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Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem

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Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem In a number of places in Chapters 2, 3 and 5, we employ simple forms of the Marcinkiewicz interpolation theorem. The purpose of this appendix is to present for the reader's convenience statements and proofs of the theorems involved and to explain the concepts associated with them. More general versions of the Mar­ cinkiewicz theorem can be found in [9], Section 13.8 and [33], Appendix B. Let us agre'e that the measure spaces appearing below are always (J-finite. A.I. The Concepts of Weak Type and Strong Type A.I.t. Strong type. Let (M, JIt, 11) and (N, .AI, v) be measure spaces and p an index in the range [1, ex)]. Suppose D is a subset of V(Il) and T is a mapping from D into the space of complex-valued measurable functions on N, or the space of nonnegative extended-real-valued measurable functions on N, or the space of equiv­ alence classes of one of these. We say T is of strong type (p, p), or simply of type (p, p), on D if there is a constant B such that (1) for allfin D. The smallest number B for which (1) holds is then termed the (p, p) norm of Ton D and is denoted II TIIp,p, when D is understood. Even if there may not exist a finite constant B for which (1) holds, it is customary to define (provided of course that D #- {On. So we may say that T is of type (p, p) on D if and only if IITllp,p < 00. A.t.2. Remarks. (i) In practice, D is usually a linear subspace of V(Il) , T takes its values in the space of complex measurable functions on N, and is linear. (ii) Our concern in the text is mostly with operators T of the form T.p intro­ duced in 1.2.2 and 2.4.1. The operators T.p are viewed as having the initial domain L2(G) and range in L2(G), G being an arbitrary LCA group. Our main interest is in knowing whether, when 1 :::;; p :::;; 00 and T.p is restricted to D = L2 n V(G), 178 App:mdix A. Special Cases of the Marcinkiewicz Interpolation Theorem T", is of type (p, p) on D; in several instances, it is the value of II T",llp,p which is more important. (iii) The definition of strong type is clearly D-dependent, in general. However, in most practical instances, this poses no difficulty. To illustrate the point, suppose cJ> E ,!l'OO(X), X being the dual group of G, T = T", and T is of type (p, p) on D. Suppose that D is a linear subspace of L2 n U(G) and that for every I in L2 n U(G) there exists a sequence (/,,) extracted from D such that lim II/" - 1112 = 0 and lim II/"Ilp ~ 1I/11p. (2) 1 (D = L n L 00 (G) for example). Then if (1) holds for every lin D, it continues to hold for every lin L2 n U(G). To see this, suppose IE L 2 n U(G) and that (/,,) is as above. Since /" -+ I in L2(G) and every operator T", is continuous on L2(G) (cf. 1.2.2), it follows that T/" -+ Tlin L2(G). Hence there is a subsequence (T/,,) which converges pointwise a.e. to Tf Now II Tilip ~ lim inf II T/")Ip (3) j-+ 00 by Fatou's lemma if p < 00, and trivially otherwise. Since T is of type (p, p) on D, (4) for allj; combining (2), (3) and (4), we deduce that Tis of type (p,p) on L2 n U(G). In the same way, Tis continuously extendable into an operator of type (p,p) on U(G). A.l.3. Weak type. Let T and D be as in A.U, and denote by A.T! the dis­ tribution function of ITII; that is, define, for t > 0, A.T/t) = v({y E N: ITI(y) I > t}). If p < 00, we say Tis 01 weak type (p, p) on D if there is a nonnegative real number A such that (5) for allfin D and all t > O. If there exist such numbers A, there is a smallest, called the weak (p, p) norm 01 T on D. If no such number exists, the weak (p, p) norm of Ton D is set equal to 00. The mapping T is said to be 01 weak type (00, 00) on D if and only if it is of type (00,00) on D; its weak (00,00) norm is declared to be the same as its (00, 00) norm. It is very simple to see that a mapping T of type (p, p) on D is also of weak type there, but the converse is false, unless of course p = 00. A.l.4. Remark. The choice of D is again to some extent immaterial. For A.2. The Interpolation Theorems 179 instance, suppose that p E [1, (0), T = Tq" D is as in A. I .2(iii) and that (5) holds for all f in D and all t > O. Then (5) holds for all f in L 2 (") U(G). To see this, adopt the notation of A. I .2(iii). Then {y: ITf(y) I > t} ~ u n {y: ITf,,/y) I > t} i j~j and hence ATlt) ~ lim v(n {y: ITf,,/y) I > t}) i-+oo j~i ::>; liminfv({y: ITf,,/y) I > t}) j-+ 00 ::>; li~ inf APt-PIIf,,)I~ J-+ co At this point, it is possible to go one step further and extend Tfrom L2 (") U(G) into a mapping from LP(G) into the set of classes of measurable functions in such a way that (5) continues to hold for every fin U(G). For, givenfin U(G), select any sequence (f,,) from L2 (") U(G) converging in U to! Write gn for any function of the class Tf". Then (5) shows that the sequence (gn) is Cauchy in measure and therefore converges in measure to some function g. It also follows from (5) that the class of g does not depend on the choice of the sequence (f,,) (provided f" -+ f in U(G), of course). So we may define Tfto be the class of g. Once again, there is a subsequence (gn) converging a.e. to g and so the same argument as before leads to (5). It follows from (5) that Tf, although it may not belong to U(G), does belong locally to Lq(G) for every q < p ([9], Exercise 13.16). A.2. The Interpolation Theorems Letfbe a measurable function and t > O. Denote by ft andr the following func­ tions. ft(x) = {f(oX) if If(x) I ::>; t otherwise ff(x) if If(x) I > t rcx) = \ 0 l otherwise. With this notation fixed, we can state and prove the first theorem. A.2.t. Theorem (Marcinkiewicz). Suppose that r E (1, (0), D is a linear sub­ space of L 1 (") L'(M) and T is an operator mapping D into the set of equivalence 180 Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem classes of complex measurable functions or of nonnegative extended-real-valued measurable functions on N. Assume that D and T satisfy the following conditions. (i) !ffE D and t > 0, thenfr andp are in D. (ii) Tis subadditive in the sense that IT(f + g)1 ~ ITfl + ITgl for f and g in D. (iii) T is of weak type (I, 1) on D with weak (I, 1) norm at most AI' so that (I) for f in D and t > o. (iv) Tis of weak type (r, r) on D with weak (r, r) norm at most A" so that (2) for fin D and t > o. Suppose that 1 < p < r. Then T is of type (p, p) on D and (3) for all fin D, where (4) In other words, the (p, p) norm of T on D is at most Ap. where Ap is given by (4). Proof SupposefE D and t > o. Sincef = fr + p, the subadditivity of T and (i) show that ITfl ~ ITfrl + ITPI, whence it follows that ATAt) ~ very: ITP(y) I > t12}) + very: ITfr(y) I > tI2}). Applying (I) and (2) top andfr respectively, we deduce that 1 r r ATAt) ~ 2A l t- L'P' dl1 + (2ArYt- J)frl dl1 = 2AI r Ifldl1 + 2rA~t-r r Iflrdll. (5) J{x: If(x)1 >t) J{x: If(x>!';;t} Now (6) A.2. The Interpolation Theorems 181 and so we deduce from (5) that + roo tP-l{2'A~t-' rill' dP,}dt Jo J{x: If(x)J .. tl = 2AI rtP-2{LI/(X)Icf>(X, t)dP,(X)}dt + 2'A~ rtP-I-r {L I/(x)I' !/lex, t) dp,(x) }dt, (7) where cf> is the characteristic function of the set E = {(x, t): I/(x) I > t} £ M x (0, (0) and !/l is the characteristic function of the set F = {(x, t): I/(x) I ~ t} £ (M x (0, oo»\E. If (Sft)ft" I is an enumeration of the positive rationals, E = U ({x: I/(x) I > Sft} x (0, Sft», n~l which shows that E is measurable in the pair of variables. The same is therefore true of F and so, by the Fubini theorem (recall that Mis u-finite) we may invert the order of the integrations in (7) to conclude that p-1IlTIII: ~ 2Al L{J~f(X)1 tP-2 dt } I/(x) I dp,(x) + 2r A~ r {rOO t p - I -, dt} I/(x)I' dJl.(x) JM Jlf(X)1 = 2AI L(p - l)-II/(x)IP-II/(x)1 dp,(x) + 2'A~ L(r - p)-ll/(x)lp-rl/(x)I' dp,(x), which is equivalent to (3) and (4).
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