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). 0 The second case of the Marcinkiewicz theorem deals with operators simulta• neously of weak types (r, r) and (00, (0), where 1 ~ r < 00.
A.2.2. Theorem (Marcinkiewicz). Suppose that 1 ~ r < 00, and that D and T satisly the assumptions in the statement 01 Theorem A.2.l save lor condition (iii). In place 01 (iii), assume that 182 Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem
(iii') Tis of (weak) type (00, (0) on D, with (weak) (00, (0) norm at most A oo ' so that
II Tfll 00 ::;; A 00 Ilfll 00 (8) for all fin D. If r < p < 00, then T is of type (p, p) on D, and
(9) for all fin D, where
P2PArAP-r AP _ r 00 p - (p - r) .
Proof Suppose fED and t > O. We may assume that A 00 > 0 and so write
f - Jt/2A",I' + f t/ 2A ", .
The condition (8) shows that
(10)
It follows from the subadditivity of T and (i) that
and then from (10) that
AT/t) ::;; v({y: ITJ./2Ajy)1 > tI2}) + v({y: ITP/2A"'(y)1 > tI2})
= v({y: ITP/ 2A"(y)1 > tI2}). (11)
By combining (11) and (2), we see that
AT/t) ::;; A~(tj2)-rIIP/2A"II~
= 2rA~t -r r Iflr d/l. J{x: If(x)l >t/2A.,j
Using once more the formula (6), we deduce that
! II Tfll~ ::;; roo t p - 1 {2r A~t -r r If I' d/l} dt p Jo J{x: If(x)l >t/2A.,j
= 2'A~ p - r 1 (12) 00 t - {i If I' d/l }dl. 1o {x: If(x)1 >t/2A.,j A.3. Vector-Valued Functions 183
Now apply a Fubini-type argument again to (12), as in the final stages of the proof of A.2.1. The conclusion is that
1 2Aro lf(X)1 -Ii TIII~ ::::; 2' A~ J l/(x)l' J, tP-,-l dt dJ1(x) p M 0
= 2' A~(2A",)P-' J I/(xW dJ1(x) (p - r) M o
A.3. Vector-Valued Functions
The ideas and results of A.I and A.2 apply with only notational changes to the cases in which complex-valued or extended-real-valued functions are replaced by vector• valued functions of the type discussed in Chapter 3. In these cases, T is assumed to map (suitably restricted) functions with values in one Hilbert space into functions or equivalence classes of functions with values in a second Hilbert space. In formu• lating the concepts of weak type and strong type, absolute values are replaced by the appropriate Hilbert space norms. We leave the reader to write down the trans• lations of the definitions and the theorems and in particular to check that the proofs of the Marcinkiewicz theorems given in A.2.l and A.2.2 go through for vector• valued functions with no more than the obvious notational changes. Appendix B. The Homomorphism Theorem for Multipliers
Let G and H be LeA groups with duals X and Y respectively. Suppose that n is a continuous homomorphism of Y into X and it. its dual homomorphism from G into H, defined by the requirement that
it.(x)(y) = x(n(y))
for y in Yand x in G. Notice that (it.)" = n. Our aim in this appendix is to give a self-contained proof of what we have called the homomorphism theorem for multipliers. While elementary in the strict sense of the word, the proof is nonetheless quite intricate.
Theorem. Suppose that I ~ p ~ 00, 4> E MiX) and 4> is continuous. Then
4> 0 n E Mp(Y) and
(1)
B.I. The Key Lemmas
The first two lemmas are purely technical, involve only standard procedures, and amount to showing that it is en.ough to prove (1) for very "good" functions 4>. B.1.1. Lemma. In order to prove the theorem it suffices to show that if l/I E Mp n Cc(X), then l/I 0 n E Mp(Y) and
Proof Suppose that 4> E Mp n C(X) and let (ka) be a net of functions on G with the following properties:
(i) ka E L l(G) and Ilkall i ~ I; (ii) ka E CC
If the hypotheses of the lemma are satisfied, then by (ii),
Therefore, if hand k are in Ll(H) and have transforms in CcCY),
(2)
Since 7r is continuous, and ¢ka ~ ¢ locally uniformly (see (iii)), it follows that
(¢ka) 0 7r ~ ¢ 0 n locally uniformly. Moreover,
by (i); and Ii k ELI (Y). So the left side of (2) tends to
\t ¢ 0 7r(y)li(y)k(y) dY\
as rx ~ 00; therefore the result follows from 1.2.2 if we take the limit on rx in (2). 0 B.1.2. Lemma. The theorem will be established if it is shown that
whenever ¢ E 'iJe tl Ll(X).
Proof Notice that, in any case, if ¢ E 'iJLI(X), then ¢ 0 n E 'iJM(Y) = MI(y) ~ M p( Y). We prove the present lemma by using 8.1.l. To this end, suppose tfJ E Mp tl Cc(X) and let (Fa) be an approximate identity on X consisting of functions in L 1 tl L OO(X) for which II Fa. II 1 ~ 1. Then (i) tfJ * Fa. E 'iJLl tl Ll(X); and (ii) tfJ * Fa. ~ tfJ uniformly on X. We claim that, furthermore, (iii) IltfJ * Fallp,p ~ IltfJllp,p' To see this, observe that by 1.2.2, it suffices to prove that
(3)
wheneverfand g are integrable and have compactly supported Fourier transforms. 186 Appendix B. The Homomorphism Theorem for Multipliers
By I.2.2(iii) again, however, since t/J E Mp(X),
More importantly,
for all XO in X. Hence
and by using Fubini's theorem, we conclude finally that
This is just (3); so (iii) holds. To finish off the proof, suppose that (1) holds for all ¢ in ljL 1 II L leX). By (i) and (iii),
Consequently, if hand k are integrable on H and have transforms in Cc( Y), 1.2.2(iii) shows that
(4)
Now notice that, by (ii) and the continuity of TC, (t/J * Fa) 0 TC ~ t/J 0 TC uniformly on Y; moreover, hk E C/Y). So we can deduce, by taking the limit on IY. in (4), that
By 1.2.2(iii), this completes the proof. 0 The next lemma establishes a generalised Parseval formula. B.1.3. Lemma. Suppose that ¢ E ljLl II Ll(X), and EE ljLl II Ll(H). Then
JG E 0 ft(x)c$(x) dx = t ¢ 0 TC(I')£(I') dl'. (5) B.2. The Homomorphism Theorem 187
Proof Let j1 be the bounded measure on X such that
j1(S) = £(-"1) dy. J,,-.(S)r
It is a simple matter to check that
(6)
for say all bounded Borel functionsf on X. In particular if x E G andf(x) = X(x), we deduce from (6) that
flex) = t n(y)(x) £( - "I) dy
= Jy y(ft(x» £( -"I) dy
= E(ft(x» by the definition of ft and the inversion formula. So the left side of (5) is just
t flex) c/J(x) dx, which, by the inversion formula again, is equal to
Yet, by (6),
t = Jy B.2. The Homomorphism Theorem B.2.1. Theorem. Let X and Y be LeA groups and n a continuous homomorphism from Y to X. Then if 1 ~ p ~ 00 and (1) Proof By Lemma B.l.2, it will suffice to prove the inequality (1) for functions ¢ which are integrable and have integrable transforms. Let ¢ be such a function. Then by 1.2.2(iii), (1) is equivalent to the inequality (2) for hand k integrable functions on H with compactly supported Fourier transforms. We proceed to establish (2). By Lemma B.l.3, t ¢ a n(y)h(y)k;(y) dy = t (h * k) a ft(x)$(x) dx. (3) Since cl> E LI(G), we may, given E > 0, choose a compact set Kin G so that (4) An examination of the proof of Theorem (31.37) of [20] shows that we can choose functions f and 9 in L I 1\ L 00 (G) so that (i) 1;;;. f * 9 ;;;. °and f * 9 = 1 on K; (ii) Ilfllpligli p'::::; 1 + E. In order to prove (2), it will suffice to prove that For if (5) holds, we conclude from (4) and (i) that (6) Since E is arbitrary, (2) follows from (3) and (6). In establishing (5), the key step is the following integral expression for the pointwise product (h * k) a ft.f * 9 (h * k) a ft(x)f * g(x) = Iff [(t"uh) a ft.f] * [(Luk) a ft. g](x) dx. (7) In (7), the function t"uh is the u-translate of h defined by the formula t"uh(x) = hex - u). B.2. The Homomorphism Theorem 189 The equality (7) is established as follows. By the translation-invariance of the Haar measure on H, h * k(ft(x»f * g(x) = t h(ft(x) - u)k(u) du t f(x - y)g(y) dy = IG IH h(ft(x) - ft(y) - u)k(u + ft(y»f(x - y)g(y) dy duo (8) The right side of (8) can be rewritten and, by the Fubini-Tonelli theorem, this is the same as IH fa [(-ruh) 0 n.f](x - Y)[(Luk) 0 n. g](y) dy du = t [(-ruh) 0 n.f] * [(Luk) 0 n. g](x) duo So (7) is established. Return now to the left side of (5) and write it which, by the Fubini-Tonelli theorem, is By using the Parseval formula, this can be written which is at most By the definition of 11¢llp,p, this last expression is at most 190 Appendix B. The Homomorphism Theorem for Multipliers which, by Holder's inequality, is bounded above by /P 11cf>lIp,p( t II(ruh) a ft.fll~ du riP (t II(Luk) 0 ft. gll~: dUr ' = 11cf>llp,p( t t Ih(ft(x) - u)f(x)IP dx dU) lip (t t Ik(n(x) + u)g(x)IP' dx dU) lip' /P = 11cf>llp,p( t t Ih(ft(x) - u)f(x)IPdu dXr (t t Ik(n(x) + u)g(x)IP' du dxYIP' = 1Icf>llp,pllfllpllhllpllkllp,llgllp' :::;; (l + e)IIcf>llp,pllhllpllkll p' by Fubini's theorem and (ii), provided of course that 1 < p < 00. In case p = 1 or p = 00 the reasoning is quite analogous and leads to the estimate (5) again. 0 B.2.2. Corollary. Let XI and X 2 be LeA groups and cf> an element of MiXI), where 1 :::;; p :::;; 00. Then the function cP on Xl x X 2 defined by the formula belongs to MiXI x X 2 ), and Proof Notice that the crux of the matter is that cf> need not be continuous. The proof consists in combining Theorem B.2.1 with a standard regularisation procedure, as follows. Let 1C I be the canonical projection of XI x X 2 onto XI' so that cP = cf> 0 1C I' Denote by Gi the dual group of Xi (i = 1, 2). We wish to prove that if hand k are functions on GI x G2 with compactly supported Fourier transforms, then (9) Let Kbe the compact support of Tif{. and write KI = nl(K). It follows from (20.15) of [20] that, for every e > 0, there exists a functionfin Ll(X!) such that IIflll = 1 (10) and (11) Now by virtue of (10) it is also the case that B.2. The Homomorphism Theorem 191 and (12) For if u and v are integrable functions on G 1 whose Fourier transforms have com• pact supports, then IIx I ¢ * f(X)u(X)D(X) dX I = IIx .f(x') Ix I ¢(X - X')u(X)D(X) dX dx'i ~ Ix, If(x') II Ix, ¢(X)u(X + X')D(X + X') dxl dX' ~ II¢IIp,p 1If(x')1 IIX'ullpllX'vll p' dx' XI because of the transIation-invariance of Haar measure and (10). Since (10) implies that II¢ *flloo ~ II¢IIoo, we can assert, thanks to (II) and (12), that there is a sequence (¢i)';' of functions on G such that (i) each ¢ i is continuous; (ii) ¢i E Mp(X1) and (13) (iii) II¢illoo ~ II¢IIoo; (14) and (iv) ¢i -+ ¢ a.e. on K!. Returning now to the proof of (9) we notice that, by Fubini's theorem and (iv), on K. Yet by Theorem B.2.1 (which applies here because of (i» and (13) (15) Now apply the dominated convergence theorem to (15), taking note of (14). 0 Remarks. The idea of using the integral representation (7) for the pointwise product of convolutions in establishing results about multipliers was first intro• duced by Herz [19]. It has been subsequently highly developed and used to great effect by many authors. See, for instance, [27], [28] and [29], [14] and [15], and most recently [8]. All of these authors use systematically the properties of the space Ap introduced by Figa-Talamanca [13] (and its variants); in particular, the fact that 192 Appendix B. The Homomorphism Theorem for Multipliers the dual of Ap is Mp. We have deliberately avoided introducing the space Ap since our aim here has been more modest. While the framework of the duality between the spaces Ap and Mp is undeniably useful and suggestive of ideas, it seemed valuable, and in the present context more appropriate, to give a proof of the homo• morphism theorem which relies on little more than the Parseval formula and the Fubini-Tonelli theorem. Appendix C. Harmonic Analysis on [!Jl2 and Walsh Series on [0, 1] The fact that harmonic analysis on the Cantor group 1012 is "the same as" the theory of Walsh series on [0, I] is well known to all practising harmonic analysts. Since, however, it is difficult to cite a reference where the appropriate identifica• tions are carried out in detail, it seems worthwhile to carry out some of the details for the sake of beginners unfamiliar with this piece of folklore. The Rademacher functions ro, r» ... on [0, I] are defined by the formulas ro(t) == I; and for j > °and t not a dyadic rational, r/t) = sgn sin(2j n:t); rj is extended to all of [0, 1] by requiring that it be right-continuous at each dyadic rational in [0, 1] and ieft-continuous at 1. The set of functions {r j } is orthonormal on [0, 1] with Lebesgue measure. For clearly and if j > k, then on each of the dyadic intervals where r k is constant, rj takes the value 1 on half the set and - 1 on the other half (measurewise). So it is evident that However, the Rademacher system is not complete. The easiest way to see this is to check that the function r 1r2 is orthogonal to all the Rademacher functions. The characters of 1012. The Cantor group 1012 is defined in the introduction to Chapter 4. To each character X of 1012 corresponds a unique element a = (a)'f of the weak direct product 194 Appendix C. Harmonic Analysis on [)2 and Walsh Series on [0, I] The value of X at the point x of [Dz is given by the formula (1) Notice that the series appearing in the exponent in (1) converges since it is a finite series: at most finitely many of the entries aj are nonzero. Conversely, each element a of IliO')Z(2) determines a character of [Dz via the formula (I). Notice that the group [)z = Ili"'Z(2) is generated by the elements Po = (0,0, ... ) and (j ~ 1). (2) If we agree to write a(x) in place of the left side of (\) and pix) for the value at x of the character determined by Pj' then (3) where aN is the last nonzero entry in a. Integration on [DZ" We have not yet given an explicit construction of the Haar measure on [Dz. This we now do. In the course of the construction we show that [Dz is essentially Borel isomorphic to [0, I]. The group [Dz is of course not homeo• morphic to [0, I]. Let S be the countable set of points (x) in [Dz with the property that Xj = from a certain stage on. Denote by IjJ the following mapping of [Dz into [0, I]: 00 ljJ(x) = I x)2j. j=1 Then IjJ is a continuous mapping of [Dz onto [0, I], but it is not one-one. In fact, every dyadic rational in (0, I) has two pre-images under 1jJ, one corresponding to the terminating dyadic expansion, the other to the repeating expansion. To get over this minor problem, remove from [Dz all the points of S. Then 1jJ, restricted to [Dz\S, is a one-to-one continuous mapping of [D2\S onto [0, I). For each positive integer N, denote by GN the closed subgroup of [D2 consisting of those elements having °in the first N places. The mapping IjJ carries the 2N cosets of GN onto the dyadic intervals of length r N: [0, 1/2N], [1/2N, 2/2N], ... [(2N - l)/2N, I]. If U is a coset of GN in [Dz, IjJ maps U\S to an interval of the form [r/2N, (r + 1)/2N), °~ r < 2N; since every open set in [Dz is a countable union of cosets of the groups GN (N ~ I), it follows that IjJ maps open sets of [Dz\S onto Borel sets in [0, I). The inverse mapping IjJ - I carries open sets to open sets. There• fore, by a standard argument ([34], Theorem 1.12), IjJ is a Borel isomorphism of [Dz\S and [0, I). Appendix C. Harmonic Analysis on [)2 and Walsh Series on [0, 1] 195 Let BB([D 2 \S) be the Borel cr-alge bra on II} 2 \S, and m the ordinary Lebesgue measure on [0, 1]. Define the measure /l' on BB(1I}2\S) by the rule /l'(E) = m(t/J(E)). Now extend /l' to a function on BB(1I}2) by agreeing that S is null, viz. set /leE) = /l'(E\S). Since E\S is Borel in II} 2\S whenever E is Borel in 1I}2' /l is well-defined. It is routine to check that /l is a Borel measure on []) 2' Since t/J carries S to a countable and hence null subset of [0, 1], we see that /l(E) = m(t/J(E)) (4) for all Borel sets E in 1I}2' Now complete (/l, BB) ([34], Theorem 1.36) and check that the relation (4) holds true for all E in the completion. We continue to use the letter /l to denote the complete measure. We claim that /l is the normalised Haar measure on II} 2' To verify this claim, we have to show that /l is regular and translation-invariant. The measure f..l assigns finite mass to every Borel set; every open set in 1I}2 is a countable union of compact sets. A general theorem of measure theory ([34], Theorem 2.18) shows that f..l is regular both on BB and on the completion. As to translation-invariance, if E is an open set in 1I}2, E is a countable, pairwise disjoint union of co sets of the groups (GN)'f; and for a coset x + GN , Consequently, /l(E + x) = /leE) whenever E is open, and by regularity, this rela• tion continues to hold for arbitrary measurable E. We have therefore set up an "identification" of (1I}2, /l) and ([0, 1], m). As a consequence the Lebesgue spaces U(1I}2) and U([O, 1]) are identified. Characters on 1I}2 and Walsh functions. If (Pn)O' are the characters of 1I}2 defined in (2) and which generate 1D2' then the n-th Rademacher function rn and the character Pn are related by the formula Pn(x) = rn 0 t/J(x) (5) If a = (a) is a character of 1I}2' then (3) and (5) show that a is identified, by t/J, with the function r~l '" rif on [0, 1). It is clear therefore that the set of characters of [))2 can be identified, by using the mapping t/J, with the set of all finite products of Rademacher functions 196 Appendix C. Harmonic Analysis on [)2 and Walsh Series on [0, 1] on [0, I]. This latter set of functions is called the Walsh system on [0, J]. It is a complete orthonormal system in LZ[O, J]. By now it should have been made clear that questions of a measure-theoretic or integration-theoretic character concerning Walsh series on [0, 1] are "the same as" the corresponding questions about Fourier series on [liz. One must of course always keep in mind that there may be a world of difference between the two set• ups when the question is of a topological character. Appendix D. Bernstein's Inequality In Chapter 7, we use the L l-norm version of Bernstein's inequality for the groups IR, lr and lL. The precise statements and proofs of the inequality in the first two cases are set down in Sections 0.1 and 0.2 respectively. In Section 0.3, a state• ment is given of a more general result, applicable to any LCA group; from this the Bernstein inequality for G = lL is easily deduced. D.l. Bernstein's Inequality for [R D.1.I. Theorem. There is a number A > 0 such that fIJi If(x - a) - f(x)1 dx ~ AAlal fIJi If(x) I dx for every A > 0, every a in IR and every integrable function f such that support (]) ~ [- A, A]. Proof Choose and fix K in C';(IR) such that K(Y) = 1 for Iyl ~ 1, and define 1 k(x) =27t JIJir K(y)e'Yx . dy. Then k has bounded and continuous derivatives of all orders, all of which are integrable. Also, by the Fourier inversion formula, k = K. Put k;,(x) = Ak(h); then k), is integrable, and k;Jy) = keY/A) = 1 for Iyl ~ A. (1) From (1) it follows that 'Caf-f= k), * ('Caf-f) = ('Cak), - k),) *J, 198 Appendix D. Bernstein's Inequality where, as usual, 7:a f(x) = f(x - a). Consequently, (2) On the other hand, II7:ak;. - kJ I = J~ Ik;.(x - a) - k;.(x) I dx = J~ Ik(x' - Aa) - k(x') I dx' = I(Aa), say. (3) To majorise I(Aa), let g be an arbitrary measurable function with bounded support such that IIg II co ~ 1. Consider F(s) = J~ k(x - s)g(x) dx. Since k has a bounded, continuous derivative, it follows at once that F is differ• entiable, and F(s) = - J~ k'(x - s)g(x) dx. Hence IF(s)1 ~ IIk'III. By the mean value theorem, there is a real number s in ~ such that IJ (k(x - Aa) - k(x»g(x) dxl = IF(Aa) - F(O) I ~ IAaF(s)I ~ Alalllk'lll· (4) If we take the supremum of the left side of (4) over the set of functions g of the kind specified earlier, we conclude that I(Aa) ~ Alalllk'lll. (5) On collecting (2), (3) and (5) together, we obtain the desired result with A = IIk'III. 0 D.2. Bernstein's Inequality for 11" 199 D.2. Bernstein's Inequality for If D.2.t. Theorem. There is a number A > 0 such that Ill" If(xe- iQ ) - f(x)1 dm(x) ~ ANlal Ill" If(x)1 dm(x) for every positive integer N, every real number a, and every trigonometric polynomial with spectrum in [ - N, N]. Proof If we define g = L Xn. Inl';;N and 1 k = 2N + 1 gh. where Xn(e it) = eint• then it is easily verified that k(n) = 1 for Inl ~ N. Hence when b E If; so (1) On the other hand. Holder's inequality shows that 1 "'bk - kill = 2N + I 1I('bh - h)g + ('bg - g)'bhlll I ~ 2N + I (lI'bh - hll211g112 + lI'bg - gIl211'bhIl2). (2) while Parseval's formula shows that IIg112 = 110112 = (2N + 1)1/2; (3) 1I,~1I2 = IIhll2 = IIhll2 = (4N + 1)1/2; (4) and lI'bg - gll~ = 1I('bgY - Oll~ = L Ibn - W Inl';;N ~ (2N + I) sup W - 112 Inl';;N ~ (2N + I)N2Ib - W, (5) 200 Appendix D. Bernstein's Inequality by the mean value theorem. Similarly, It follows from (1)-(6) that Ilrd - fill ~ 2N 1+ IllflllNlb - 1I {(2N + 1)1/22(4N+ 1)1/2 + (2N + 1)1/2(4N + Il12} (4N + 1)1/2 = 3 (2N + 1)1/211flllNlb - 11 ~ 3.j2Nlb - 1111flll ~ 3.j2Nlai IIfll" by the mean value theorem, if b = e- ia • This completes the proof and shows that A can be taken to be 3..12. 0 D.3. Bernstein's Inequality for LeA Groups The method we used to prove Theorem 0.2.1 can be modified to yield a form of Bernstein's inequality for any LeA group G. D.3.1. Theorem. Let G be an LeA group with character group X; let K be a relatively compact subset of X, and M a relatively compact open neighbourhood of 0 in X. Then mx(K + M- M») 1/2 IIraf - fill ~ 3( mx(M) wK+M-M(a)llflll for all integrable functions fan G with J supported in K, and all a in G. Here wD(a) = sup {Ix(a) - 11: XED} whenever a E G and D £; X; mx denotes the Haar measure on X. Remarks. (a) When G = ~, we may take K = [-..1., ..1.], M = (-..1., ..1.) and deduce Theorem D.1.1 by observing that le iya - 11 ~ Iyal when y and a are real. (b) Similarly, Theorem 0.2.1 can be deduced by taking K = M = [-N, N]. D.3. Bernstein's Inequality for LeA Groups 201 (c) When G = 71., take K = {e it : \t\ ~..1.} and M = {eit : \t\ Chapter 2. The systematic study of singular integrals of the type considered in Chapter 2 has its origins in the fundamental paper of Calderon and Zygmund [5]. The subsequent literature is very extensive, and has given rise to many important generalisations and refinements of their methods and results. For a glimpse at this vast literature, see the references in [38] and [39]. Chapter 5. As far as we can determine, the martingale approach to Littlewood• Paley theory is due originally to D. L. Burkholder [4]. A little later, an alternative treatment of the martingale version of the Littlewood-Paley theorem was given by R. F. Gundy [17]. Chapter 6. (i) The conjugate function theorems (6.2.3, 6.3.3 and 6.4.3) and the Hilbert transform theorems (6.7.4 and 6.7.6) were discovered by M. Riesz. Two approaches were used by Riesz, one based on complex variables (the Math. Z. paper), the other on his then-new convexity methods (the Acta paper); see the references below. The case p = 2 of 6.7.4(ii), expressed bilineariy, is older, and due to Hilbert. In the Acta paper, Riesz also discussed Fourier multipliers, referring back to earlier work by S. Sidon and W. H. Young, on M 1 (Z) and M oc,(Z), and M. Fekete. Fekete had dealt with the multipliers of various classes of functions: continuous functions, Riemann integrable functions, etc. The papers referred to are as follows. Fekete, M.: Uber Faktorenfolgen welche die "Klasse" einer Fourierschen Reihe unveriindert lassen. Acta Sci. Math. Szeged 1, 148-166 (1923). Riesz, M.: Sur les maxima des formes bilineaires et sur les fonctionnelles lineaires. Acta Math. 49,465-497 (1926). Riesz, M.: Sur les fonctions conjuguees. Math. Z. 27, 218-244 (1927). Sidon, S.: Reihentheoretische Siitze und ihre Anwendungen in der Theorie der Fourierschen Reihen. Math. Z. 10, 121-127 (1921). Young, W. H.: On Fourier series of functions of bounded variation. London Roy. Soc. Proc. 88,561-568 (1913). Young, W. H.: On the Fourier series of bounded functions. Proc. London Math. Soc. 12, 41-70 (1913). (ii) The proof which we have given of Theorem 6.5.2 (the vector Riesz theorem) consists in applying the singular integral techniques of Chapters 2 and 3 to a particular vector-valued kernel. However, if the scalar M. Riesz theorem• equivalently, the continuity of the conjugate function operator Tc on LP (1 < p < oo)-is granted, then it is possible to deduce Theorem 6.5.2 (cf. 6.5.2(2» im- Historical Notes 203 mediately from an important principle due to 1. Marcinkiewicz and A. Zygmund. The Marcinkiewicz-Zygmund principle states roughly that a linear operator T, continuous on U(X) (0 < p < 00), has a natural extension which is continuous on LP(X, £), the space of p-th power integrable functions on X with values in the Hilbert space £. It seems worthwhile to give a precise statement and a proof of the Marcinkiewicz• Zygmund theorem. Theorem (Marcinkiewicz-Zygmund). Let (X, .,It, /1) and (Y, .;Y, v) be measure spaces, and assume that 0 < p < 00. If S is a vector subspace of U(X, JIt, p) and T is a linear mapping from S into U(Y,.;Y, v) such that (I) for all fin S, then, for every positive integer N and every N-tuple (/1' ... ,fN) of elements of S, we have (2) Proof Denote by L the unit sphere in eN and by a the normalised surface measure on L. (If Ulf denotes the compact group of unitary transformations of eN, and m the normalised Haar measure on Ulf, then a can be defined, in terms of m, by the requirement that t g da = I'/l g(U(po» dm(U) (3) for all continuous functions g on L, where Po is an arbitrarily chosen point of L.) The main property of a that we shall utilise is that a is "unitarily invariant" in the sense that a(U£) = aCE) (4) for (say) all Borel subsets £ of L and all unitary transformations U of eN. The property (4) follows immediately from (3). If WI and w2 are points on L, there is a unitary transformation U of eN such that (5) Let (x, y) denote the standard inner product in eN. Then, thanks to (4) and (5), cp = IL I(s, W2W da(s) = tI(s, UwlW da(s) = IL 1(U*s, wtW da(s) = 11(S, (1)IP da(s), (6) 204 Historical Notes U* denoting the adjoint of U. Observe also that cp > °since the function s --+ I(s, Q)z)1 is continuous, and positive at Q)z. To come now to the proof proper, let (fl' ... ,IN) be as in the enunciation, and let s = (SI' ... ,SN) be an arbitrary point of I:. Then by (1), (7) For each point y of Y for which (Til (y), ... , TIN(Y)) =I 0, we can write a unit vector in eN, i.e. an element of I:. Similarly, for each point x of X for which (f1(X), ... IN(X)) =I 0, we can write ¢(x) for the point of I:. It then follows from (7) that r{r . I(s, I/J(y))IP (.f ITlb)12) P'2 dv(Y)}da(S) JI: J(Y.(Tf,(y), ••• ,TfN(Y)),cO) .-1 ~ MP r{r . I(s, ¢(x))IP (.f 1/;(X)lz)PIZ dfl(X)}da(S). (8) JI: J{X.(f,(X), •.• .rN(X)),cO) .-1 We deduce, by applying Fubini's theorem to (8), that r (.f IT/;(YW)PI2{ r I(s, I/J(y)IP da(S)}dV(Y) J{Y:(Tf,(y), ••. ,TfN(Y»,cO} 1= 1 JI: ~ MP r . (.f If;CXW)P'Z { r I(s, ¢(x))IP da(S)}dfl(X). (9) J{x.(f,(X), ••• .rN(X»,cO) I-I JI: But, because of the invariance property (6), (9) reduces to the inequality which, since cp > 0, is what we had to prove. 0 Remark. The Marcinkiewicz-Zygmund theorem is, in a sense, a cultural antecedent of later work of A. Grothendieck on extensions of continuous linear operators from domains of scalar-valued functions to domains of Hilbert-space• valued functions. Marcinkiewicz, J., Zygrnund, A.: Quelques inegalites pour les operations Iineaires. Fund. Math. 32, 115-121 (1939). Historical Notes 205 (iii) The analogue for the circle group of (22) in 6.7.6 is due to S. B. Steckin. It appears as Theorem 2 in the paper referred to below. Steckin's proof rests essentially on the appropriate form of the Hilbert transform theorem. Further developments of Steckin's work have been given by 1. 1. Hirschman, lr.. See also 16.4.7 in [9]. Hirschman, 1. I., Jr.: On multiplier transformations. Duke Math. J. 26, 221-242 (1959). MR 21#3721. Steckin, S. B.: On bilinear forms. Doklady Akad. Nauk. SSSR (N.S.) 71, 237-240 (1950) (Russian). MR 11 p. 504. Chapter 8. (i) The (strong) Marcinkiewicz mUltiplier theorem is due to 1. Marcinkiewicz. Marcinkiewicz, J.: Sur les multiplicateurs des series de Fourier. Studia Math. 8, 78-91 (1939). (ii) For the first versions of the Littlewood-Paley theorem, the reader should consult the following three fundamental papers. Littlewood, J. E., Paley, R. E. A. c.: Theorems on Fourier series and power series (I). J. Lon• don Math. Soc. 6, 230-233 (1931). Littlewood, J. E., Paley, R. E. A. C.: Theorems on Fourier series and power series (II). Proc. London Math. Soc. 42, 52-89 (1936). Littlewood, J. E., Paley, R. E. A. C.: Theorems on Fourier series and power series (III). Proc. London Math. Soc. 43,105-126 (1937). References 1. Bloom, W. R.: Bernstein's inequality for locally compact Abelian groups. J. Austral. Math. Soc. XVII, 88-101 (1974). 2. Bourbaki, N.: Elements de mathematique. XIII. Premiere partie: Les structures fonda• mentales de l'analyse. Livre VI: Integration. Actualites Sci. Ind. No. 1175. Paris: Hermann 1952. MR 14, p. 960. 3. Brainerd, B., Edwards, R. E.: Linear operators which commute with translations. l. Repre• sentation theorems. J. Austral. Math. Soc. VI, 289-327 (1966). MR 34#6542. 4. Burkholder, D. L.: Martingale transforms. Ann. Math. Statist. 37, 1494-1504 (1966). MR 34#8456. 5. Calderon, A. P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85-139 (1952). MR 14, p. 637. 6. Chung, K. L.: A course in probability theory, 2nd ed .. New York: Academic Press 1974. 7. Coifman, R. R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogenes. Lecture Notes in Mathematics, No. 242. Berlin-Heidelberg-New York: Springer 1971. 8. Cowling, M. G.: Spaces AZ and U-£" Fourier multipliers. Doctoral Dissertation, The Flin• ders University of South Australia 1974. 9. Edwards, R. E.: Fourier series: a modern introduction. Vols. I, II. New York: Holt, Rine• hart and Winston 1967 and 1968. MR 35#7062 and 36#5588. 10. Edwards, R. E.: Functional analysis: theory and applications. New York: Holt, Rinehart and Winston 1965. MR 36#4308. 11. Edwards, R. E.: Changing signs of Fourier coefficients. Pacific J. Math. 15,463--475 (1965). MR 34#564. 12. Fefferman, C.: The multiplier problem for the ball. Ann. of Math. (2) 94, 330-336 (1971). MR 45#5661. 13. Figa-Talamanca, A.: Translation invariant operators in U. Duke Math. J. 32, 495-501 (1965). MR 31#6095. 14. Figa-Talamanca, A., Gaudry, G. l.: Multipliers of U which vanish at infinity. J. Functional Analysis 7, 475--486 (1971). MR 43#2429. 15. Figa-Talamanca, A., Gaudry, G. I.: Extensions of multipliers. Boll. Un. Mat. Ital. (4) 3, 1003-1014 (1970). MR 43#5255. 16. Garsia, A. M.: Martingale inequalities: Seminar notes on recent progress. Reading, Mass.: W. A. Benjamin 1973. 17. Gundy, R. F.: A decomposition for £I-bounded martingales. Ann. Math. Statist. 39, 134-138 (1968). MR 36#4625. 18. Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities. Cambridge: Cambridge University Press 1934. 19. Herz, C.: Remarques sur la note precedente de Varopoulos. C. R. Acad. Sci. Paris 260, 6001-6004 (1965). MR 31 #6096. 20. Hewitt, E., Ross, K. A.: Abstract harmonic analysis. Vols. I, II. Berlin-G6ttingen-Heidel• berg 1963 and 1970. MR 28#158 and 41#7378. 21. H6rmander, L.: Estimates for translation invariant operators in LP spaces. Acta Math. 104, 93-140 (1960). MR 22#12389. References 207 22. Inglis, I. R.: Martingales, singular integrals and approximation theorems. Doctoral disser• tation, The Flinders University of South Australia 1975. 23. Kahane, J.-P.: Some random series of functions. Lexington, Mass.: D. C. Heath 1968. MR 40#8095. 24. Knapp, A. W., Stein, E. M.: Singular integrals and the principal series I. Proc. Nat. Acad. Sci. U.S.A. 63, 281-284 (1969). MR 41#8588. 25. Larsen, R.: An introduction to the theory of multipliers. Berlin-Heidelberg-New York: Springer 1970. 26. de Leeuw, K.: On Lp multipliers. Ann. of Math. (2) 81, 364-379 (1965). MR 30#5127. 27. Lohoue, N.: Algebres Ap(G) et convoluteurs de U(G). These de Doctorat es Sciences Mathe• matiques. Orsay 1971. 28. Lohoue, N.: Sur certains ensembles de synthese dans les algebres A.(G). C. R. Acad. Sci. Paris Ser. A-B 270, A589-A591 (1970). MR 41#8933. 29. Lohoue, N.: Sur Ie critere de S. Bochner dans les algebres Bp(Rn) et I'approximation des con• voluteurs. C. R. Acad. Sci. Paris Ser. A-B 271, A247-A250 (1970). MR 42#8180. 30. Meyer, Y.: Endomorphismes des ideaux fermes de V(G), classes de Hardy, et series de Fou• rier lacunaires. Ann. Sci. Ecole Norm. Sup. (4) 1, 499-580 (1968). MR 39#1910. 31. Paley, R. E. A. c.: A remarkable system of orthogonal functions. Proc. London Math. Soc. 34,241-279 (1932). 32. Phillips, K., Taibleson, M.: Singular integrals in several variables over a local field. Pacific J. Math. 30, 209-231 (1969). MR 40#7886. 33. Riviere, N. M.: Singular integrals and multiplier operators. Ark. Mat. 9, 243-278 (1971). 34. Rudin, W.: Real and complex analysis. New York: McGraw-Hili 1966. MR 35#1420. 35. Rudin, W.: Fourier analysis on groups. New York: John Wiley and Sons 1962. MR 27#2808. 36. Schwartz, L.: Sur les multiplicateurs de .?PU. Kungl. Fysiografiska Sallskapets i Lund Forhandlingar, 22, no. 21, 5 pp. (1953). MR 14, p. 767. 37. Spector, R.: Sur la structure locale des groupes abeliens localement compacts. Bull. Soc. Math. France Suppl. Mem. 24 (1970). MR 44#729. 38. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton Math• ematical Series, No. 30. Princeton, N. J.: Princeton University Press 1970. MR 44#7280. 39. Stein, E. M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63. Princeton, N. J.: Princeton University Press 1970. MR 40 #6176. 40. Zygmund, A.: Trigonometric series, 2nd ed .. Vols. I, II. New York: Cambridge University Press 1969. MR 21 #6498. Terminology We list below the definitions of some terms which are used, but not defined, in the text. Those terms which are defined in the text are listed, with page references, in the Index of Authors and Subjects; any terms used without definition are con• sidered by us to be so standard as to require no explanation. Borel isomorphism. Let (X,.A) and (Y, %) be measurable spaces. A Borel isomorphism t/I of (X, .A) and (Y, %) is a one-one mapping of X onto Y such that t/I(E) E % if and only if E E .A. Borel a-algebra. Let X be a topological space. The Borel a-algebra on X is the a-algebra generated by the family of all open subsets of X. First countable topological space. A topological space X is said to be first countable if each point x of X has a neighbourhood base which is countable. Locally integrable/unction. A function/on a measure space (X,.A, fJ.) is said to be locally integrable if (i) / is measurable for .A; and (ii) feE is integrable with respect to fJ. for every set E in .A which is of finite fJ.-measure. Locally null set. A measurable subset E of a measure space (X, .A, fJ.) is said to be locally null if fJ.(E n F) = 0 for all sets F in .A which are of finite fJ.-measure. Pseudomeasure. Let G be an LeA group, X its dual, and Define the norm on A(G) by the rule A pseudomeasure a on G is a continuous linear functional on A(G). Relatively compact set. Let X be a topological space. A subset E of X is said to be relatively compact if its closure is compact. Spectrum 0/ a trigonometric polynomial. Let G be a compact Abelian group, and / a trigonometric polynomial on G. The spectrum of / is the support of the function]. Index of Notation We list below those symbols which may not be in common use, or may be subject to more than one interpretation, and which are used systematically through the book. In a few cases, the symbols listed are not defined in the text, and we indicate briefly the interpretation intended. In all other cases, the definitions of the symbols are to be found on the page(s) indicated. (a, b) [inner product] 50 J(K) 55 At..B [symmetric difference of the sets A and J8(k) 44 B) . kz [set of integral multiples of k] B(.1'('h .1'('2) 50 ~ [Fourier transform of vector-valued kernel] BMO(Q) 95 54 dI(X) [X a topological space; dI(X) the q• Lk [convolution operator] 41 algebra of Borel subsets of X] LK [vector-valued convolution operator] 50 c [conjugate function] 104 ~'(G) 6 Cc(G) [space of continuous functions with L'(G) 6 compact supports on G] £1'(G,.1'(') 51 C~(IR) [space of k times continuously differ• m [Haar measure] entiable, compactly supported, functions m,(z) 172 on IR] Mf[rnaximal function] 33 c/ g [class generated by g) 21 Mf[martingale maximal function] 84 D./[dilation of Iby amount a] 129 M(G) [space of regular, complex, Borel On [Dirichlet kernel of order n] 113 measures on G] 02 [Cantor group] 51 M,(X) [space of Fourier multipliers of L'(G)] D2 [dual of Cantor group] 51, 194 7 Enl=E(fI§n) [conditional expectation] 78 III [set of natural numbers] 32 i [Fourier transform of vector-valued func- 11111, 6,51 tion] 53 IILtll,., 41 II> 95 IILKf/,., 54 f,,f' 179 11<611,., 1 1"1. 21 IITII,., 177 1"'2 21 I!T4>II", 7 Ft , FA [Fejer kernel] 140, 141 Q 32 a£1(X)[G an LCA group; Xits dual. a£1(X) 01 [square function] 90 = (i:/e £1(G)}] rJ U-th Rademacher function] 193 aM(X) [G an LCA group;Xits dual. aM(X) SJ 2, 11 = (f:1 = fl, for some tl in M(G)}] SJI 2,8 Gd [G with the discrete topology] 71 SJf[martingale difference] 80 h [Hilbert kernel] 121 SJ,t 23 .1'(' [Hilbert space] S" [Fourier partial sum/integral operator] 8 Hf[Hilbert transform of f] 122 Tc [conjugate function operator] 104 H1(Q) 95 T4> [multiplier operator] 6, 55 J(k) 45 U1 ® U2 20 210 Index of Notation Var ,p [total variation of ,p] A [distribution function] 45 Var,p [total variation of ,p over the interval LI] ATf [distribution function of IT/Il 178 ~ Var ,p [total variation of,p over the closure ~~ [characteristic function of LI] ;; fr [dual homomorphism] 184 of the interval LI] p J [generator of O2] 57, 194 VN , VA [de la Vallee-Poussin kernel] 137, P., u. [Rudin-Shapiro polynomials] 174 139, 143 u(Jt") [u-algebra generated by the family Jt"] 7l(a) [cyclic group {O, ... , a - I} of order a] r./[translate of/by amount a] 128 7l(2) [cyclic group {O, I} of order 2] 57 X. [character e" -+ e1n1 of lJ"] 4 1 [identity mapping] X, [character n -+ e1n1 of 7l; also character 01} [Kronecker's delta] x -+ elXl of IHI] 60 (LI),}) ["diagonal product" family] 27 Da [group of a-adic numbers] LI,p(n) [first difference] 149 (D, F,p) 77 8 30, 31 Index of Authors and Subjects arc 114. 135. 141 dyadic interval 4, 134, 145, 148 Bernstein's inequality 139. 140. 142. 197. Fefferman, C. 19 199-201 Fejer kernel 137, 140, 142, 147 bilinear form 105. 106. 127 Fekete, M. 202 Bohr compactification 71.72.76 Figa-Talamanca. A. 166, 191 bounded variation 148. 157 Fourier partial sum 81.82 Burkholder. D.L. 76.202 Fourier partial integral/sum operator 4, 8. 134 Calder6n. A. 3. 30. 35. 44. 202 Fourier transform of Jf"-valued function 53 Calder6n-Zygmund' technique/theory 44. Fourier transform of operator-valued kernel 105. 118. 120 54 Cantor group 57. 81. 193 Fournier. J. 135, 146 Cantor group: Haar measure on 194. 195 conditional expectation 3.76-78.80.82.92. Garsia. A. M. 76, 95 102 Gaudry. G. I. 162, 166 conditional expectation operator 78 g-function 5 conjugate function operator 104, 106, 108, group of a-adic numbers 60, 70 115, 118, 122,202 Grothendieck, A. 204 conjugate function theorem 105-107. 111, Gundy. R. F. 76, 202 114, 115,202 convolution (of vector-valued functions) 54 Hadamard block 166 convolution operator 30,41. 105 Hadamard decompositions 3,73, 148, 153- convolution operator (vector-valued) 50,54, 155,159-161,169.170 55 Hadamard sequence/set 148, 153, 161, 162. corona 58, 59, 64, 65, 68 168.170-172 countable (index set) 8 Hardy. G. H. 34 covering family 30-32, 55, 58, 61, 75, 108, Herz, C. 191 112.113.116,141.144 Hewitt, E. 60 covering lemma 32-34 Hilbert, D. 202 Hilbert kernel 118, 121 decomposition 6,8, 142, 164 Hilbert transform 105, 106, 118, 120-122, decomposition theorem 35, 49, 52 125, 126, 128, 129. 132, 133,202 de la Vallee-Poussin kernel 137,139-141,143 Hirschman. I. I., Jr. 205 differentiation theorem 33 homomorphism theorem for multipliers 26, dilation 106, 128, 129 70,73, 184. 187 dilation operator 128 Hormander, L. 152, 173 Dirichlet kernel 113, 137, 164, 176 Hormander-Mihlin theorem 152 distribution 12, 108 distribution function 86, 178 Inglis, I. R. 76 Doob. J. 87 intersection of decompositions 28 dual homomorphism 184 dyadic decomposition 73. 134, 135, 148. kernel 41. 112. 116. 118, 121. 122. 127, 136 155, 159. 173 kernel (operator-valued) 52, 54, 55, 202 212 Index of Authors and Subjects A(p) set 19,24,138,147, 168-171 R (Riesz) property 19,20,27,104,120,136, Littlewood, J. E. 34 138, 141, 143, 150 LP (Littlewood-Paley) decomposition/prop• Riesz (M.) multiplier theorem 104-106,110, erty 8-11, 13-15, 17, 19,20,23,26,58- 111,114,115,121,126-128,140,141,202 61,64,65,67,68,70-72,74,102,103,136, Riesz (M.) theorem (vector version) 118, 142, 143, 146, 155, 159, 161-164, 166-168, 120,202 172 Riviere, N. M. 41 LP (Littlewood-Paley) theorem 2-6, 9, 19, Ross, K. A. 60 42, 57, 59, 67, 72-74, 76, 91, 95, 99, 100, Rudin-Shapiro polynomials 174 102, 134-136, 143, 145, 148, 149, 156, 160, 161,166,202,205 Schwartz, L. 19 LP (Littlewood-Paley) theory 4, 5, 202 Set of type (2, p) 14 Sidon, S. 202 Marcinkiewicz, J. 3, 5, 6, 203-205 Sidon set 146, 147, 168-170 Marcinkiewicz interpolation theorem 35, singular integral 3, 35, 202 42,43,47,92,96,100,135,177,179,181, singular kernel 42, 106 183 singular multiplier 172, 173, 175 Marcinkiewicz multiplier theorem (strong Spector, R. 74, 75 Marcinkiewicz theorem) 105, 148, 151, square function 90 153-155, 159, 160,205 Steekin, S. B. 3, 105, 110, 114, 118, 127, 128, Marcinkiewicz-Zygmund principle/theorem 205 203, 204 Stein, E. M. 33 martingale 3, 6, 76, 80, 95, 202 strong type 91,97,100, 177, 178,180,182 martingale (associated with f) 80, 82 subadditive operator 91, 180, 182 martingale condition/property 78, 88 suitable family of compact open subgroups martingale difference 80 59,64,68,69, 73, 81 martingale difference series 80, 90, 91 suitable family of open subgroups 69-72 maximal function 33, 34, 82, 84 supermartingale 92 measurable vector-valued function 50, 52 surface measure 203 Meyer, Y. 166 Mihlin, S. G. 152 translation 128 multiplier 3, 5-7, 104, 105, 120, 135, 148, 156,162,174,191 unconditionaJly convergent series 10, 12, 15, multiplier norm 7 16 multiplier operator 7, 106, 122, 128, 129, uniformly of type A(P) 167-170, 172 131-133, 146 up-crossing 87 up-crossing arrangement 87, 88 Paley, R. E. A. C. 57, 76 Paley's theorem 58, 76 Walsh functions/series/system 57, 193, 195, Parseval formula 2,7, 186, 192 196 Parseval relation (for conditional expecta• WM (weak Marcinkiewicz) multiplier tions) 78 theorem 5,8 Plancherel formula (for martingales) 97 WM (weak Marcinkiewicz) property 8, 12- Plancherel-Riesz-Fischer theorem I, 7, II, 14, 17, 26, 58, 64, 67, 68, 70-74, 135, 148, 12 159, 161 Poisson integral 5 weak (p, p) norm 44, 178, 180, 182 (p,p)norm 177,180 weak type 48, 49, 91, 93, 96, 97, 100, 135, product decomposition 23 136, 138, 146, 147, 178, 180-182 product diagonal family 27 Young, W. H. 202 Rademacher functions 13, 14, 16, 24, 29, 138, 143, 193, 195 (0-1) property 76 rectangle 120 Riesz, M. 3, 104-106,202 Zygmund, A. 3, 5, 30, 35, 202, 203 Riesz convexity theorem 7 Ergebnisse der Mathematik und ihrer Grenzgebiete 1. Bachmann: Transfinite Zahlen 2. Miranda: Partial Differential Equations of Elliptic Type 4. Samuel: Methodes d'algebre abstraite en geometrie algebrique 5. Dieudonne: La geometrie des groupes classiques 7. Ostmann: Additive Zahlentheorie. 1. Teil: Allgemeine Untersuchungen 8. Wittich: Neuere Untersuchungen iiber eindeutige analytische Funktionen 10. Suzuki: Structure of a Group and the Structure of its Lattice of Subgroups. Second edition in preparation 11. Ostmann: Additive Zahlentheorie. 2. Teil: Spezielle Zahlenmengen 13. Segre: Some Properties of Differentiable Varieties and Transformations 14. Coxeter/Moser: Generators and Relations for Discrete Groups 15. Zeller/Beckmann: Theorie der Limitierungsverfahren 16. Cesari: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations 17. Severi: II teorema di Riemann-Roch per curve, superficie e varieta questioni collegate 18. Jenkins: Univalent Functions and Conformal Mapping 19. Boas/Buck: Polynomial Expansions of Analytic Functions 20. Bruck: A Survey of Binary Systems 21. Day: Normed Linear Spaces 23. Bergmann: Integral Operators in the Theory of Linear Partial Differential Equations 25. Sikorski: Boolean Algebras 26. Kiinzi: Quasikonforme Abbildungen 27. Schatten: Norm Ideals of Completely Continuous Operators 30. Beckenbach/Bellman: Inequalities 31. Wolfowitz: Coding Theorems of Information Theory 32. Constantinescu/Cornea: Ideale Rander Riemannscher Flachen 33. Conner/Floyd: Differentiable Periodic Maps 34. Mumford: Geometric Invariant Theory 35. Gabriel/Zisman: Calculus of Fractions and Homotopy Theory 36. Putnam: Commutation Properties of Hilbert Space Operators and Related Topics 37. Neumann: Varieties of Groups 38. Boas: Integrability Theorems for Trigonometric Transforms 39. Sz.-Nagy: Spektraldarstellung linearer Transformationen des Hilbertschen Raumes 40. Seligman: Modular Lie Algebras 41. Deuring: Algebren 42. Schiitte: Vollstandige Systeme modaler und intuitionistischer Logik 43. Smullyan: First-Order Logic 44. Dembowski: Finite Geometries 45. Linnik: Ergodic Properties of Algebraic Fields 46. Krull: Idealtheorie 47. Nachbin: Topology on Spaces of Holomorphic Mappings 48. A. lonescu Tulcea/C. lonescu Tulcea: Topics in the Theory of Lifting 49. Hayes/Pauc: Derivation and Martingales 50. Kahane: Series de Fourier absolument convergentes 51. Behnke/Thullen: Theorie der Funktionen mehrerer komplexer Veranderlichen 52. Wilf: Finite Sections of Some Classical Inequalities 53. Ramis: Sous-ensembles analytiques d'une variete banachique complexe 54. Busemann: Recent Synthetic Differential Geometry 55. Walter: Differential and I!\tegrallnequalities 56. Monna: Analyse non-archimedienne 57. Alfsen: Compact Convex Sets and Boundary Integrals 58. Greco/Salmon: Topics in m-Adic Topologies 59. L6pez de Medrano: Involutions on Manifolds 60. Sakai: C·-Algebras and W*-Algebras 61. Zariski: Algebraic Surfaces 62. Robinson: Finiteness Conditions and Generalized Soluble Groups, Part 1 63. Robinson: Finiteness Conditions and Generalized Soluble Groups, Part 2 64. Hakim: Topos anneles et schemas relatifs 65. Browder: Surgery on Simply-Connected Manifolds 66. Pietsch: Nuclear Locally Convex Spaces 67. Dellacherie: Capacites et processus stochastiques 68. Raghunathan: Discrete Subgroups of Lie Groups 69. Rourke/Sanderson: Introduction to Piecewise-Linear Topology 70. Kobayashi: Transformation Groups in Differential Geometry 71. Tougeron: Ideaux de fonctions differentiables 72. Gihman/Skorohod: Stochastic Differential Equations 73. Milnor/Husemoller: Symmetric Bilinear Forms 74. Fossum: The Divisor Class Group of a Krull Domain 75. Springer: Jordan Algebras and Algebraic Groups 76. Wehrfritz: Infinite Linear Groups 77. Radjavi/Rosenthal: Invariant Subspaces 78. Bognar: Indefinite Inner Product Spaces 79. Skorohod: Integration in Hilbert Space 80. Bonsall/Duncan: Complete Normed Algebras 81. Crossley/Nerode: Combinatorial Functors 82. Petrov: Sums of Independent Random Variables 83. Walker: The Stone-tech Compactification 84. Wells/Williams: Embeddings and Extensions in Analysis 85. Hsiang: Cohomology Theory of Topological Transformation Groups 86. Olevskii: Fourier Series with Respect to General Orthogonal Systems 87. Berg/Forst: Potential Theory on Locally Compact Abelian Groups 88. Weil: Elliptic Functions according to Eisenstein and Kronecker 89. Lyndon/Schupp: Combinatorial Group Theory 90. Edwards/Gaudry: Littlewood-Paley and MUltiplier Theory