. sum following from i comma homogenization j = 1 to N integral theorem sub Capital . } Omega a to the power of epsilon ij partialdiff u sub epsilon divided by partialdiff x sub j to the power of hline partialdiff v divided by partialdiff x\noindent sub i dx = angbracketleftTheorem \ h f icomma l l 5 .hline 8 . sub\ h v f right i l l angbracketUnder the preceding hypotheses , \ h f i l l l e t u $ = ( u for{ all0 } v in, H sub u 0 to{ the1 } power) $ of 1\ openh f i l parenthesis l be uniquely Capital de Omega− closing parenthesis period .. By taking in particular v = u sub epsilon and making use of the properties \noindentof the matrixf i n open e d by parenthesis\quad ( a sub 5 . ij 8 closing ) , parenthesis\quad and sub f o 1 r less each or equal\quad i commar e a l j less $ \ orvarepsilon equal N comma> we see0 that , the $ sequence\quad l opene t $parenthesis u {\ varepsilon u sub epsilon} closing$ \quad parenthesisbe \quad sub epsilonthe \ greaterquad 0unique i s bounded\quad in Hs sub o l u 0 t i to o n the\quad powero of f 1 ( open 5 . 1 ) . parenthesisThen , \ Capitalquad as Omega $ \ closingvarepsilon parenthesis\ periodrightarrow 0 , $ Consequently comma given an arbitrary fundamental sequence E comma .. appeal to the W to the power of 1 comma 2 hyphen \ beginproperness{ a l i g nof∗} A open parenthesis see in particular property open parenthesis P closing parenthesis 2 closing parenthesis yields au subsequence{\ varepsilon E to the power}\rightarrow of prime from E u { 0 } in H ˆ{ 1 } { 0 } ( \Omega ) − weak , \ tag ∗{$ ( 5and . some 9 u = ) open $}\\\ parenthesisf r a c {\ upartial sub 0 commau u sub{\ 1varepsilon closing parenthesis}}{\ inpartial F sub 0 to thex power{ j of}}\ 1 suchrightarrow that comma as\ Ef to r a c {\ partial theu power{ 0 }}{\ of primepartial ni epsilon rightx { arrowj }} 0 comma+ we\ partial have open{ parenthesisj } u 5 period{ 1 } 9 closingin parenthesis L ˆ{ 2 } hyphen( \ openOmega parenthesis) −5 periodweak 1 0 closing\Sigma parenthesis( period 1 \ ..leq Thus commai , j \ leq N). \ tag ∗{$ ( 5 . 1 0 ) $} \end{ a l i g n ∗}

\noindent Proof . \quad For f i x e d $ \ varepsilon > 0 , $ we have

\ begin { a l i g n ∗} \sum ˆ{ i , j = 1 } { N }\ int {\Omega } a ˆ{\ varepsilon }{ i j }\ f r a c {\ partial u {\ varepsilon }}{\ partial x { j }}ˆ{\ r u l e {3em}{0.4 pt }}\ f r a c {\ partial v }{\ partial x { i }} dx = \ langle f , \ r u l e {3em}{0.4 pt } { v }\rangle \ tag ∗{$ ( 5 . 1 1 ) $} \end{ a l i g n ∗}

\noindent f o r a l l $ v \ in H ˆ{ 1 } { 0 } ( \Omega ) . $ \quad By taking in particular $ v = u {\ varepsilon }$ and making use of the properties of the matrix $ ( a { i j } ) { 1 \ leq i , j \ leq N } , $ we see that the sequence $ ( u {\ varepsilon } ) {\ varepsilon > 0 }$ i s bounded in $Hˆ{ 1 } { 0 } ( \Omega ) . $ Consequently , given an arbitrary fundamental sequence $ E , $ \quad appeal to the $Wˆ{ 1 , 2 } − $

\noindent properness of $ A ( $ see in particular property ( P ) 2 ) yields a subsequence $ E ˆ{\prime }$ from $ E $ andsomeu $= ( u { 0 } , u { 1 } ) \ in F ˆ{ 1 } { 0 }$ such that , as $Eˆ{\prime } \ ni \ varepsilon \rightarrow 0 ,$ wehave(5.9) − ( 5 . 1 0 ) . \quad Thus , Capital Sigma hyphen CONVERGENCE .. 1 33 \ hspacethe theorem∗{\ f i i l s l } proved$ \Sigma if we can check− $ that CONVERGENCE u verifies the\ variationalquad 1 33 equation in open parenthesis 5 period 8 closing parenthesis .. open parenthesis attention i s drawn to .. Remark .. 3 period 6 closing Σ− CONVERGENCE 1 33 parenthesis\noindent periodthe .. theorem For this purpose i s proved comma if .. take we incan .. open check parenthesis that u 5 verifies period 1 ..1 the closing variational parenthesis .. equation the in ( 5 . 8the ) \ theoremquad ( attention i s proved if i we s candrawn check to that\quadu Remarkverifies\ thequad variational3 . 6 ) equation . \quad inFor ( 5 this purpose , \quad take in \quad ( 5 . 1 \quad 1 ) \quad the particular. 8 function ) ( attention v = Capital i Phi s drawn sub epsilon to with Remark 3 . 6 ) . For this purpose , take in particularLine 1 Capital function Phi sub epsilon $ v open = parenthesis\Phi x{\ closingvarepsilon parenthesis} =$ psi with 0 open parenthesis x closing parenthesis plus epsilon ( 5 . 1 1 ) the particular function v = Φ with psi 1 open parenthesis x comma x divided by epsilon to the powerε of closing parenthesis open parenthesis x in Capital Omega closing\ [ \ begin parenthesis{ a l i g n e comma d }\Phi Line{\ 2 wherevarepsilon psi 0 in D} open( parenthesis x ) Capital = \ Omegapsi closing0 ( parenthesis x ) comma + psi\ varepsilon 1 in D open parenthesis\ psi 1 Capital ( Omega x , closing\ f r aparenthesis c { x }{\ oslashvarepsilon open parenthesis}ˆ{ ) } A infinity( x slash\ in C closing\Omega parenthesis), with A\\x infinity) slash where \ psi 0 \ in D( \Omega ), \ psi 1 \ in Φ D((x) = ψ0(x)\Omega + εψ1(x, ) (x\∈otimesΩ), C = open brace psi in A infinity : M open parenthesis psi closing parenthesis = 0 closing braceε period Line 3 Clearlyε we have to (A \ infty / C ) with A \ infty / C = \{\ psi \ in A \ infty : the power of Capital Phi sub Capital Phi sub epsilonwhere toψ the0 ∈ power D(Ω), of ψ1 epsilon∈ D(Ω) to the⊗ ( powerA∞/C of)with in DA∞ right/C = arrow{ψ ∈ openA∞ parenthesis: M(ψ) = 0} psi. M(0 Capital Omega\ psi closing) parenthesis = 0 period\} Furthermore. \\ in sub H sub 0 to the power of 1 open parenthesis Capital Omega closing ∂Φ easy ∂ψ0 parenthesisC l e a r l y hyphenClearly{ weΦ weak have toε the∈D} powerˆ{\( Ω)Phi of. commaFurthermorein}ˆ{\ subvarepsilon comma, to the} powerit{\isanandPhi of it i{\ sε anvarepsilon andexercise partialdiffto}} Capital∂showˆ{\inin Phi that} sub,D epsilonas{\ε→ dividedrightarrow0 , } we haveΦε → ψ0 H1(Ω)−weak, 1 L2(Ω)−strong Σ 0 ∂x ∂x iψb by( partialdiff{\ psi x sub0 }\ i to theOmega power of) easy . sub right Furthermore arrow to the{ powerin } ofˆ{ exercise, } i{→ partialdiffH ˆ{ 1 psi} i { 0+ divided0 } ( by partialdiff\Omega x sub) i sub− plusweak to} partialdiffˆ{ i t } sub{ , i psi-hatwide} i s sub an 1 to{ theand power}\ off r a show c {\ inpartial that L to the\Phi power{\ of 2varepsilon open parenthesis}}{\ Capitalpartial Omega subx { i }}ˆ{ easy } {\rightarrow }ˆ{ e x e r c i s e }\ f r a c {\ partial (1 ≤ i ≤ N). From the latter convergence result together with ( 5 . 1 closing\ psi parenthesis0 }{\ partial hyphen to thex power{ i of}} comma{ + } subto strong{\ partial to the power} { ofi as to\widehat the power{\ ofpsi epsilon}}ˆ right{ show arrow} 0{ sub1 Capital} in 0 ) ( where E0 3 ε → 0) we deduce using Corollary 3 . 1 9 that , as E0 3 ε → 0, Sigmathat comma{ L ˆ{ 2 } ( \Omega }ˆ{ , } { ) − }ˆ{ as } { strong }ˆ{\ varepsilon \rightarrow } 0 open{\Sigma parenthesis} 1, less\end or{ equala l i g n i e less d }\ or] equal N closing parenthesis period .. From .. the .. latter .. convergence .. result .. ∂uε ∂Φε ∂u0 ∂ψ0 2 together .. with .. open parenthesis 5 period→ ( 1 .. 0+ closing∂ju1)( parenthesis+ ∂iψ ..b1) open inL parenthesis(Ω) − weakΣ where ∂xj ∂xi ∂xj ∂xi E to the power of prime ni epsilon right arrow 0 closing parenthesis we deduce using Corollary .. 3 period 1 9 .. that comma as E\noindent to the powerfor $ of1 ( prime≤ i, j1 ni≤ epsilonN.\ leqWe right cani arrow now\ leq pass0 comma toN the ) l imit . $ (\ asquadE0 3Fromε → 0)\quadin ( 5the . 1\ 1quad ) usingl a t t e r \quad convergence \quad r e s u l t \quad t o g e t h e r \quad with \quad ( 5 . 1 \quad 0 ) \quad ( where $ E ˆ{\prime }\ ni \ varepsilon \rightarrow 0 ) $ we deduce using Corollary \quad 3 . 1 9 \quad that , as partialdiffProposition u sub epsilon 3 divided . 7 by partialdiff x ( sub it j i partialdiff s clear that Capitalaij Phimay sub be epsilon viewed divided as a by function partialdiff in x sub i right arrow $ E ˆ{\prime }\ ni2,∞ \ varepsilon \rightarrow 0 , $ parenleftbiggC( partialdiffΩ; X uA sub) = 0 divided by partialdiff x sub j plus partialdiff sub j u sub 1 parenrightbigg parenleftbigg partialdiff 2,∞ psi 0 dividedK by( partialdiffΩ; XA x sub)). i plusThe partialdiff result is sub that i psi-hatwide 1 parenrightbigg in L to the power of 2 open parenthesis Capital Omega\ [ \ f r a closing c {\ partial parenthesis hyphenu {\ weakvarepsilon Capital Sigma}}{\ partial x { j∞}}\ f r a c {\ partial \Phi {\ varepsilon }}{\ partial baΩ( u , Φ) = l(Φ) for all Φ ∈ F0 . x for{ ....i }}\ 1Thanks less orrightarrow equal to i ( comma 5 . 7 j( ) less ,\ oritf r a equal c follows{\ Npartial period that ....u Weui can s{ the now0 solution}}{\ pass topartial the of l( imit 5 . 8....x ) open .{ Hencej parenthesis}} + the as\ partial E to the power{ j } ofu prime{ 1 ni} theorem epsilon)( right\ arrowf r a c {\ 0 closingpartial parenthesis\ psi .... in0 open}{\ parenthesispartial 5 periodx { 1i 1 closing}} + parenthesis\ partial using{ i }\widehat{\ psi } 1 ) in L ˆ{ 2 } ( \Omega ) − weak \Sigma \ ] Propositionfollows 3 period.  7 .... open parenthesis it i s clear that a sub ij may be viewed as a function in C parenleftbig to the power of hline Capital Omega semicolon XAt sub the A presentto the power t ime of 2, forcomma each infinity1 ≤ j parenrightbig≤ N, let = K parenleftbig to the power of hline Capital Omega semicolon X sub A to the power of 2 comma infinity parenrightbig closing parenthesis\noindent periodf o r ..\ h The f i l result l $ is 1 that\ leq i , j \ leq N . $ \ h f i l l We can now pass to the l imit \ h f i l l ( as N 1 $ E ˆ{\prime }\ ni \ varepsilonZ forall\rightarrowv∈H#(∆(A)) 0 ) $ \ h f i l l in ( 5 . 11 ) using hatwide-a sub Capital Omega openj parenthesisX u comma Capital Phi closingkj parenthesis = l open parenthesis Capital Phi closing { a(χ , v) = χj ∈H1 (∆(A)):a, (s) ∂kv(s)dβ(s) (5.12) parenthesis .. for all Capital Phib in F sub 0 to the power of infinity period# b k=1 ∆(A) \noindentThanks toProposition .... open parenthesis 3 . 7 5\ periodh f i l l 7( closing it i parenthesis s clear comma that .... $a it ....{ i follows j }$ that may .... be u viewed i s the solution as a function of open in $ C ( ˆ{\ r u l e {3em}{0.4 pt }}\Omega ; X ˆ{ 2 , \ infty } { A } ) = $ parenthesis 5 period 8 closing parenthesis period .... Hence the1 theorem 1 where ba(., .) is the sesquilinear form on H#(∆(A)) × H#(∆(A)) given by follows period square \noindentAt the present$ K t ime ( comma ˆ{\ r for u l e each{3emi,j=1 1}{ lessZ 0.4 or pt equal}}\ j lessOmega or equal N; comma X ˆ let{ 2 , \ infty } { A } ) ) . $ \quad The result is that X 1 Equation: open parenthesisba(u, 5 v period) = 1 2 closingbaij( parenthesiss)∂ju(s) ∂iv ..(s bracelefttp-braceleftmid-braceleftbt)dβ(s), u, v ∈ H#(∆(A)). a-hatwide open parenthesis chi to the power of j comma v closing parenthesisN ∆(A) = sum sub k = 1 to the power of N integral sub Capital Delta open parenthesis \ centerline { $ \widehat{a} {\Omega } ( $ u $ , \Phi ) = l ( \Phi ) $ \quad f o r a l l A closing parenthesisFor obvious from reasons for all v , in H ( 5 sub . 1 hash 2 ) to uniquely the power determines of 1 open parenthesisχj. Let Capitalthen Delta open parenthesis A closing parenthesis$ \Phi closing\ in parenthesisF ˆ{\ infty to chi} to{ the0 } power. $ of j} in H sub hash to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesisZ closing parenthesisk=1 : hatwide-aZ comma to the power of kj open parenthesis s closing parenthesis \noindent Thanks to \ h f i l l ( 5 . 7 ) , X\ h f i l l i t \ h fj i l l follows that \ h f i l l u i s the solution of ( 5 . 8 ) . \ h f i l l Hence the theorem overbar partialdiff sub k vq openij = parenthesisbaij(s)dβ s closing(s) − parenthesisbaik( ds) beta∂kχ ( opens)dβ( parenthesiss), 1 ≤ i,s j closing≤ N. parenthesis where hatwide-a open parenthesis∆(A) period comma periodN ∆(A closing) parenthesis is the sesquilinear form on H sub hash to the power \noindent f o l l o w s $ . \ square $ of 1 open parenthesisIt can be Capital shown Delta that open the matrix parenthesis(qij A)1 closing≤i,j≤N parenthesishas the usual closing symmetry parenthesis and times ellipt H sub icity hash to the power of 1 open parenthesis Capital Delta open parenthesis A closing parenthesis closing parenthesis given by \ centerlineproperties{At the present ( proceed t ime as , in for each[ 2 6 $ ] ) 1 .\ Finallyleq j , the\ leq l imitN , function $ l e t } hatwide-au0 openin parenthesis ( 5 . 9 ) u comma i s v the closing ( unique parenthesis ) weak = sum solution from i of comma j = 1 to N integral sub Capital Delta open parenthesis A closing parenthesis a-hatwide sub ij open parenthesis s closing parenthesis partialdiff sub j u open parenthesis s closing \ begin { a l i g n ∗} parenthesis overbar partialdiff sub i v open parenthesisi,j=1 2 s closing parenthesis d beta open parenthesis s closing parenthesis comma u \{\widehat{a} ( \ chi ˆ{ Xj } ,∂ u0 v ) = \sum1 ˆ{ N } { k = 1 }\ int {\Delta comma v in H sub hash to the power of 1− open parenthesisqij Capital= finΩ, Delta u0 ∈ openH0 (Ω) parenthesis, A closing parenthesis closing parenthesis ∂xi∂xj period(A) }ˆ{ f o r a l l v \ inN H ˆ{ 1 } {\# } ( \Delta (A)) } {\ chi ˆ{ j } \ inFor obviousH ˆ{ 1 reasons} {\ comma# } ..( open\Delta parenthesis(A)): 5 period 1 2 closing parenthesis}\widehat uniquely{a} determines{ , }ˆ chi{ tokj the} power( of s j period ) \ overline {\}{\ partial } { k } v ( sas i ) s immediate d \beta by a( s imple s adaptation ) \ tag ∗{$ of ( the 5 analogous . 1 result 2 ) $ in} the periodic .. Let then setting ( see , e . g . , [ 2 6 ] ) . \endq sub{ a l ij i g = n integral∗} sub Capital Delta open parenthesis A closing parenthesis hatwide-a sub ij open parenthesis s closing parenthesis d beta open parenthesis s closing parenthesis minus sum from k = 1 to N integral sub Capital Delta open parenthesis A closing parenthesis\noindent a-hatwidewhere sub $ \ ikwidehat open parenthesis{a} ( s closing . parenthesis , . partialdiff)$ is sub the k chisesquilinear to the power of formon j open parenthesis $Hˆ{ s1 closing} {\# } parenthesis( \Delta d beta(A)) open parenthesis s closing\times parenthesisH ˆ{ comma1 } {\ 1 less# } or equal( i\ commaDelta j less( or equal A N ) period )$ givenby It can be shown that the matrix open parenthesis q sub ij closing parenthesis sub 1 less or equal i comma j less or equal N has the\ [ usual\widehat symmetry{a} and( ellipt u icity , v ) = \sum ˆ{ i , j = 1 } { N }\ int {\Delta ( A)properties}\widehat .. open parenthesis{a} { i proceed j } ( .. as s .. in ) .. open\ partial square bracket{ j 2} .. 6u closing ( square s bracket ) \ overline closing parenthesis{\}{\ partial period } { i } ..v Finally ( comma s ) .. the d .. l\ imitbeta .. function( .. s u sub ) 0 .. , in .. u open , parenthesis v \ in 5 periodH ˆ9{ closing1 } {\ parenthesis# } ( .. i s\ ..Delta the ( A)).open parenthesis\ unique] closing parenthesis weak solution of minus sum from i comma j = 1 to N q sub ij partialdiff to the power of 2 u sub 0 divided by partialdiff x sub i partialdiff x sub j = f in Capital Omega comma u sub 0 in H sub 0 to the power of 1 open parenthesis Capital Omega closing parenthesis comma \noindentas i s immediateFor obvious by a s imple reasons adaptation , \ ofquad the analogous( 5 . 1 result 2 ) .. uniquely in the periodic determines $ \ chi ˆ{ j } . $ \quad Let then setting open parenthesis see comma e period g period comma .. open square bracket 2 .. 6 closing square bracket closing parenthesis\ [ q { i period j } = \ int {\Delta (A) }\widehat{a} { i j } ( s ) d \beta ( s ) − \sum ˆ{ k = 1 } { N }\ int {\Delta (A) }\widehat{a} { i k } ( s ) \ partial { k }\ chi ˆ{ j } ( s ) d \beta ( s ) , 1 \ leq i , j \ leq N. \ ]

\noindent It can be shown that the matrix $ ( q { i j } ) { 1 \ leq i , j \ leq N }$ has the usual symmetry and ellipt icity

\noindent p r o p e r t i e s \quad ( proceed \quad as \quad in \quad [ 2 \quad 6 ] ) . \quad F i n a l l y , \quad the \quad l imit \quad f u n c t i o n \quad $ u { 0 }$ \quad in \quad ( 5 . 9 ) \quad i s \quad the ( unique ) weak solution of

\ [ − \sum ˆ{ i , j = 1 } { N } q { i j }\ f r a c {\ partial ˆ{ 2 } u { 0 }}{\ partial x { i }\ partial x { j }} = f in \Omega , u { 0 }\ in H ˆ{ 1 } { 0 } ( \Omega ), \ ]

\noindent as i s immediate by a s imple adaptation of the analogous result \quad in the periodic setting ( see , e . g . , \quad [ 2 \quad 6 ] ) . 1 34 .. G period NGUETSENG comma N period SVANSTEDT \noindent5 period 31 period 34 \quad .... ConcludingG . NGUETSENG remarks period , N ..... SVANSTEDT Each structure hypothesis on a sub ij open parenthesis 1 less or equal i comma j less or equal N closing parenthesis ex hyphen 1 34 G . NGUETSENG , N . SVANSTEDT \noindenthibited above5 . open 3 . parenthesis\ h f i l l Concluding s ee Examples remarks .. 5 period . \ 1h hyphen f i l l Each 5 period structure 7 closing hypothesis parenthesis can on be $reduced a { toi j open} ( 1 \5leq . 3 . Concludingi , j remarks\ leq N . )Each $ ex structure− hypothesis on aij(1 ≤ i, j ≤ N) ex - parenthesishibited 5 period above 6 closing ( s parenthesis ee Examples for a suitable 5 . 1 -W 5 to . 7 the ) can power be of reduced 1 comma to 2 hyphen( 5 . 6 ) for a suitable proper H hyphen algebra A period .. By way of i llustration comma the appropriate H hyphen algebras for Ex hyphen \noindentW hibited1,2− proper above H - algebra ( s ee ExamplesA. By way\quad of i5 llustration . 1 − 5 , . the 7 ) appropriate can be reduced H - algebras to ( 5 . 6 ) for a suitable amples ....for 5 Ex period - 1 comma .... 5 period 3 comma .... 5 period 4 .... and .... 5 period 5 .... are respectively the H hyphen algebra$ W ˆ{ of1 Example , 2 2 period} − 1$ 4 with proper Hamples− algebra 5 . 1 , $A 5 . 3 , . 5 $ . 4\quad and 5By . 5 way are of respectively i llustration the H -, algebra the appropriate of Example H 2− algebras for Ex − A sub 1. = 1 C 4 sub with per open parenthesis Y to the power of prime closing parenthesis comma Y to the power of prime = parenleftbig minus 1 divided by 2 sub comma 1 divided by 2 parenrightbig N minus 1 sub comma .... the H hyphen algebra A = AP sub R A = C (Y 0),Y 0 = (− 1 1 )N − 1 the H - algebra A = AP ( N ) for a suitable parenleftbig\noindent R1amples to theper power\ h f i of l l N5 parenrightbig . 1 , \2 ,h2 f.... i l l for5, a .suitable 3 , \ h f i l l 5 . 4 \ h fR i l lR and \ h f i l l 5 . 5 \ h f i l l are respectively the H − algebra of Example 2 . 1 4 with R open parenthesisR ( see subsection see subsection 2 . 2 3 period ) , the 3 closing H - algebra parenthesisA = commaB∞,per ....(Y )( the HExample hyphen algebra 2 . 1 A = 1 B ) sub, infinity comma per\noindent open parenthesisand$ the A Y{ closing1 } = parenthesis C { openper parenthesis} ( Y Example ˆ{\prime 2 ....} period) 1 .... , 1 closingY ˆ{\ parenthesisprime } comma= ( .... and− the \ f r a c { 1 }{ 2 } { , }\ f r a c { 1 }{ 2 } )N − 1 { , }$ \ h f i l lN the H − algebra $A = AP { R } ( R ˆ{ N } ) $ \ h f i l l for a suitable H hyphenH algebra - algebra A =A B= subB∞ infinity,R(R )( commaExample R parenleftbig 2 . 1 2R )to for the apower suitable of N parenrightbigR. For further open details parenthesis Example 2 .... period 1 2 closingsee [ 3 parenthesis 0 , 3 2 .... ] . for Thus a suitable, Σ R− periodconvergence .... For further theory details s eems to be the r ight tool that \noindentsee openi square s needed$R bracket ( to $ extend 3 0 see comma subsection .. 3 2 closing 2 square . 3 ) bracket , \ h f period i l l the H − algebra $A = B {\ infty , perThus} comma(homogenization Y Capital ) Sigma ($ theory hyphen Example2 beyond convergence\ h the f i ltheory usual l . 1 s periodic eems\ h f i to l l be setting1 the ) r , ight and\ h ftool ithereby l l thatand i brigdes the needed the to extend homogenizationgap between theory beyond classical the usual periodic periodic homogenization setting and thereby and brigde stochastic the homogenization . \noindentgap .... betweenForH − thealgebra .... sake classical of clearness $A .... periodic = we .... haveB homogenization{\ choseninfty a s imple ....,R and PDE ....} stochastic( to il R lustrate ˆ{ ....N homogenization} the) large ( $ part period Example2 \ h f i l l . 1 2 ) \ h f i l l for a suitable $ RFor the . $ sakeΣ−\convergence ofh f clearness i l l For we furtheris have destined chosen details to a s play imple in PDE homogenization to il lustrate the . large For part the homogenization by Capital SigmaΣ− convergence hyphen convergence of rather is destined sophisticated to play PDE in homogenization ’ s we refer period , e . g .. . For , to the [ 3homogenization 3 , 3 4 by , \noindentCapital Sigma2 4see ] . hyphen [ 3 0 convergence , \quad 3 of 2 rather ] . sophisticated PDE quoteright s we refer comma e period g period comma to open squareThus bracket $ , 3 3\ ..Sigma comma ..− 3$ 4 comma convergence .. 2 4 closingReferences theory square s bracket eems period to be the r ight tool that i s needed to extend References[ 1 ] G . Allaire , Homogenization and two - s cale convergence , SIAM J . Math . Anal . 23 ( 1 \noindentopen square992homogenization ) bracket , 1 closing square theory bracket beyond .... G the period usual Allaire periodic comma Homogenization setting and and thereby two hyphen brigde s cale the convergence comma SIAM J period Math period Anal period 23 open1 482 parenthesis – 1 5 1 8 . 1 992 closing parenthesis comma \noindent1 482 endash[ 2gap ] 1 N 5\ .1h Bakhvalov f8 i period l l between and G\ . Panasenkoh f i l l c l a , sHomogenization s i c a l \ h f i l l : Averagingp e r i o d i c processes\ h f i l l inhomogenization periodic media , \ h f i l l and \ h f i l l s t o c h a s t i c \ h f i l l homogenization . open square( Mathematics bracket 2 closing and its square Applications bracket ..: BVil N period 36 ) Bakhvalov, Kluwer Acad and G. Pub period . ,Panasenko Dordrecht ,comma 1 989 . Homogenization [ 3 ] A : Averaging processes\noindent in. periodic BensoussanFor the media sake, J comma . L .of Lions clearness and G . Papanicolaou we have chosen , Asymptotic a s Analysis imple PDE for Periodic to il Struc lustrate - tures the, large part open parenthesisNorth Holland Mathematics , Amsterdam and its , Applications 1 978 . [ 4 ] : BVil N . Bourbaki 36 closing , parenthesisInt e´ gration comma, Chap Kluwer . 1 - Acad 4 , Hermann period Pub period comma Dordrecht\noindent comma, Paris$ \ 1,Sigma 1 989 966 period . [ 5− ]$ N .convergence Bourbaki , Int ise´ gration destined, Chap to . 5 play , Hermann in homogenization , Paris , 1 967 . [ 6 . ]\quad N For the homogenization by $open\Sigma square. Bourbaki bracket− $ , 3Topologie convergence closing square G e´ n bracket ofe´ rale rather .., A Chap period sophisticated . V Bensoussan - X , Hermann comma PDE , Paris J ’ period s , 1 we 974 L refer period. [ 7 ] Lions , N e . Bourbaki . and g G . period , to Papanicolaou [ 3 3 \quad , \quad 3 4 , \quad 2 4 ] . comma Asymptotic, Topologie Analysis G e´ n fore´ Periodicrale , Chap Struc . hyphenI - IV , Hermann , Paris , 1 971 . [ 8 ] N . Bourbaki , Espaces \ centerlinetures commaVectoriels{ References North Topologiques Holland} comma, Chap Amsterdam . I - II , Hermanncomma 1 978, Paris period , 1 965 . [ 9 ] A . Bourgeat , A . Mikelic open squareand S bracket . Wright 4 closing , Stochastic square two bracket - s cale .. convergence N period Bourbaki in the mean comma and Int applications acute-e gration, J . comma Reine Angew Chap period 1 hyphen 4 comma\noindent Hermann. Math[ 1 comma . ] 456\ h fParis( i l 1 l 994G comma ) . , 1Allaire 9 1 – 966 5 1 period . [, 1 Homogenization 0 ] H . Bre ´ zis , Analyse and two Fonctionnelle− s cale, convergence Masson , Paris , , SIAM J . Math . Anal . 23 ( 1 992 ) , open square1 983 bracket . [ 1 1 5 ] closing J . Casado square - bracket Diaz , ..Two N period - s cale Bourbaki convergence comma for Int nonlinear acute-e Dirichlet gration comma problems Chap in period 5 comma Hermann\ centerline commaperforated{1 Paris 482 do comma−− - mains1 15 967 1, 8Proc period . } . Roy . Soc . Edinburgh Sect . A 130 ( 2000 ) , 249 – 276 . [ 1 2 ] open squareG . W bracket . Clark 6 andclosing L . square A . 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Papanicolaou , Sulla convergenza , diAsymptotic alcune succession Analysis for Periodic Struc − tures , North Holland , Amsterdam , 1 978 . open squaredi integrali bracket del 8 closing tipo dell square ’ area bracket, Rend .. N . periodMat . 8 Bourbaki( 1 975 comma ) , 277 Espaces – 294 . Vectoriels [ 1 5 ] J Topologiques . Dieudonne ´comma, Chap period I hyphen[ 4 ] II\quadEl commae´ mentsN Hermann . Bourbaki d ’ Analyse comma ,, Paris T Int . I comma I ,$ Chap\acute 1 . 965 XII{ periode -} XV$ , Gauthier gration - Villars , Chap , Paris . 1 ,− 14 968 , . Hermann [ 1 6 ] J ., Paris , 1 966 . [ 5 ] \quad N . Bourbaki , Int $ \acute{e} $ gration , Chap . 5 , Hermann , Paris , 1 967 . open squareDieudonn brackete ´, El 9 closinge´ ments square d ’ Analyse bracket, .. T . A VI period , Chap Bourgeat . XXII , comma Gauthier A - period Villars Mikelic , Paris , and 1 975 S . period [ 1 Wright comma Stochastic[ 6 ] \ twoquad7 ] hyphen NN . Dunford. s Bourbaki cale convergence and J ,. T Topologie . Schwartz in the mean , GLinear and $ \ Operatorsacute{e}, Part$ n I , Interscience $ \acute{ Pube} $ . , Inc rale . , New , Chap . V − X , Hermann , Paris , 1 974 . [applications 7 ] \quadYork comma ,N . Bourbaki J period Reine , TopologieAngew period G Math $ \ periodacute 456{e} open$ parenthesis n $ \acute 1 994{ closinge} $ parenthesis rale , Chap comma . 1 I 9− endashIV , Hermann , Paris , 1 971 . 5[ 1 period 8 ] \quad N . Bourbaki , Espaces Vectoriels1 957 . Topologiques , Chap . I − II , Hermann , Paris , 1 965 . [open 9 ] square\quad[ 1 8 bracket ]A R . . Bourgeat 1 D 0 . closing Edwards square , ,AFunctional . bracket Mikelic .. Analysis H and period S,Br New . acute-eWright York etzis , al comma Stochastic . , Holt Analyse - Rinehart twoFonctionnelle− - Winstons cale comma , 1 convergence Masson comma in the mean and p q Parisapplications comma965 1 983. [ 1 ,period 9 J ] . J Reine . J . F . Angew Fournier . and Math J . Stewart. 456 (, Amalgams 1 994 ) of , 1L 9and−− l5, Bull 1 . . Amer . Math . [open 1 0 square ] Soc\quad . bracket13 H( 1 . 1 985 Br 1 closing ) , $ 1\ –acute square 2 1 . { brackete} $ .. zis J period , Analyse Casado hyphen Fonctionnelle Diaz comma , .. Masson Two hyphen , Paris s cale , convergence 1 983 . for nonlinear[ 1 1 ]Dirichlet[\ 20quad ] problems AJ . .Guichardet Casado in perforated− , AnalyseDiaz do ,Harmonique hyphen\quad Two Commutative− s cale, convergence Dunod , Paris , for 1 968 nonlinear . [ 2 1 ] V Dirichlet . problems in perforated do − mainsmains comma,V Proc . Jikov Proc . , Roy Speriod . M . . Roy SocKozlov period . and Edinburgh Soc O . period A . Oleinik Sect Edinburgh , .Homogenization A Sect 130 period ( 2000 A of 130 Differential ) open , 249 parenthesis−− Operators276 2000 . and closing parenthesis comma 249[ endash1 2 ] Integral 276\quad period FunctionalsG .W. Clark, Springer and - L Verlag . A , . Berlin Packer , 1 994 , \ .quad [ 22 ]Two R− . Larsens cale , Banach homogenization Algebras , of implicit degenerate evo lution equationsopen squareMarcel , bracket J Dekker . Math 1 2 , closing New . Anal York square , . 1 bracket973 Appl . . .. 2G period 14 ( W1 997period ) Clark , 420 and−− L period438 . A period Packer comma .. Two hyphen s cale[ 1 homogenization 3 ] \quad G of .W. implicit Clark degenerate and evoL . lution A . Packer , Two − s cale homogenization of nonlinear degenerate evo lution equationsequations comma , J . J Math period . Math Anal period . Appl Anal . period 238 Appl ( 1 period999 ) 2 , 14 3 open 1 6 parenthesis−− 328 . 1 997 closing parenthesis comma 420 endash[ 14 438 ] \ periodquad E . De Giorgi , Sulla convergenza di alcune succession di integrali del tipo dell ’ area , Rend . Matopen . square 8 ( 1 bracket 975 ) 1 3 , closing 277 −− square294 bracket . .. G period W period Clark and L period A period Packer comma Two hyphen s cale[ 1 homogenization 5 ] \quad J of . nonlinear Dieudonn degenerate $ \acute evo lution{e} { , }$ El $ \acute{e} $ ments d ’ Analyse , T . I I , Chap . XII − XV , Gauthier − Villars , Paris , 1 968 . [equations 1 6 ] \quad commaJ J period. Dieudonn Math period $ \acute Anal period{e} Appl{ , period}$ El 238 $open\acute parenthesis{e} $ 1 999 ments closing d parenthesis ’ Analyse comma , T . 3 1VI 6 , Chap . XXII , Gauthier − Villars , Paris , 1 975 . endash[ 1 7 328 ] period\quad N . Dunford and J . T . Schwartz , Linear Operators , Part I , Interscience Pub . , Inc . , New York , open square bracket 14 closing square bracket .. E period De Giorgi comma Sulla convergenza di alcune succession di integrali del\ centerline tipo dell quoteright{1 957 area . } comma Rend period Mat period 8 open parenthesis 1 975 closing parenthesis comma 277 endash 294 period \noindentopen square[ bracket 1 8 ] 1\quad 5 closingR square . D . bracket Edwards .. J period , Functional Dieudonn acute-e Analysis sub comma , New El York e-acute et ments al . d ,quoteright Holt − AnalyseRinehart − Winston , 1 965 . comma[ 1 9 T ] period\quad I I commaJ . J Chap . F period. Fournier XII hyphen and XV J .comma Stewart Gauthier , Amalgams hyphen Villars of comma $ L ˆ Paris{ p } comma$ and 1 968 $ period l ˆ{ q } , $open Bull square . bracket Amer . 1 6Math closing . square Soc . bracket 13 (.. 1 J 985 period ) Dieudonn , acute-e sub comma El e-acute ments d quoteright Analyse comma1 −− T2 period 1 . VI comma Chap period XXII comma Gauthier hyphen Villars comma Paris comma 1 975 period open square bracket 1 7 closing square bracket .. N period Dunford and J period T period Schwartz comma Linear Operators comma\noindent Part I[ comma 20 ] Interscience\quad A . Pub Guichardet period comma , Inc Analyse period commaHarmonique New York Commutative comma , Dunod , Paris , 1 968 . [1 2 957 1 period ] \quad V . V . Jikov , S . M . Kozlov and O . A . Oleinik , Homogenization of Differential Operators and open square bracket 1 8 closing square bracket .. R period D period Edwards comma Functional Analysis comma New York et al\noindent period commaIntegral Holt hyphen Functionals Rinehart hyphen , Springer Winston− commaVerlag 1 965 , period Berlin , 1 994 . [open 22 ] square\quad bracketR . 1 Larsen 9 closing , square Banach bracket Algebras .. J period , Marcel J period DekkerF period Fournier , New York and J ,period 1 973 Stewart . comma Amalgams of L to the power of p and l to the power of q comma Bull period Amer period Math period Soc period 13 open parenthesis 1 985 closing parenthesis comma 1 endash 2 1 period open square bracket 20 closing square bracket .. A period Guichardet comma Analyse Harmonique Commutative comma Dunod comma Paris comma 1 968 period open square bracket 2 1 closing square bracket .. V period V period Jikov comma S period M period Kozlov and O period A period Oleinik comma Homogenization of Differential Operators and Integral Functionals comma Springer hyphen Verlag comma Berlin comma 1 994 period open square bracket 22 closing square bracket .. R period Larsen comma Banach Algebras comma Marcel Dekker comma New York comma 1 973 period Capital Sigma hyphen CONVERGENCE .. 1 35 \ hspaceopen square∗{\ f i bracket l l } $ 23\Sigma closing square− $ bracket CONVERGENCE .. D period\quad Lukkassen1 35 comma G period Nguetseng and P period Wall comma Two hyphen s cale convergence comma Int period J period Pure Appl period Math period comma Σ− CONVERGENCE 1 35 \noindent2 open parenthesis[ 23 ] 2002\quad closingD . parenthesis Lukkassen comma , G 35 . endash Nguetseng 86 period and P . Wall , Two − s cale convergence , Int . J . Pure Appl . Math . , [ 23 ] D . Lukkassen , G . Nguetseng and P . Wall , Two - s cale convergence , Int . J . Pure Appl 2open ( 2002 square ) bracket , 35 −− 24 closing86 . square bracket .. D period Lukkassen comma G period Nguetseng comma H period Nnang and . Math . , 2 ( 2002 ) , 35 – 86 . [ 24 ] D . Lukkassen , G . Nguetseng , H . Nnang and P . Wall , P[ period 24 ] Wall\quad commaD . Reiterated Lukkassen homogenization , G . Nguetseng of nonlinear , H . Nnang and P . 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Rajesh , Homogenization of a nonlinear degenerate parabolic Sigma - convergence of parabolic differential operators , Multiple s cales pro b - differentialdifferential equation equation comma ,Electron Electron period . J J period . Diff Diff .period Equns Equns , 1 comma 7 ( 200 1 7 open 1 ) parenthesis , 1 −− 1 200 9 1. closing parenthesis lems in b i omathematics , mechanics , physics and numerics , Gakuto Intern . Ser . Math . comma[ 26 1 ] endash\quad 1G 9 period . Nguetseng , Sigma − convergence of parabolic differential operators , Multiple s cales pro b − Sc . Appl . , 3 1 ( 2009 ) , 93 – 1 32 . [ 27 ] G . Nguetseng , A general convergence result for open square bracket 26 closing square bracket .. G period Nguetseng comma Sigma hyphen convergence of parabolic differential a functional related to the theory of homog - enization , SIAM J . Math . Anal . , 20 ( 1 989 ) , operatorslems in comma b i omathematics Multiple s cales pro , mechanics b hyphen , physics and numerics , Gakuto Intern . Ser . Math . Sc . 608 – 623 . [ 28 ] G . Nguetseng , Asymptotic analysis for a s tiff variational problem arising in Appllems in. b , i 3 omathematics 1 ( 2009 )comma , 93 mechanics−− 1 32 comma . physics and numerics comma Gakuto Intern period Ser period Math period mechanics , SIAM J . Math . Anal . 2 1 ( 1 990 ) , 1 394 – 1 4 1 4 . [ 29 ] G . Nguetseng , Sc[ period 27 ] \quad G . Nguetseng , A general convergence result for a functional related to the theory of homog − Almost periodic homogenization : asymptotic analysis of a s econd order enizationAppl period , comma SIAM 3 J 1 .open Math parenthesis . Anal 2009 . , closing 20 ( parenthesis 1 989 ) comma , 608 −− 93 endash623 . 1 32 period e l lip ti c equation , Preprint . [ 30 ] G . Nguetseng , Homogenization Structures and [open 28 ]square\quad bracketG . 27 Nguetseng closing square , bracket\quad ..Asymptotic G period Nguetseng analysis comma for A general a s tiff convergence variational result for problema functional arising related in mechanics , Applications I , Zeit . Anal . Anwend . 22 ( 2003 ) , 73 – 1 7 . [ 3 1 ] G . Nguetseng , toSIAMJ the theory .Math of homog . Anal hyphen . 2 1 ( 1 990 ) , 1 394 −− 1 4 1 4 . Mean value on locally compact Abelian groups , Acta Sci . Math . ( Szeged ) 69 ( 2003 ) 203 - 22 1 [enization 29 ] \quad commaG. SIAM\quad J periodNguetseng Math period , \quad Anal periodAlmost comma periodic 20 open homogenization parenthesis 1 989 closing : \quad parenthesisasymptotic comma analysis 608 \quad of a s econd order . [ 32 ] G . Nguetseng , Homogenization Structures and Applications II , Zeit . Anal . Anwend endash 623 period . 23 ( 2004 ) , 483 – 508 . [ 33 ] G . Nguetseng and H . Nnang , Homogenization of nonlinear \noindentopen squaree bracket l lip28 ti closing c equation square bracket , Preprint .. G period . Nguetseng comma .. Asymptotic analysis for a s tiff variational monotone operators beyond the periodic s e tting , Electron . J . Diff . Equns 36 ( 2003 ) , 1 – 24 . problem[ 30 ] arising\quad in mechanicsG. \quad commaNguetseng , \quad Homogenization Structures \quad and Applications I , \quad Z e i t . \quad Anal . \quad Anwend . \quad 22 [ 34 ] G . Nguetseng and J . L . Woukeng , Sigma - convergence of nonlinear parabolic operators , (SIAM 2003 J ) period , 73 Math−− 1 period 7 . Anal period 2 1 open parenthesis 1 990 closing parenthesis comma 1 394 endash 1 4 1 4 period Non - linear Anal . 66 ( 2007 ) , 968 – 1 4 . [ 35 ] E . Sanchez – Palencia , Nonhomogeneous [open 3 1 square ] \quad bracketG 29. Nguetseng closing square , bracket Mean .. value G period on .. locally Nguetseng compact comma .. Almost Abelian periodic groups homogenization , Acta Sci : .. asymptotic. Math . ( Szeged ) 69 Media and Vibration Theory , Springer - Verlag , analysis( 2003 .. of ) a 203 s econd− 22 order 1 . Berlin , 1 980 . [ 36 ] S . Spagnolo , Sulla convergenza di s o luzioni di equazioni paraboliche e e l [e 32 l lip ] ti\ cquad equationG . comma Nguetseng Preprint , period\quad Homogenization Structures and Applications II , Zeit . Anal . Anwend . \quad 23 litiche , Ann . Scul . Norm . Sup . Pisa 22 ( 1 968 ) , 571 – 597 . [ 37 ] S . Spagnolo , Convergence (open 2004 square ) , bracket 483 −− 30508 closing . square bracket .. G period .. Nguetseng comma .. Homogenization Structures .. and Applications in energy for e l lip ti c operators , in Numerical Solutions of Partial Differential Equations III , I comma[ 33 ] ..\ Zeitquad periodG . .. Nguetseng Anal period ..and Anwend H .Nnang period .. , 22 Homogenization of nonlinear monotone operators beyond the Acad . Press , New York , 1 976 . [ 38 ] L . Tartar , Cours Peccot , Colle ` ge de France , Paris , periodicopen parenthesis s e tting 2003 closing , Electron parenthesis . J comma . Diff 73 endash . Equns 1 7 period 36 ( 2003 ) , 1 −− 24 . 1 977 . [ 39 ] V . V . Zhikov , On an extension and application of the two - s cale convergence [open 34 ] square\quad bracketG . 3 Nguetseng 1 closing square and bracket J . L .. . G Woukeng period Nguetseng , Sigma comma− convergence Mean value on of locally nonlinear compact parabolic Abelian groups operators , Non − method , Sb . Math . 191 ( 2000 ) , 973 – 1 0 1 4 . [ 40 ] V . V . Zhikov , On two - s cale commalinear Acta Anal Sci period . 66 Math ( 2007 period ) open, 968 parenthesis−− 1 4 Szeged . closing parenthesis 69 convergence , J . Math . Sc . , 120 ( 2004 ) , 1 328 – 1 352 . [ 41 ] V . V . Zhikov and E . V . [open 35 ] parenthesis\quad E. 2003\quad closingSanchez parenthesis−− 203Palencia hyphen 22 ,1 period\quad Nonhomogeneous \quad Media \quad and \quad Vibration \quad Theory , \quad Springer − Verlag , Krivenko , Homogenization of s ingularly perturbed e l lip tic operators , Matem . Zametki 33 : 4 ( open square bracket 32 closing square bracket .. G period Nguetseng comma .. Homogenization Structures and Applications II 1 983 ) , 571 – 582 , Engl . transl . : Math . Notes 33 ( 1 983 ) , 294 – 300 . comma\noindent Zeit periodBerlin Anal , period 1 980 Anwend . period .. 23 1 University of Yaounde 1 , Department of Mathematics , P . O . Box 8 1 2 [open 36 ] parenthesis\quad S 2004 . Spagnolo closing parenthesis , Sulla comma convergenza 483 endash di 508 s period o luzioni di equazioni paraboliche e e l litiche , Ann . Scul . Yaounde , Normopen . square Sup bracket . Pisa 33 22 closing ( 1 square 968 ) bracket , 571 ..−− G period597 Nguetseng . and H period Nnang comma Homogenization of nonlinear Cameroon . monotone[ 37 ] operators\quad S beyond . Spagnolo the , Convergence in energy for e l lip ti c operators , in Numerical Solutions of Partial E - mail address : n guetsen g @ uy 1 . unine t . c m Differentialperiodic s e tting Equations comma Electron III , period Acad J . period Press Diff , period New York Equns , 36 1 open 976 parenthesis . 2003 closing parenthesis comma 1 2 University of Gothenburg , Department of Mathematical Sciences , SE - endash[ 38 24 ] period\quad L . Tartar , Cours Peccot , Coll $ \grave{e} $ ge de France , Paris , 1 977 . 4 1 2 [open 39 ] square\quad bracketV . V34 closing. Zhikov square , \ bracketquad On .. 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V period V period Zhikov and E period V period Krivenko comma Homoge- nization of s ingularly perturbed e l lip tic operators comma Matem period Zametki 33 : 4 open parenthesis 1 983 closing parenthesis comma 571 endash 582 comma Engl period transl period : Math period Notes 33 open parenthesis 1 983 closing parenthesis comma 294 endash 300 period 1 University of Yaounde 1 comma Department of Mathematics comma P period O period Box 8 1 2 Yaounde comma Cameroon period E hyphen mail address : n guetsen g at uy 1 period unine t period c m 2 .. University of Gothenburg comma .. Department of Mathematical Sciences comma .. SE hyphen 4 1 2 96 Gothenburg comma Sweden period E hyphen mail address : n i l s s at c h al m ers period se