Isomorphisms Between H1 Spaces

M o n o g r a f i e M a t e m a t y c z n e Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) Volume 66 ( New Ser...

Author: Paul F.X. Müller

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M o n o g r a f i e

M a t e m a t y c z n e

Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)

Volume 66 ( New Series )

Founded in 1932 by S. Banach, B. Knaster, K. Kuratowski, S. Mazurkiewicz, W. Sierpiński, H. Steinhaus

Managing Editor: Przemysław Wojtaszczyk, IMPAN and Warsaw University Editorial Board: Jean Bourgain (IAS, Princeton, USA) Tadeusz Iwaniec (Syracuse University, USA) Tom Körner (Cambridge, UK) Krystyna Kuperberg (Auburn University, USA) Tomasz Łuczak (Poznań University, Poland) Ludomir Newelski (Wrocław University, Poland) Gilles Pisier (Université Paris 6, France) Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences) Grzegorz Świątek (Pennsylvania State University, USA) Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences)

Volumes 31– 62 of the series Monografie Matematyczne were published by PWN – Polish Scientific Publishers, Warsaw

Paul F.X. Müller

Isomorphisms between H 1 Spaces

Birkhäuser Verlag Basel • Boston • Berlin

Author: Paul F.X. Müller Institute of Analysis Johannes Kepler University Linz Altenbergerstr. 69 Austria e-mail: [email protected]

2000 Mathematics Subject Classification 30D55, 42C10, 46B03, 46B07, 46-99, 47B38, 60G46

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN: 3-7643-2431-7 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-2431-7 ISBN-13: 978-3-7643-2431-5 987654321

www.birkhauser.ch

To Joanna

Contents Preface

xi

1 The Haar System: Basic Facts and Classical Results 1.1 Bases in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . The Haar basis . . . . . . . . . . . . . . . . . . . . . . . . Khintchine’s inequality . . . . . . . . . . . . . . . . . . . Burkholder’s inequality . . . . . . . . . . . . . . . . . . . The Walsh system . . . . . . . . . . . . . . . . . . . . . . 1.2 Dyadic H 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feﬀerman’s inequality . . . . . . . . . . . . . . . . . . . . Sharp maximal functions . . . . . . . . . . . . . . . . . . 1.3 Bounded square functions and large deviation inequalities . Square functions in Lp . . . . . . . . . . . . . . . . . . . . Multipliers into SL∞ . . . . . . . . . . . . . . . . . . . . 1.4 Martingales and biorthogonal systems . . . . . . . . . . . . Martingale inequalities . . . . . . . . . . . . . . . . . . . Biorthogonal systems . . . . . . . . . . . . . . . . . . . . 1.5 Basic operators . . . . . . . . . . . . . . . . . . . . . . . . . Multipliers and paraproducts . . . . . . . . . . . . . . . . Rearrangement operators and Calder´ on–Zygmund kernels Orthogonal projections . . . . . . . . . . . . . . . . . . . 1.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 . 1 . 1 . 6 . 13 . 19 . 34 . 35 . 45 . 52 . 52 . 68 . 73 . 74 . 81 . 88 . 88 . 92 . 104 . 112

2 Projections, Isomorphisms and Interpolation 2.1 Complemented subspaces . . . . . . . . Three-valued martingale diﬀerences . Rosenthal’s space . . . . . . . . . . . Weighted intersections . . . . . . . . . 2.2 Pelczy´ nski’s decomposition method . . . Inﬁnite direct sums . . . . . . . . . . H 1 with values in 2n . . . . . . . . . .

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117 117 124 130 137 139 140 142

viii

Contents 2.3

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144 148 153 156 159 166

3 Combinatorics of Colored Dyadic Intervals 3.1 The Carleson packing condition . . . . . . . . . . . . . . . . . Generations of dyadic intervals . . . . . . . . . . . . . . . . The Gamlen–Gaudet construction . . . . . . . . . . . . . . 3.2 Orthogonal projections and colored intervals . . . . . . . . . . Jones’s compatibility condition and colored intervals . . . . The ﬁrst step towards the uniform approximation property 3.3 Rearrangement operators . . . . . . . . . . . . . . . . . . . . Rearrangement operators on BMO . . . . . . . . . . . . . . Rearrangement operators on L1 . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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169 169 170 176 181 181 193 196 198 224 228

4 Martingale H 1 Spaces 4.1 Maurey’s isomorphism . . . . . . . . . . Operations on martingale diﬀerences . Martingale transform techniques . . . 4.2 Isomorphic classiﬁcation . . . . . . . . . Classiﬁcation of martingale H 1 spaces Classiﬁcation of weighted intersections 4.3 More on subsystems of the Haar system The theorem of Gamlen and Gaudet . Related open problems . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . .

2.4

Interpolation of operators . . . Calder´ on’s product . . . . . . Pisier’s norm on H 1 . . . . . Dual estimates . . . . . . . . Analytic families of operators Notes . . . . . . . . . . . . . .

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229 229 230 233 242 242 252 257 257 260 265

5 Isomorphic Invariants for H 1 5.1 Complemented copies of Hilbert spaces . . . . Existence, abundance . . . . . . . . . . . . Johnson’s factorization . . . . . . . . . . . Intrinsic description . . . . . . . . . . . . . 5.2 Complemented copies of Hn1 . . . . . . . . . . Dichotomies . . . . . . . . . . . . . . . . . Intrinsic description . . . . . . . . . . . . . H 1 with values in 2 . . . . . . . . . . . . . 5.3 The uniform approximation property of BMO Splitting the Haar support . . . . . . . . . UAP data with large Haar coeﬃcients . . . UAP data with small Haar coeﬃcients . . .

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267 267 267 271 277 282 283 295 301 308 312 314 316

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Contents

5.4

ix

General UAP data . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

6 Atomic H 1 Spaces 1 . . . . . 6.1 Basic similarities between H 1 and Hat Maximal functions and atoms . . . . . . . . . Square functions . . . . . . . . . . . . . . . . 6.2 Carleson’s biorthogonal system . . . . . . . . . Deﬁnition . . . . . . . . . . . . . . . . . . . . Carleson coeﬃcients versus Haar coeﬃcients The compensation argument . . . . . . . . . 1 . . . . . . . . . . Unconditional basis in Hat 6.3 Spaces of homogeneous type . . . . . . . . . . . Lipschitz partitions of unity . . . . . . . . . . Estimates for molecules . . . . . . . . . . . . 6.4 Orthogonal projections in atomic H 1 spaces . . The square root of the Gram matrix . . . . . 6.5 Martingale approximation in atomic H 1 spaces 6.6 Notes . . . . . . . . . . . . . . . . . . . . . . .

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347 347 351 355 360 362 366 374 391 397 404 405 407 418 421 430

Bibliography

433

List of Symbols

449

Index

451

Preface The theory of Hardy spaces is ﬁrmly rooted in the ﬁelds of complex analysis, trigonometric series, and probability. Its history began in the decade between 1906 and 1916. In 1906 appeared P. Fatou’s treatise on power series and trigonometric series in Acta Mathematica. G. H. Hardy proved in 1915 that integral means of power series in the unit disk are log-convex, just in time to be included in the ﬁrst edition of E. Landau’s “Darstellung und Begr¨ undung einiger neuerer Ergebnisse der Funktionentheorie” (1916). In the same year F. and M. Riesz presented their ¨ treatise “Uber die Randwerte einer analytischen Funktion” to the Scandinavian congress of mathematicians in Stockholm. The classical theory of Hardy spaces is concerned with the boundary behavior of analytic functions on the unit disk and with estimates forthe Fourier coeﬃcients of the limiting function. Given a power series f (z) = an z n , Hardy and 1 Littlewood (1930) proved that on the unit circle the L norm of its non-tangential maximal function is equivalent to the L1 norm of its boundary values. R. E. A. C. Paley (1933) shows that the sequence of lacunary coeﬃcients is square summable provided that the boundary values are integrable on the unit circle, more precisely that 1/2 1 π ≤ |f (eit )|dt. |a2n |2 π −π Kolmogorov’s theorem (1925) asserts that the Hilbert transform is of weak type one-one. These three results, together with the F. and M. Riesz theorem, were crucial in establishing the importance of the Hardy space H 1 (T) of integrable functions on the unit circle for which the harmonic extension to the unit disk is analytic. Probability, in the 1920’s, was confronted with analogous problems of boundary convergence in the work of N. Wiener (construction of Brownian motion), A. Kolmogorov (sums of independent random variables) and J. Khintchine (the law of large numbers) to name just a few. Thus emerged martingales, martingale convergence theorems and J. L. Doob’s maximal function. By 1977 there existed two highly developed classes of H 1 spaces: these were the atomic H 1 spaces which are linked to analytic functions via Feﬀerman’s duality theorem, and the martingale H 1 spaces consisting of martingales for which Doob’s maximal function is integrable. A remarkably simple object, called dyadic H 1 , appears in both of these classes.

xii

Preface

Several of the most important theorems for atomic H 1 spaces directly correspond to a theorem for the martingale class (atomic decomposition, equivalence of square functions and maximal functions, H 1 − BMO duality, singular integrals correspond to martingale transforms, etc. ) . Given the obvious analogies between the two families of Banach spaces, A. Pelczy´ nski conjectured that there are deeper connections to be uncovered. A. Pelczy´ nski’s conjecture asked speciﬁcally whether dyadic H 1 and H 1 (T) (the L1 subspace consisting of boundary values of analytic functions) are isomorphic as Banach spaces. For a family of Banach spaces, whose members are constructed by one and the same rule, the question of its isomorphic classiﬁcation is inevitable. This applied in particular to the class of H 1 spaces, and clarifying the special role played by dyadic H 1 (with its obvious unconditional basis) posed a problem of particular interest. Its solution by B. Maurey opened a new line of investigation for the Banach space H 1 . In this treatise we give a thorough analysis of B. Maurey’s work and its consequences, including L. Carleson’s construction of an unconditional basis for H 1 (T). This monograph is a study of dyadic H 1 , its isomorphic invariants and its position within the two classes of martingale and atomic H 1 spaces. Simultaneously, we provide a detailed analysis of the Haar system and of the operators that are built from it. These include rearrangement operators, orthogonal projections, paraproducts and Calderon–Zygmund integral operators. This book was written under the conviction that the intensive occupation with the Haar system is fully justiﬁed by its impact on wide ranging areas of analysis and probability. The following list has been compiled with the aim of substantiating this point of view. 1. The Haar system is clearly the simplest martingale diﬀerence sequence, yet we have B. Maurey’s theorem to the eﬀect that every martingale is conveniently embedded in the partial sums of a Haar series. 2. Calderon–Zygmund operators can be expressed as paraproducts plus an absolutely converging series of basic operators rearranging and splitting the Haar system. This is the content of Figiel’s integral representation. 3. In establishing isomorphic invariants one is clearly free to choose the simplest object among the spaces known to be isomorphic to the Banach space under investigation. For instance, dyadic H 1 is the most unassuming example among the martingale spaces H 1 [(Fn )] and also among the atomic spaces H 1 (X, d, µ). Hence we work in dyadic H 1 to establish isomorphic invariants and simultaneously we search in the classes H 1 [(Fn )] and H 1 (X, d, µ) for the spaces that are isomorphic to dyadic H 1 . This line of development was initiated by the outstanding work of B. Maurey. It led to theresult that H 1 (X, d, µ) is isomorphic to one of the following spaces, H 1 , ( Hn1 )1 and 1 (the same conclusion holds for martingale spaces when the generating σ-algebras are purely atomic).

Preface

xiii

4. Expressing a problem using Haar functions and exploiting their relative simplicity leads us quite often straight to the point, that is, to the problem’s inherent combinatorial diﬃculty. This occurs for instance in the construction of multipliers into SL∞ , in Johnson’s factorization of operators on Lp , or in the construction of a resolving operator to establish the uniform approximation property for H 1 . 5. The Haar system is a reliable instrument for separating core ideas from their surrounding technicalities. The best known examples illustrating this point are obtained by specializing, to the dyadic setting, the classical theorems of real and harmonic analysis, such as the atomic decomposition, Feﬀerman’s inequality, Hardy–Littlewood maximal function estimates, good λ inequalities, interpolation and extrapolation of operators etc. Contents. The ﬁrst chapter introduces the Haar system. In Chapter 1 we present the classical inequalities of Khintchine, Burkholder, Feﬀerman, Hardy–Littlewood. It contains a discussion of large deviation inequalities, Schechtman’s sign-embeddings and multipliers into SL∞ . We prove that the unconditionality of the Haar basis yields convergence of the Walsh–Paley expansion and estimates for the dyadic gradient in Lp . We study counterexamples pertaining to the Walsh expansion along increasing multiplicities. General martingales and biorthogonal systems are analyzed by exploiting their relation to the Haar system. The ﬁrst chapter concludes with estimates for paraproducts, Figiel’s representation of singular integral operators and with a discussion of orthogonal projections in H 1 . In Chapter 2 we review basic concepts of functional analysis. We list the known complemented subspaces of H 1 , prove the Banach space decomposition principle of A. Pelczy´ nski and discuss analytic families of operators on H p spaces. The analysis of rearrangement operators and orthogonal projections is predominant in this book. It gives rise to hard problems involving combinatorics of colored dyadic intervals. We address these problems by constructing a coherent set of combinatorial techniques, forming the content of Chapter 3. In Chapter 4 we study martingale H 1 spaces. We present a detailed analysis of Maurey’s isomorphism, thus showing the existence of an unconditional basis. Exploiting the unconditional basis, obtained by Maurey’s isomorphism, yields the classiﬁcation theorem for martingale H 1 spaces. Chapter 5 covers the isomorphic invariants of H 1 . We establish dichotomies for complemented subspaces of H 1 , prove that H 1 and H 1 (2 ) are not isomorphic as Banach spaces, and that H 1 satisﬁes the uniform approximation property. Chapter 6 contains the presentation of atomic H 1 spaces and their rela1 in considerable detail, obtaining tion to dyadic H 1 . We treat the example of Hat equivalent norms using the Hilbert transform, the Lusin area function and the non-tangential maximal function. Chapter 6 contains a careful presentation of L. Carleson’s biorthogonal system. We show that it is an unconditional basis for

xiv

Preface

1 1 Hat and thus prove B. Maurey’s theorem that Hat and H 1 are isomorphic Banach spaces. The chapter closes with the classiﬁcation theorem for atomic H 1 spaces.

Prerequisites. I assume only that the reader knows basic real, complex and functional analysis, and some probability theory, and refer to [76] by C. Goﬀman and G. Pedrick, to the ﬁrst chapters of [163] by Z. Nehari, and to the appendix of [61] by R. Durrett. Acknowledgment. My research interests were formed by my collaboration with P. Wojtaszczyk at the Polish Academy of Sciences in Warszawa (1982–1984), with G. Schechtman at the Weizmann Institute of Science in Rehovot (1988–1991) and with P. W. Jones at Yale University in New Haven (1995–1998, and 2000). It is with great pleasure that I acknowledge the profound inﬂuence of P. Wojtaszczyk, G. Schechtman and P. W. Jones on my mathematical development and on the topics covered in this monograph. I would also like to thank those mathematicians I discussed with in Linz, M. Bl¨ umlinger, J. B. Cooper, A. Kamont, K. Kiener, B. Kirchheim, V. Pillwein, W. Schachermayer, M. Schmuckenschl¨ager, C. Sch¨ utt, C. Stegall and A. Wakolbinger. I am especially indebted to B. Kirchheim with whom I organized a series of exciting seminars (1992–1995) which focussed intensely on problems of hard analysis. My records of the seminars were the starting point for the work on this book. Special thanks go to V. Pillwein who read the entire manuscript, made many valuable suggestions and detected numerous errors. Financial support was provided by the Austrian Science Foundation (FWF) project P15907-N08.

Chapter 1

The Haar System: Basic Facts and Classical Results This chapter introduces the Haar system. It contains basic inequalities associated with the Haar system and their interpretation as boundedness properties of Haar multipliers, paraproducts, rearrangement operators and orthogonal projections.

1.1

Bases in Lp

In this section we establish that the Haar system is an unconditional basis in Lp (1 < p < ∞). Proving inequalities of Khintchine and Burkholder we obtain a square function characterization of these spaces. We show that unconditionality of the Haar system implies that the Walsh system (in Walsh–Paley order) is a Schauder basis in Lp (1 < p < ∞). We discuss the multiplicity of Walsh functions and study associated approximation problems.

The Haar basis in Lp (1 ≤ p < ∞) An interval I ⊆ [0, 1] is called a dyadic interval if there exists n ∈ N ∪ {0}, and 1 ≤ k ≤ 2n such that k−1 k I= , . 2n 2n Let I1 be the left half of I and I2 be the right half of I. Then I1 , I2 are again dyadic intervals. Explicitly they are given as 2k − 2 2k − 1 2k − 1 2k , = , I1 = and I . 2 2n+1 2n+1 2n+1 2n+1 We call I1 , I2 the dyadic successors of I. Conversely I is called the dyadic predecessor of I1 and I2 . We denote by D the collection of all dyadic intervals, and

2

Chapter 1. The Haar System: Basic Facts and Classical Results

we let Dn denote the ﬁnite sub-collections {I ∈ D : |I| ≥ 2−n }. Observe that D is nested in the following sense. If I, J ∈ D are not disjoint, then either I ⊆ J or J ⊆ I. We deﬁne the L∞ normalized Haar system indexed by dyadic intervals as follows. For I ∈ D we let ⎧ ⎨ 1 on the left half of I, −1 on the right half on I, hI = ⎩ 0 otherwise. The Haar system is biorthogonal in L2 ([0, 1]). Following are several remarks relating the expansion with respect to the Haar system to a sequence of conditional expectation operators. Let An = {I ∈ D : |I| = 2−n }. Denote by En the conditional expectation with respect to the σ-algebra generated by An . Explicitly En (f ) is given as

1I . (1.1.1) En (f ) = f dt |I| I I∈An

Haar’s identity. The conditional expectation En (f ) and the expansion of f along the Haar system are related through A. Haar’s identity, 1 hJ En (f ) = hJ . f dt + f, (1.1.2) |J| 0 J∈Dn−1

The following remarks verify the basic identity (1.1.2). Let I be a dyadic interval, let I1 be the left half of I, and let I2 be the right half of I. These identities are evident, 2 · 1I1 = 1I + hI , (1.1.3) 2 · 1I2 = 1I − hI . Multiplying the equations of (1.1.3) relations 1 f dt = |I1 | I1 1 f dt = |I2 | I2

by a function f and integrating gives the 1 hI f dt + f, , |I| I |I| (1.1.4) 1 hI f dt − f, . |I| I |I|

Recall that hI = 1 on I1 and hI = −1 on I2 . Hence the two equations of (1.1.4) may be compressed into one line as follows. Let j ∈ {1, 2}, and let x ∈ Ij . Then, 1 1 hI f dt = f dt + f, hI (x). (1.1.5) |Ij | Ij |I| I |I| Equation (1.1.5) relates the average over a dyadic interval, the Haar coeﬃcient of its dyadic predecessor and the average over the dyadic predecessor. A moment’s

1.1. Bases in Lp

3

reﬂection shows that the identity expressed in (1.1.5) may be repeatedly applied. Carrying out the iteration shows the following statement. Let K ∈ D and let C(K) = {J ∈ D : J ⊃ K}. (We use J ⊃ K to denote that K is strictly contained in J.) Then for x ∈ K we obtain 1 |K|

K

1

f dt =

f dt + 0

J∈C(K)

f,

hJ hJ (x). |J|

(1.1.6)

Comparing (1.1.6) and (1.1.1) shows that the identity (1.1.2) holds true. The linear span of the indicator functions {1I : I ∈ D} forms a dense subspace of Lp ([0, 1]), 1 ≤ p < ∞. Thus Haar’s identity (1.1.2) implies that augmenting the Haar system {hI : I ∈ D} by the constant function 1[0,1] gives a complete biorthogonal system in L2 ([0, 1]). Partial sum operators associated to the Haar system. In order to form partial sum operators we need to impose a linear ordering on the Haar system {hI : I ∈ D}. We do this by associating an integer to the binary expansion of the left endpoint of a dyadic interval. Let n ∈ N ∪ {0}, and 1 ≤ k ≤ 2n . We form I ∈ D and i ∈ N, as k−1 k I= and i = 2n + k − 1. , 2n 2n Then deﬁne hi = hI .

(1.1.7)

By the deﬁning equation (1.1.7) the sequence {hi }∞ i=1 is mapped bijectively onto the Haar system indexed by dyadic intervals {hI : I ∈ D}. We include the constant function by putting h0 = 1[0,1] . Denote by P the partial sum operator with respect to the sequence of Haar functions {hi }∞ i=0 . Thus P (f ) =

f,

i=0

hi hi .

hi 22

(1.1.8)

The representation formula (1.1.9) below implies that the partial sum operators {P }∞ =0 are given by a sequence of conditional expectations. This statement summarizes A. Haar’s original analysis [79] of the orthogonal system that is now named after him. Proposition 1.1.1. For every ∈ N there exists a collection of pairwise disjoint dyadic intervals P so that

1I f dt P (f ) = . (1.1.9) |I| I I∈P

4

Chapter 1. The Haar System: Basic Facts and Classical Results

Proof. Let ∈ N. Let L ∈ D be the dyadic interval which is uniquely deﬁned by the relation h = hL . Let n ∈ N be such that |L| = 2−n . Let An = {I ∈ D : |I| = 2−n }. Next depending on the position of L we will deﬁne a splitting of An into RL and LL . Given L we let RL = {K ∈ An : sup K ≥ sup L + 2−n }. The collection RL contains those dyadic intervals of length 2−n which are strictly to the right of L. Next let LL = {K ∈ An : sup K ≤ sup L}. Thus LL contains the interval L and the dyadic intervals of length 2−n which lie to the left of L. For K ∈ LL we let K1 be the left half of K and K2 be the right half of K. Next deﬁne the collection P by taking the union P = RL ∪ {K1 , K2 }. K∈LL

We now turn to verifying that with P the identity (1.1.9) holds true. The starting point is the observation that {hi : 1 ≤ i ≤ } = {hI : I ∈ Dn−1 ∪ LL }. Hence with I = Dn−1 ∪ LL we rewrite P (f ) as P (f ) =

1

f dt + 0

I∈I

f,

hI hI . |I|

(1.1.10)

Comparing the right-hand side of (1.1.10) with the identity in (1.1.6) gives (1.1.9). Remarks. 1. The above proof shows that the length of the intervals in P is related to the index by the following two-sided estimate, 1 2 ≤ |I| ≤ 2

for

I ∈ P .

(1.1.11)

2. Let F be the σ -algebra generated by the collection of pairwise disjoint dyadic intervals P . Then F ⊂ F+1 . Hence the partial sum operators of the Haar system are given by conditional expectation operators associated to an increasing sequence of σ -algebras. Schauder’s theorem. The next theorem states that on Lp ([0, 1]) the partial sum operators {P }∞ =0 are uniformly bounded. This is J. Schauder’s contribution to the analysis of the Haar system. The representation formula (1.1.9) provided the starting point for J. Schauder in [185].

1.1. Bases in Lp

5

Theorem 1.1.2. Let 1 ≤ p ≤ ∞, let f ∈ Lp ([0, 1]). Then for every ∈ N,

P f Lp ≤ f Lp .

(1.1.12)

p For 1 ≤ p < ∞, the linear span of {hi }∞ i=0 is dense in L ([0, 1]) and

lim P f − f Lp = 0.

→∞

(1.1.13)

Proof. By Proposition 1.1.1 the partial sum operator can be written as

1I , (1.1.14) P (f ) = f dt |I| I I∈P

where P is a collection of pairwise disjoint dyadic intervals. Applying H¨ older’s inequality to the right-hand side of (1.1.14) gives that,

P f pLp

p 1−p = f dt |I| I I∈P ≤ |f |p dt. I∈P

I

The sum appearing in the last line of the above estimate is just f pLp . Thus (1.1.12) holds true for 1 ≤ p < ∞. The case of L∞ follows directly from (1.1.14). The linear span of the indicator functions of dyadic intervals forms a dense subspace of Lp ([0, 1]), 1 ≤ p < ∞. Hence by (1.1.2) the linear span of the Haar system is dense in Lp ([0, 1]), 1 ≤ p < ∞. It remains to verify (1.1.13). Let f ∈ Lp ([0, 1]) and 1 ≤ p < ∞. Given > 0 there exist 0 ∈ N and g ∈ span{hi : i ≤ 0 } such that f − g Lp < . Then for ≥ 0 it is easy to see that g = i=0 bi hi . Hence P (g) = g and the following identity holds, f − P (f ) =f − g + P (g − f ). (1.1.15) The identity (1.1.15) and the norm estimate (1.1.12) imply (1.1.13).

For f ∈ Lp ([0, 1]), 1 ≤ p < ∞, we call the series ∞ i=0

ai h i

where

ai = f,

hi ,

hi 22

the Haar expansion of f and {ai } the sequence of Haar coeﬃcients. By J. Schauder’s Theorem 1.1.2, the Haar expansion converges to f in Lp ([0, 1]) and the Haar coeﬃcients are uniquely determined.

6

Chapter 1. The Haar System: Basic Facts and Classical Results

Schauder bases. The preceding theorem leads to the notion of bases in Banach spaces. A sequence {xi }∞ i=1 in a Banach space E is called a Schauder basis if for every x ∈ E there exists a unique sequence of coeﬃcients {ai } so that lim x −

n→∞

n

ai xi E = 0.

i=1

Thus associated to a Schauder basis there is the sequence of partial sum operators deﬁned by the equation

∞ n Pn a i xi = a i xi . i=1

i=1

A theorem of St. Banach asserts that for a Schauder basis {xi }∞ i=1 , the partial sum operators are uniformly bounded, i.e., sup Pn : E → E < ∞.

(1.1.16)

n

The supremum in (1.1.16) is called the basis constant of {xi }∞ i=1 in E. We denote by E ∗ the dual Banach space to E consisting of continuous linear functionals on E. A Schauder ∞basis gives rise to continuous biorthogonal functionals as follows. For x = i=1 ai xi , deﬁne a linear functional x∗i : E → R by putting x∗i (x) = ai . Thus x∗i (x)xi = (Pi − Pi−1 )(x), hence

x∗i E ∗ xi E ≤ 2 sup Pn : E → E < ∞, n

and x∗i (xj )

=

1 0

if i = j, if i =

j.

By Theorem 1.1.2 for 1 ≤ p < ∞, the Haar system {hi }∞ i=0 is a Schauder p basis in Lp ([0, 1]). The basis constant of {hi }∞ in L ([0, 1]) is equal to 1. The i=0 biorthogonal functional to hi is hi / hi 22 .

Khintchine’s inequality The Khintchine inequality asserts that for series of Rademacher functions the norm in L2 and the norm in Lp , p < ∞, are equivalent. For n ∈ N, we let rn = hI , (1.1.17) {I:|I|=2−n+1 }

1.1. Bases in Lp

7

denote the n-th Rademacher function. The sequence {rn , n ∈ N} is the Rademacher system. It consists of statistically independent functions taking the values {+1, −1}, each with probability one-half. Following is the Khintchine inequality for L1 . Theorem 1.1.3. (Khintchine’s inequality). For every sequence of real numbers {an } and for every N ∈ N,

1 0

N 1/2

N 1 2 an rn (y) dy ≥ √ an . 2 n=1 n=1

(1.1.18)

The constant √12 , appearing on the right-hand side of Khintchine’s inequality is best possible. In this sharp form the inequality (1.1.18) is due to S. Szarek [200]. The Walsh system. While the Khintchine inequality itself involves just the Rademacher functions, the proof relies heavily on certain properties of the Walsh system. We recall the deﬁnition of the Walsh system. Let A be a ﬁnite set of nonnegative integers. Then the Walsh function wA is deﬁned as a product of Rademacher functions as follows, wA = rk . k∈A

Using the empty set as index we denote the constant function as 1[0,1] = w∅ . The Walsh system {w∅ } ∪ {wA : A ⊂ N, A ﬁnite} is complete and orthonormal in L2 ([0, 1]). Hence every X ∈ L2 ([0, 1]) can be expanded in its Walsh series, X= X, wA wA . The Haar system and the Walsh system are related by the basic identity of J.L. Walsh, (1.1.19) span{wA : max A ≤ n} = span{hI : |I| ≥ 2−n+1 }. Hence if X ∈ L2 ([0, 1]) is constant on all dyadic intervals of length = 2−n , then X ∈ span{w∅ } ∪ {wA : max A ≤ n}, and conversely. The dyadic derivative. Given a ﬁnite linear combination of Walsh functions X, we deﬁne the dyadic derivative of X by the equation D(X) = |A|X, wA wA . Proof of Theorem 1.1.3. Fix real numbers {an }, let N ∈ N and form N X(t) = an rn (t) . n=1

8

Chapter 1. The Haar System: Basic Facts and Classical Results

Proving Kintchine’s Inequality amounts to showing that the following estimate holds, 1 2 1 X 2 (t)dt ≤ 2 X(t)dt . 0

0

N Overview. Below we will see that the function X(t) = | n=1 an rn (t)| is distinguished by satisfying the following two very peculiar properties: If |A|X, wA = 0, then |A| ≥ 2.

(1.1.20)

For every t ∈ [0, 1], we have X(t) ≥ D(X)(t).

(1.1.21)

Let us accept temporarily that X(t) satisﬁes the properties (1.1.20) and (1.1.21) and let us show how Kintchine’s inequality follows. 1 1 2 X (t)dt ≥ X(t)D(X)(t)dt 0 0 = |A|X, wA 2 ≥ 2X, wA 2 |A|≥2

1

X (t)dt − 2

=2 0

Now subtract 2 −1. This gives

1 0

2

1

2

X(t)dt

.

0

X 2 (t)dt on both sides and multiply the resulting inequality by

1

X (t)dt ≤ 2 0

2

1

2

X(t)dt

,

0

as claimed. Veriﬁcation of (1.1.20). We prove now that X, wA = 0 for |A| = 1. The Rademacher functions are those Walsh functions wA for which |A| = 1. Hence (1.1.20) is just stating that X, rn = 0, for every Rademacher function. Note that each Rademacher function is odd around the point 1/2. That is for 0 < t ≤ 1/2 we have that 1 1 rn (−t + ) = −rn (t + ). 2 2 By taking absolute values we ﬁnd that N X(t) = an rn (t) n=1

1.1. Bases in Lp

9

is even around the same point 1/2. The interval [0, 1] is clearly symmetric around the point 1/2. Hence, 1 X(t)rn (t)dt = 0. 0

Veriﬁcation of (1.1.21). Next we prove the pointwise estimate X ≥ DX. Note that X is constant on all intervals of length = 2−N . In other words, X depends only on (products and sums of) the Rademacher functions r1 (t), . . . , rN (t). Consequently, there exists a function F such that X(t) = F (r1 (t), . . . , rN (t)). The next identity for the dyadic derivative is easy to check for individual Walsh functions wA , hence by linear extension it holds in general. It should be thought of as the dyadic chain rule. 2 · D(X) = F (r1 , . . . , rN ) − F (−r1 , . . . , rN ) + F (r1 , r2 , . . . , rN ) − F (r1 , −r2 , . . . , rN ) .. . + F (r1 , . . . , rN ) − F (r1 , . . . , −rN ). Clearly the N summands on the left column of the above sum are identical and equal to X. Now we let Rj denote the operator acting on the j-th coordinate of F by reversing the sign. That is (Rj F )(r1 , . . . , rj , . . . , rN ) = F (r1 , . . . , −rj , . . . , rN ). We may rewrite the dyadic chain rule for X in the following way, 2 · D(X) = N · X −

N

(Rj F )(r1 , . . . , rN )

j=1

=N ·X −

N

Rj (|a1 r1 + · · · + aN rN |).

j=1

Now we continue by ﬁnding a minorization of the sum on the right-hand side. By the triangle inequality we have N N Rj (|a1 r1 + · · · + aN rN |) ≥ Rj ((a1 r1 + · · · + aN rN )) . j=1 j=1 The sum on the right-hand side is clearly a linear combination of the Rademacher functions r1 , . . . , rN . Observe that each Rademacher function appears N −2 times.

10

Chapter 1. The Haar System: Basic Facts and Classical Results

Hence N

Rj ((a1 r1 + · · · + aN rN )) = (N − 2)(a1 r1 + · · · + aN rN ).

j=1

Combining we observe the following lower bound, N

Rj (|a1 r1 + · · · + aN rN |) ≥ (N − 2)|a1 r1 + · · · + aN rN |.

j=1

It remains to insert this estimate in the identity expressed by the dyadic chain rule. This gives that, 2 · D(X) ≥ N · X − (N − 2) · |a1 r1 + · · · + aN rN | = 2X.

Kahane’s inequality. We may now review the proof of Theorem 1.1.3. We showed that for any choice of scalar coeﬃcients (an ), the function an rn (t) X(t) = satisﬁes the following surprising estimate, 1 X 2 (t)dt ≤ 2 0

2

1

X(t)dt

.

0

J. P. Kahane extended Khintchine’s inequality in the following way. For any normed linear space E and any sequence (xn ) in E the non-negative function Z(t) = xn rn (t) E

satisﬁes

1

Z 2 (t)dt ≤ C 0

2

1

Z(t)dt

,

0

where C is a universal constant. The preceding estimate is known as Kahane’s inequality. We point out that the proof of Theorem 1.1.3 not only shows Kahane’s inequality, but also that the constant in Kahane’s inequality is the same as in the real case. Indeed, we have that 1 2 1 2 Z (t)dt ≤ 2 Z(t)dt . 0

0

Next we prove Khintchine’s inequality for Lp , where p ≥ 2. The constant we √ obtain in Khintchine’s inequality is of order p. For p → ∞, this gives the right asymptotics.

1.1. Bases in Lp

11

Theorem 1.1.4. Let p ≥ 2. Let {an } be a sequence of real numbers. Then the following inequality holds,

1

| 0

N

1/p p

an rn (t)| dt

N 1/2 √ 2 ≤A p an .

n=1

n=1

Proof. Let a ∈ R. Then for each Rademacher function rn the following inequality holds, 1 1 exp{arn }dt = (exp{a} + exp{−a}) 2 0 (1.1.22) 2 a ≤ exp . 2 2 Next ﬁx {an } and normalize such that N n=1 an = 1. Then we form F (t) =

N

an rn (t).

n=1

Let µ > 0. As the Rademacher functions {rn } are independent we may iterate inequality (1.1.22) to obtain that

1

exp{µF } ≤ 0

N

exp{

n=1

µ2 µ2 a2n } = exp{ }. 2 2

Next observe that exp{µ|F (t)|} ≤ exp{µF (t)} + exp{−µF (t)}. Hence,

1

exp{µ|F |} ≤ 2 exp{ 0

µ2 }. 2

By the inequality of Chebyshef, the following estimate holds for the distribution function of |F |. For every α > 0, 1

exp{µ|F (t)|}dt exp α µ2 − α}. ≤ 2 exp{ 2

|{µ|F | > α}| ≤

0

(1.1.23)

We specialize the estimate (1.1.23) by taking α = µ2 , then |{|F | > µ}| ≤ 2 exp{−

µ2 }. 2

(1.1.24)

Integrating inequality (1.1.24) gives an upper bound for ||F ||pp . After a change of variables we ﬁnd an estimate for ||F ||pp in terms of the Gamma function, for which

12

Chapter 1. The Haar System: Basic Facts and Classical Results

standard inequalities exist. Indeed,

1

|F (t)|p dt = p 0

∞

sp−1 |{|F | > s}|ds 0 2 ∞ s sp−1 exp − ≤ 2p ds 2 0 ∞ p−2 p−2 r 2 exp{−r}dr = 2p2 2 0 p−2 p−2 = 2p2 2 Γ . 2

At this point we use a simple estimate for the Gamma function. Namely, p p−2 Γ ≤ (Ap) 2 . 2

Remark. With a surprising trick we obtain from Theorem 1.1.4 the√L1 version of Khintchine’s inequality (albeit with a constant that is larger that 2). Deﬁne N a n rn . X= n=1

1/3

2/3

H¨ older’s inequality implies that X 2 ≤ X 1 X 4 . Apply Theorem 1.1.4 2/3 1/3 with p = 4. This gives X 4 ≤ 2A X 2 . Cancelling X 2 we get X 2 ≤ 1/3 2/3 1 (2A) X 1 hence the L version of Khintchine’s inequality with constant 4A2 . Basic sequences. In a Banach space E, a sequence {yi }∞ i=1 is called a basic sequence if it is a Schauder basis in its closed linear span. Two basic sequences ∞ {yi }∞ i=1 in E and {zi }i=1 in F are called equivalent if there exist c > 0 and C > 0 so that n n n ai zi F ≤

ai yi E ≤ C

ai zi F , c

i=1

i=1

i=1

for every n ∈ N and every choice of scalars a1 . . . an . Let 1 ≤ p < ∞. Khintchine’s inequality asserts that the Rademacher functions in Lp are equivalent to the unit vector basis in the Hilbert space 2 (in particular the closed linear span of the Rademacher functions is isomorphic to 2 ). If moreover 1 2 and f ∈ L we obtain from Theorem 1.1.4 that

∞

∞ 1/2 √ 2 f, rn rn ≤ A p f, rn . n=1

Consequently,

(1.1.25)

n=1

p

√ √

R1 (f ) p ≤ A p f 2 ≤ A p f p .

Next choose 1 < q < 2 and let p be the H¨older conjugate exponent deﬁned by 1/p + 1/q = 1. Then p > 2. Let g ∈ Lq be a ﬁnite linear combination of Haar functions. Then R1 (g) is a ﬁnite linear combination of Rademacher functions. Determine f ∈ Lp so that f p = 1 and R1 (g) q = R1 (g), f . Using that R1 is self adjoint, we obtain from (1.1.25) that R1 (g), f = g, R1 (f ) ≤ g q R1 (f ) p √ ≤ A p g q .

(1.1.26)

√ Hence R1 (g) q ≤ A p g q , as claimed.

Burkholder’s inequality We give a proof of Burkholder’s inequality, and obtain the square function characterization of Lp by merging the inequalities of Burkholder and Khintchine. Theorem 1.1.5. Let 1 < p < ∞. Let f ∈ Lp ([0, 1]) with

1

f=

f dt + 0

Let I ∈ {−1, 1}, then

g=±

aI h I .

I∈D

1

f dt + 0

I aI hI ,

I∈D

satisﬁes the estimate ||f ||p ≤ (p∗ − 1)||g||p , where p∗ = max{p, p/(p − 1)}. Proof. By a simple duality argument to prove the case when p > 2. it suﬃces Then p∗ = p. We assume that f = f + aI hI is a ﬁnite linear combination of Haar functions. We will show that ||f ||p ≤ (p − 1)||g||p .

14

Chapter 1. The Haar System: Basic Facts and Classical Results

With the function v(x, y) = |x|p − (p − 1)p |y|p , we have that

||f ||pp

− (p − 1)

p

||g||pp

=

1

v(f (t), g(t))dt. 0

Hence we have to show that

1

v(f (t), g(t))dt ≤ 0.

(1.1.27)

0

To this end we deﬁne Burkholder’s most famous function, 1 u(x, y) = p(1 − )p−1 (|x| + |y|)p−1 (|y| − (p − 1)|x|). p

(1.1.28)

This function became famous because of the following four properties: (B 1) u dominates v, v(x, y) ≤ u(x, y). (B 2) For every a ∈ R and ∈ {−1, 1}, u(x + a, y + a) + u(x − a, y − a) ≤ 2u(x, y). (B 3) u is symmetric, u(x, y) = u(−x, −y). (B 4) u(0, 0) = 0. We denote the mean value of f over the dyadic interval J by fJ . That is, 1 f. fJ = |J| J Correspondingly we denote the mean value of g over J by gJ . Let J1 be the left half of J and let J2 be the right half of J. We claim that for Burkholder’s function u the following inequality holds: |J1 |u(fJ1 , gJ1 ) + |J2 |u(fJ2 , gJ2 ) ≤ |J|u(fJ , gJ ).

(1.1.29)

Note that for t ∈ J1 we have that hJ (t) = 1 whereas for t ∈ J2 , hJ (t) = −1. Hence by (1.1.3) for f the following identities hold, fJ1 = fJ + aJ , fJ2 = fJ − aJ ,

1.1. Bases in Lp

15

and analogously we have for g, gJ1 = gJ + J aJ , gJ2 = gJ − J aJ . Inserting the above identities and using property (B 2) of Burkholder’s function we have that u(fJ1 , gJ1 ) + u(fJ2 , gJ2 ) = u(fJ + aJ , gJ + J aJ ) + u(fJ − aJ , fJ − J aJ ) ≤ 2u(fJ , gJ ). It remains to multiply both sides with |J|/2 to see that the claimed inequality (1.1.29) holds true. Next ﬁx k ∈ N and sum the inequality (1.1.29) over those dyadic intervals for which the length is equal to 2−k . Thus we obtain from inequality (1.1.29) this crucial rescaling estimate, |K|u(fK , gK ) ≤ |J|u(fJ , gJ ). (1.1.30) {K:|K|=2−k }

{J:|J|=2−k+1 }

Note that the right-hand side of the estimate (1.1.30) is a rescaled version of the left-hand side; the diﬀerence is that the length of the dyadic intervals we are summing over has increased by a factor of 2. The assertion of (1.1.30) is that by rescaling from intervals of length 2−k to larger intervals of length 2−k+1 we are increasing the value of the sum. Now we turn to verifying (1.1.27). As f and g are ﬁnite linear combinations there exists a (large) constant k0 so that f and g are constant on dyadic intervals of length ≤ 2−k0 . Hence, 1 v(f (t), g(t))dt = v(fK , gK )|K|. 0

{K:|K|=2−k0 }

Recall that Burkholder’s function u is a pointwise majorant of v, hence, v(fK , gK ) ≤ u(fK , gK ), and we obtain the following line from which we start an inductive procedure, ||f ||pp − (p − 1)p ||g||pp ≤ u(fK , gK )|K|. (1.1.31) {K:|K|=2−k0 }

Now we exploit the rescaling estimates (1.1.30). We start with the right-hand side of the inequality (1.1.31) where the summation is taken over 2k0 intervals of length = 2−k0 . Call the resulting sum σ(k0 ). Repeatedly applying the rescaling estimate (1.1.30) shows that the following chain of inequalities holds true, σ(k0 ) ≤ σ(k0 − 1) ≤ · · · ≤ σ(1) ≤ σ(0).

16

Chapter 1. The Haar System: Basic Facts and Classical Results

To form σ(0) we take the sum over one interval of length =1. Thus σ(0) = u(f[0,1] , g[0,1] ). Finally we recall that the Burkholder function u is symmetric, by (B 3), and that u(0, 0) = 0. This gives that u(fK , gK )|K| ||f ||pp − (p − 1)p ||g||pp ≤ {K:|K|=2−k0 }

≤ u(f[0,1] , g[0,1] ) 1 u(f[0,1] , g[0,1] ) + u(−f[0,1] , −g[0,1] ) = 2 ≤ u(0, 0) = 0.

Given f ∈ Lp ([0, 1]) we deﬁne the square function of f as follows, S(f ) =

a2I h2I

1/2 ,

where

aI = f,

hI . |I|

Khintchine’s inequality together with Burkholder’s theorem imply a square function characterization of Lp , when 1 < p < ∞. The original proof of Theorem 1.1.6, with diﬀerent constants is due to R. E. A. C. Paley. Theorem 1.1.6. Let 1 < p < ∞. Let f ∈ Lp ([0, 1]) such that f = 0. Then √ √ (2 2)−1 (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p ≤ 2(p∗ − 1)||f ||p , where p∗ = max{p, p/(p − 1)} denotes the constants appearing in Burkholder’s inequality. Proof. Let f ∈ Lp ([0, 1]), and expand it in its Haar series, f= aI h I . Let {rI } be an enumeration of the Rademacher system indexed by dyadic intervals. For y ∈ [0, 1], we deﬁne fy = aI hI rI (y). Then by Burkholder’s inequality, Theorem 1.1.5, for each y ∈ [0, 1], the following inequality holds, 1 1 p ∗ p |fy (t)| dt ≤ (p − 1) |f (t)|p dt. 0

0

Now we pass from the above pointwise estimate, holding for every y ∈ [0, 1], to the average over y ∈ [0, 1]. Clearly the average satisﬁes the same upper bound, that is, 1 1 1 |fy (t)|p dt ≤ (p∗ − 1)p |f (t)|p dt. 0

0

0

1.1. Bases in Lp

17

Next we ﬁx t ∈ [0, 1] and we apply Minkowski’s inequality followed by Khintchine’s inequality, Theorem 1.1.3, 1 p 1 |fy (t)|p dy ≥ |fy (t)|dy 0 0 p 1 ≥ √ S(f )(t) . 2 It remains to use Fubini’s theorem and string together the above estimates. p 1 1 1 1 p √ S (f )(t)dt ≤ |fy (t)|p dydt 2 0 0 0 1 1 (1.1.32) = |fy (t)|p dtdy 0

0

≤ (p∗ − 1)p

1

|f (t)|p dt. 0

This proves the right-hand side inequality in Theorem 1.1.6. We now turn to the left-hand side. Here we distinguish between the cases p ≥ 2 and p ≤ 2. We start with the latter case, p ≤ 2. We apply Burkholder’s inequality in the following way. For each y ∈ [0, 1], 1 1 |f (t)|p dt ≤ (p∗ − 1)p |fy (t)|p dt. (1.1.33) 0

0

Consequently the same estimate holds true if we take the average over y ∈ [0, 1] of the right hand side in (1.1.33). Thus 1 1 1 p ∗ p |f (t)| dt ≤ (p − 1) |fy (t)|p dtdy. (1.1.34) 0

0

0

Next observe that for p ≤ 2, Minkowski’s inequality gives the estimate

1

p/2

1

|fy (t)| dy ≤

|fy (t)| dy 2

p

0

.

(1.1.35)

0

Finally we combine the estimates (1.1.34) and (1.1.35) and apply Fubini’s theorem as follows, 1

1

|f (t)|p dt ≤ (p∗ − 1)p

0

|fy (t)|p dtdy 0

∗

≤ (p − 1)

1

0

1

|fy (t)| dy 0

= (p∗ − 1)p

p/2

1 2

p

0 1

S p (f )(t)dt. 0

dt

18

Chapter 1. The Haar System: Basic Facts and Classical Results

Summing up we showed that for 1 < p ≤ 2 the left-hand side estimate in Theorem 1.1.6 holds, that is, (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p . Next we turn to the case when p ≥ 2. Here we use duality. Let q ≤ 2 be the older conjugate index to p, thus 1/p + 1/q = 1. Let h ∈ Lq with ||h||q ≤ 2, and H¨ h = 0 so that ||f ||p = f, h. By the inequalities of Cauchy–Schwarz and H¨ older we obtain that

1

f, h ≤

S(f )(t)S(h)(t)dt 0

≤ ||S(f )||p ||S(h)||q . We established above (1.1.32). Thereby we proved the right-hand √ side estimate of Theorem 1.1.6. Now we apply (1.1.32) obtaining ||S(h)||q ≤ 2(q ∗ − 1)||h||q . As ||h||q ≤ 2, this gives √ ||f ||p ≤ 2 2(q ∗ − 1)||S(f )||p . Finally we observe that for p ≥ 2, we have q ≤ 2, and also q ∗ − 1 = p − 1 = p∗ − 1. Summing up for p ≥ 2, √ ||f ||p ≤ 2 2(p∗ − 1)||S(f )||p . Unconditional bases. In a Banach space E let {xi }∞ i=1 be a Schauder basis with ∞ biorthogonal functionals {x∗i }∞ . We say that {x } i i=1 i=1 is an unconditional basis for E if the following condition holds: There exists K > 0 so that for each x ∈ E, n ∗ sup sup i xi (x)xi ≤ K x E . (1.1.36) n i ∈{−1,1} i=1

E

The inﬁmum over all constants K > 0 satisfying (1.1.36) is the unconditional basis constant of {xi }∞ i=1 in E. By Schauder’s Theorem 1.1.2 the Haar system is a basis in Lp ([0, 1]), 1 ≤ p < ∞. Burkholder’s inequality asserts that for 1 < p < ∞ the Haar system is actually an unconditional basis and that the unconditional basis constant of the Haar system in Lp is bounded by max{p, p/(p − 1)} − 1. The constant max{p, p/(p − 1)} − 1 is known to be best possible (see for instance Chapter 8 in [165] by I. Novikov and E. M. Semenov). The UMD property of Banach spaces. Before examining further consequences of Burkholder’s Theorem 1.1.5 we will devote a paragraph to the deﬁnition of the Banach spaces for which the vector-valued extension of Burkholder’s theorem

1.1. Bases in Lp

19

holds true. A Banach space satisﬁes the UMD property if for 1 0 so that the following estimate holds, p p 1 1 I aI hI (t) dt ≤ Cpp aI hI (t) dt, (1.1.37) 0 0 I∈H

I∈H

E

E

for every ﬁnite collection H of dyadic intervals, every {aI ∈ E : I ∈ H} and every choice of signs {I ∈ {+1, −1} : I ∈ H}. Basic properties of UMD spaces are recorded in B. Maurey’s seminal paper on the Haar system [139]. These include: (a) If E has the UMD property, then every closed subspace of E is reﬂexive. (b) The observation of G. Pisier that if (1.1.37) holds for some p0 (1 < p0 < ∞) then (1.1.37) holds for all 1 < p < ∞. In particular to verify the UMD property for E it suﬃces to establish (1.1.37) for L2 . B. Maurey’s work on the Haar system was the starting point for intensive investigations of Banach spaces with the UMD property. We pay close attention to the methods developed for the study of UMD spaces and present in detail the following: (a) B. Maurey’s proof that martingale diﬀerences are unconditional in the reﬂexive Lp spaces (Theorem 1.4.1). (b) J. Bourgain’s proof of E. M. Stein’s martingale inequality (Theorem 1.4.2). (c) T. Figiel’s representation of singular integral operators and T. Figiel’s expansion of biorthogonal systems (Theorem 1.5.3 and Theorem 1.4.3). (d) The proof by A. Naor and G. Schechtman of G. Pisier’s inequality on dyadic gradients (Theorem 1.1.7).

The Walsh system in Lp (1 < p < ∞) Here we use the Haar basis to study the Walsh system. In particular we establish estimates for Walsh series by applying the unconditionality of the Haar basis in Lp . The best known application is Paley’s proof that the Walsh system (in Walsh– Paley order) is a Schauder basis in Lp . A recent connection between the Haar basis and the Walsh system is established in the work of A. Naor and G. Schechtman on Lp estimates for dyadic gradients. Besides expansion in Walsh–Paley order we test a diﬀerent and natural method of approximation using Walsh functions. The method is brieﬂy described as follows: First project a given function f onto the span of Walsh functions wA with |A| = m. Denote the result by Rm (f ). To recover f use the associated partial sum operators n Rm (f ), m=0

20

Chapter 1. The Haar System: Basic Facts and Classical Results

and let n tend to inﬁnity. Clearly this method works well for the Hilbert space L2 . Below however we will present the surprising theorem of B. Roider to the eﬀect that for any p = 2 there is f ∈ Lp which is not approximated by the above partial sums. The proof presented here is due to K. Kiener, who builds on core ideas of P. Enﬂo’s construction of a separable Banach space without the approximation property. The following link between the Walsh system and the Haar system was ob1 served by J. L. Walsh in 1923. Let g ∈ Lp ([0, 1]) with 0 gdt = 0. Let k ∈ N, and put dk = g, wA wA . {A⊂N : max A=k}

By deﬁnition, the sequence {dk } is disjointly supported over the Walsh system. Simultaneously, and this is the point of Walsh’s observation, {dk } is also disjointly supported over the Haar system. Indeed, the following identity holds, hI dk = g, hI . |I| −k+1 {I : |I|=2

∞

}

Consequently g = k=1 dk , with convergence in Lp . Moreover with Burkholder’s theorem the observation of Walsh implies that ∞ ±dk ≥ (p∗ − 1)−1 g Lp . (1.1.38) k=1

Lp

The Poincar´e inequality for the dyadic gradient We apply the unconditionality of the Haar basis in Lp to prove a dyadic version of Poincar´e’s inequality. The dyadic gradient is deﬁned as follows. Let N ∈ N, and 1 ﬁx f so that f − 0 f dt ∈ span{wA : max A ≤ N }. Hence the Walsh expansion of f is of the form 1 f= f dt + cA wA , where cA ∈ R. (1.1.39) 0

A⊆{1,...,N }

Thus f is expressed through the Rademacher functions {r1 , . . . , rN } and can be represented as f (t) = F (r1 (t), . . . , rN (t)). Then the dyadic partial derivative ∂i is deﬁned by the equation 1 [F (r1 (t), . . . , ri (t), . . . , rN (t)) − F (r1 (t), . . . , −ri (t), . . . , rN (t))] . 2 In the course of proving Khintchine’s inequality we observed that the dyadic derivative D(f ), deﬁned through the Walsh expansion of f by Df = |A|wA , f wA , ∂i f (t) =

1.1. Bases in Lp

21

satisﬁes Df =

N

∂i f.

i=1

This identity holds true since ∂i acts on individual Walsh functions by the relation ∂i wA =

if i ∈ A, if i ∈

A.

wA 0

(1.1.40)

Finally deﬁne the dyadic gradient ∇f = (∂i f )N i=1 and put

|∇f | =

N (∂i f )2

1/2 .

i=1

Theorem 1.1.7. Let 1 < p < ∞. Let f ∈ Lp have Walsh expansion (1.1.39). Then

1

f dt Lp ≤ Cp ∇f Lp .

f − 0

Proof. We start by isolating an algebraic identity relating ∂i to the Walsh expansion of f. Let ai ∈ R, and let i ≤ N. Put g=

N

ai ∂i f.

i=1

Equation (1.1.40), the deﬁning relation for ∂i , gives the following useful representation of g. For t ∈ [0, 1],

g(t) =

A⊆{1,...,N }

ai

f, wA wA (t).

(1.1.41)

i∈A

Next we split the Walsh expansion of g using the basic observation of Walsh. For k ≤ N, put

dk (t) = ai f, wA wA (t). (1.1.42) {A:max A=k}

i∈A

By the identity of Walsh, the sequence {dk } is disjointly supported over the Haar system. Hence Burkholder’s Theorem 1.1.5 gives the lower bound N ±dk k=1

≥ (p∗ − 1)−1 g Lp . Lp

(1.1.43)

22

Chapter 1. The Haar System: Basic Facts and Classical Results

Now we focus on ﬁnding estimates for ∇f Lp . Recall that {ri } denotes the Rademacher system. Fix s ∈ [0, 1] and i ≤ n. We will apply the above observations using ai = ri (s). Displaying the dependence on s ∈ [0, 1] we write

dk (s, t) = ri (s) f, wA wA (t). {A:max A=k}

i∈A

Note that the integral identity below holds, since the integrands on the right-hand side and on the left-hand side coincide. p p 1 1 1 1 N N ri (s)∂i f (t) dsdt = dk (s, t) dsdt. (1.1.44) 0 0 0 0 i=1

k=1

Khintchine’s inequality gives pointwise estimates for the length of the dyadic gradient. For t ∈ [0, 1], we have that 1 |∇f (t)| ≥ (Ap)p/2

1

p

0

N p ri (s)∂i f (t) ds. i=1

Integrate over t ∈ [0, 1] and invoke the identity (1.1.44). This gives

∇f pLp

1 ≥ (Ap)p/2

1

0

1

0

N p dk (s, t) dsdt.

(1.1.45)

k=1

Next we apply Burkholder’s inequality in the form (1.1.43). Clearly rk (s) = ±1 for each ﬁxed s ∈ [0, 1]. Hence the following lower bound holds, N dk (s, ·)

Lp

k=1

N ≥ (p∗ − 1)−1 rk (s)dk (s, ·)

.

(1.1.46)

Lp

k=1

Integrate the estimate (1.1.46) over s ∈ [0, 1]. By triangle inequality, the integral of the norms ≥ norm of the integral. Thus we arrive at the following lower bound for ∇f Lp . N 1 rk (s)dk (s, ·)ds , (1.1.47) Cp ∇f Lp ≥ 0 Lp

k=1

where Cp = (Ap)1/2 (p∗ − 1). It remains to evaluate the expression appearing on the right-hand side of (1.1.47) . A glance at the deﬁnition of dk (s, t) gives that

1

rk (s)dk (s, t)ds = 0

f, wA wA (t).

{A:max A=k}

1.1. Bases in Lp

23

Hence for Cp = (Ap)1/2 (p∗ − 1), we found the lower bound N f, wA wA (t) Cp ∇f Lp ≥ k=1 {A:max A=k}

.

Lp

Finally observe that 1the right-hand side of the above estimate coincides with the p norm in L of f − 0 f. Pisier’s inequality. Estimates for the dyadic gradient were established ﬁrst by G. Pisier [178] in the context of vector-valued coeﬃcients cA ∈ E, where E is a Banach space. Following is the statement of G. Pisier’s original inequality. Let

1

f=

f dt + 0

where cA ∈ E.

cA wA ,

(1.1.48)

A⊆{1,...,N }

Equation (1.1.40) deﬁnes the dyadic partial derivatives {∂i f } by linear extension, and {ri } denotes the Rademacher functions, then

1

f (x) − 0

0

1

f pE dx

≤ (2e log N )

1

p 0

N

ri (t) ∂i f (x) pE dtdx.

(1.1.49)

i=1

In the vector-valued case the constant on the right-hand side depends on N. M. Talagrand [201] shows that 2e log N, appearing in (1.1.49) is of the correct order. Moreover in [201] M. Talagrand proves the scalar case with a constant independent of N. The proof of Theorem 1.1.7 presented above is due to A. Naor and G. Schechtman [162]. Verify, by inspection, that their proof carries over to the case where the coeﬃcients in the Walsh expansion of f belong to a Banach space with the UMD property. Convergence and divergence of Walsh series So far we were concerned with estimates for ﬁnite linear combinations of Walsh functions, hence questions of convergence did not arise. Passing from ﬁnite sums to inﬁnite series gives rise to interesting and subtle problems of ordering the Walsh system. For the Walsh–Paley enumeration {wj }∞ j=0 we show that for 1 < p < ∞, n ≤ p∗ − 1, f, w w sup sup j j

f p ≤1 n j=0

(1.1.50)

p

where p∗ = max{p, p/(p − 1)}. Clearly this estimate implies convergence of the corresponding partial sums in Lp . On the other hand, ordering the Walsh system

24

Chapter 1. The Haar System: Basic Facts and Classical Results

along increasing multiplicity leads to divergence of the corresponding partial sums in Lp for p = 2. We prove below that for 1 < p = 2 < ∞, n Rm (f ) = ∞, (1.1.51) sup sup n

f p ≤1 m=0

p

where Rm denotes the orthogonal projection onto Walsh functions of multiplicity m. The estimate (1.1.50) is due to R.E.A.C Paley, the particular constant p∗ − 1 is a consequence of Burkholder’s Theorem 1.1.5. The lower bound (1.1.51) is a result of B. Roider. The Walsh–Paley order. We now deﬁne the Walsh–Paley order in {wA }, and show Paley’s theorem that (thus ordered) the Walsh system is a Schauder basis in Lp , 1 < p < ∞. Let n ∈ N, with dyadic expansion n=

∞

j 2j ,

j=0

where j ∈ {0, 1}. (Clearly only ﬁnitely many j are = 0.) Then we deﬁne wn =

∞

j rj+1 .

(1.1.52)

j=0

We also deﬁne w0 = 1. Note that w1 = r1 , w2 = r2 and that in general w2m = rm+1 . Moreover the sequence {wn }∞ n=1 is an enumeration of the Walsh system {wA : A ⊂ N, A ﬁnite }. Paley’s identity. We prepare the proof of Paley’s theorem by stating identities relating the Walsh system to the Haar system. First we have that k 2 −1

j=0

wj =

k

(1 + rn )

n=1

(1.1.53)

k

= 2 1[0,2−k [ . Let I be a dyadic interval with |I| = 2−k . The identity (1.1.53) leads to a formula for the Walsh expansion of the Haar function hI . Moreover for t ∈ I and x ∈ [0, 1], 2k+1 −1

wj (t)wj (x) = 2k hI (t)hI (x).

(1.1.54)

j=2k

The above formulas were known to (and used by) Walsh in 1923. Following is Paley’s identity from 1931 which is the key to his proof that the Walsh system

1.1. Bases in Lp

25

can be ordered a Schauder basis in Lp , 1 < p < ∞ : Let x, t ∈ [0, 1], ∞to become k and let n = k=0 k 2 . Then wn (x)wn (t)

n−1

wj (x)wj (t) =

j=0

∞

k

k=0

2k hI (x)hI (t).

(1.1.55)

{I:|I|=2−k }

Next we show that the partial sum operators with respect to the Walsh system in Walsh–Paley order are convergent in Lp (1 < p < ∞). By the identity of Walsh it suﬃces to prove that they form a sequence of uniformly bounded operators. Theorem 1.1.8. Let 1 < p < ∞, and let {wj }∞ j=0 denote the Walsh system in Walsh–Paley order. Then the partial sum operator, Tn (f ) =

n−1

f, wj wj ,

j=0

satisﬁes the norm estimate

Tn f p ≤ (p∗ − 1) f p ,

(1.1.56)

where p∗ = max{p, p/(p − 1)}. Proof. Fix n ∈ N. Instead of working directly with Tn we consider the operator T˜n : f → wn Tn (wn f ). Clearly in Lp the norms of the operators Tn and T˜n coincide. Hence it suﬃces to show the estimate (1.1.56) with Tn replaced by T˜n . The following observation is crucial to Paley’s proof. It identiﬁes T˜n as a {0, 1} multiplier on the Haar(!) system. Precisely, we claim that for a given n there exist coeﬃcients δI ∈ {0, 1} so that hI T˜n (f ) = δI f, hI . (1.1.57) |I| I∈D

To verify (1.1.57) we observe that T˜n (f )(x) =

n−1

wn f, wj wj (x)wn (x)

j=0

1

=

f (t)wn (x)wn (t) 0

n−1

(1.1.58) wj (x)wj (t)dt.

j=0

The kernel appearing in the second line of (1.1.58) has been identiﬁed in (1.1.55). It is equal to ∞ j 2j hI (t)hI (x), j=0

{I:|I|=2−j }

26

Chapter 1. The Haar System: Basic Facts and Classical Results

where j ∈ {0, 1} is determined by the dyadic expansion of n. Replacing the kernels and carrying out the integration gives that T˜n (f ) =

∞

j

f,

{I:|I|=2−j }

j=0

hI hI . |I|

Having established (1.1.57) it suﬃces to apply Theorem 1.1.5 to conclude that

T˜n (f ) p ≤ (p∗ − 1) f p .

The multiplicity of Walsh functions. The multiplicity of a given Walsh function is deﬁned to be the number of Rademacher functions used in its deﬁnition. Formally, let A be a ﬁnite subset of N. Then the multiplicity of wA = j∈A rj , is the cardinality of A. For w∅ = 1[0,1] we deﬁne the multiplicity to be equal to 0. Let Wm denote the Walsh functions of multiplicity m. Let {wm,k }∞ k=1 be any enumeration of the elements in Wm . Let Rm denote the orthogonal projection onto the span of Wm . Thus for f ∈ L2 , Rm (f ) =

∞

wm,k , f wm,k .

(1.1.59)

k=1

For m = m the sets Wm and Wm are disjoint, hence the spaces span{Wm } are orthogonal and ∞

Rm (f ) 22 .

f 22 = m=0

It follows that the partial sums ∞ n

wm,k , f wm,k

(1.1.60)

m=0 k=1

form a convergent sequence in L2 approximating f. Hence elements in L2 are well approximated by adding Walsh functions of increasing multiplicity. We now study this method of approximation in Lp , when p = 2. The starting point is a theorem obtained by A. Bonami [15] and K. Kiener [112] independently. It extends Khintchine’s inequality to Walsh functions of multiplicity m. Theorem 1.1.9. Let 1 0 and 2 Cm,p > 0 so that for every sequence {ak }∞ k=1 ∈ ,

cm,p

∞

k=1

1/2 a2k

∞

∞ 1/2 ≤ ak wm,k ≤ Cm,p a2k . k=1

p

√ m (a) If p > 2, then cm,p = 1 and Cm,p ≤ C p .

k=1

(1.1.61)

1.1. Bases in Lp

27

−2 (b) If p < 2, then Cm,p = 1 and cm,p ≥ Cm,4 .

Moreover Rm , the orthogonal projection onto the span of Walsh functions with multiplicity m, is a bounded and self-adjoint operator on Lp , 1 < p < ∞. Proof. First we treat p ≥ 2. The proof proceeds by induction. For m = 1 the theorem holds by Khintchine’s inequality. Next assume that the theorem holds for a given m ∈ N. We will now show that it holds for m + 1. Let Km+1 = {A ⊂ N : |A| = m + 1}. The sets {A ∈ Km+1 : max A = n},

n ∈ N,

: A ∈ Km+1 } be a sequence where only ﬁnitely form a partition of Km+1 . Let {aA many entries are = 0. Deﬁne h = A∈Km+1 aA wA . Let n ∈ N and put aA wA\{n} rn . (1.1.62) dn = A∈Km+1 max A=n

We get h = dn , and by the identity of Walsh, the sequence {dn } is disjointly supported over the Haar system. The square function characterization of Lp implies that 1/2 . d2n

h p ≤ Cp (1.1.63) p √ (We prove in Theorem 1.3.4 below that Cp ≤ C p for p ≥ 2.) Note that rn is a common factor of each of the Walsh functions appearing in (1.1.62). Hence ⎞2 ⎛ ⎟ ⎜ d2n = ⎝ aA wA\{n} ⎠ . A∈Km+1 max A=n

Since p ≥ 2, the triangle inequality in Lp/2 and the above formula for d2n imply that the right-hand side of (1.1.63) is bounded by ⎛ 2 ⎞1/2 ⎟ ⎜ ⎟ Cp ⎜ a w . (1.1.64) A A\{n} ⎠ ⎝ n∈N A∈Km+1 max A=n

p

Observe that for A ∈ Km+1 and max A = n the Walsh function wA\{n} has multiplicity m. In fact {wA\{n} : A ∈ Km+1 , max A = n} equals {wB : B ∈ Km , max B ≤ n − 1}. Using the induction hypothesis we obtain that 2 2 aA wA\{n} ≤ Cm,p a2A . A∈Km+1 A∈Km+1 max A=n

p

max A=n

28

Chapter 1. The Haar System: Basic Facts and Classical Results

Recall that the collections {A ∈ Km+1 , max A = n}, n ∈ N form a partition of Km+1 . Thus we obtained that ⎛

h p ≤ Cm+1,p ⎝

⎞1/2 a2A ⎠

.

(1.1.65)

A∈Km+1

Let Rm be the orthogonal projection onto the span of Walsh functions with multiplicity m. Clearly Rm f 2 ≤ f 2 , and (since p ≥ 2) f 2 ≤ f p . Hence (1.1.65) implies that for f ∈ Lp ,

Rm f p ≤ Cm,p Rm f 2 ≤ Cm,p f p .

(1.1.66)

We pass to p ≤ 2 using a well-known trick based on H¨ older’s inequality. Clearly it suﬃces to consider p = 1. Deﬁne h= a A wA . A∈Km

1/3

2/3

H¨older’s inequality implies that h 2 ≤ h 1 h 4 , and by the ﬁrst part we get 2/3 1/3 2/3 1/3

h 4 ≤ Cm,4 h 2 . Cancelling h 2 gives h 2 ≤ Cm,4 h 1 which ﬁnishes the proof for p ≤ 2. Given Theorem 1.1.9, K. Kiener conjectured that the convergence of the partial sums (1.1.60) could be extended to Lp (1 < p = 2 < ∞). This conjecture was disproved by B. Roider (1974) in a letter to K. Kiener. Below we present K. Kiener’s proof [113] of B. Roider’s result that for p = 2 there exists g ∈ Lp which is not approximated by its partial sums ∞ n

wm,k , gwm,k .

m=0 k=1

The proof of K. Kiener [113] starts with the combinatorial identity (1.1.76) which played a central role in the construction of the ﬁrst Banach space without the approximation property. Speciﬁcally (1.1.76) appears in P. Enﬂo’s classical work [62]. Roider’s example. Note that Rm , the orthogonal projection onto the span of Walsh functions with multiplicity m, is a self-adjoint operator. Hence it suﬃces to prove (1.1.51) for p < 2. To this end we ﬁx n ∈ N, and consider f = 22n 1[0,2−2n [

(1.1.67)

1.1. Bases in Lp

29

Observe that by (1.1.53) f can be expanded using Walsh functions of multiplicity ≤ 2n. Thus f ∈ span{W0 , . . . , W2n }. Next project f onto the space of Walsh functions with multiplicity ≤ n, and denote the resulting function by X. Thus n

X=

Rm (f ).

(1.1.68)

m=0

Clearly in L2 we have X 2 ≤ f 2 . The next theorem shows that for p = 2 the norm of X in Lp is considerably larger than that of f. Theorem 1.1.10. Let 1 0 is a universal constant independent of n or p. Proof. We begin by deﬁning a partition of the unit interval. (On each set of this partition we will later identify the value of X.) Let 0 ≤ r ≤ 2n. Deﬁne Pr ⊂ [0, 1] by the following rule: Let t ∈ [0, 1], then put t ∈ Pr if there exists A ⊆ {1, . . . , 2n} so that |A| = r and so that for each m ∈ {1, . . . , 2n} −1, rm (t) = 1,

if m ∈ A, if m ∈ / A.

(1.1.70)

The proof splits naturally into three parts. First we show that the function X is constant on the sets {Pr } and we represent the value of X on {Pr } by an elegant complex integral formula. This part of the proof is based on Enﬂo’s identity. Second we parameterize the complex integrals and ﬁnd closed formulas for their value. Third we use this information to obtain a lower bound for X p . The ﬁrst two parts of the proof consist in identifying the values of X(t) deﬁned by (1.1.68) on the sets {Pr }. First, by deﬁnition, |X(t)| = 1 + 2n 1 + · · · + 2n , for t ∈ P ∪ P . We will show next that for t ∈ P , we have 2|X(t)| = 2n 0 2n 1 n n and that in general for k = 1, 2, . . . n − 1, −1 2n n 2n 2|X(t)| = n k 2k

for t ∈ P2k ∪ P2k+1 .

(1.1.71)

We obtain the identities (1.1.71) from an elegant formula due to K. Kiener. X(t) =

1 2πi

(1 − z)r−1 (1 + z)2n−r

dz z n+1

for t ∈ Pr ,

where the contour integral is taken over the boundary of the unit disk.

(1.1.72)

30

Chapter 1. The Haar System: Basic Facts and Classical Results

Applications of Cauchy’s integral formula. We turn to proving (1.1.72). Recall 22n −1 that expanding f = 22n 1[0,2−2n [ in its Walsh series gives f = j=0 wj . Hence for m ≤ 2n, we have Rm (f ) = w∈Wm w. Now ﬁx t ∈ [0, 1] and z ∈ D. Then deﬁne 2n F (z, t) = z m Rm (f )(t). (1.1.73) m=0

Verify the following identity by multiplying the factors of the product on the right-hand side of (1.1.74), F (z, t) =

2n

(1 + zrm (t)).

(1.1.74)

m=1

The product representation of F (z, t) yields that for ﬁxed r ≤ 2n and z ∈ D the the function t → F (z, t) is constant on Pr . Indeed for t ∈ Pr , 2n

(1 + rm (t)z) = (1 − z)r (1 + z)2n−r .

(1.1.75)

m=1

We apply the Cauchy integral formula to (1.1.73) and use the equation established in (1.1.75). This gives Enﬂo’s identity that for t ∈ Pr , Rm (f )(t) =

1 2πi

(1 − z)r (1 + z)2n−r

dz z m+1

,

(1.1.76)

where the contour integral is taken over the boundary of the unit disk. Take the sum of the equations (1.1.76) using that n

z −m−1 =

m=0

(1 − z n+1 ) . (1 − z)z n+1

Hence by summing (1.1.76) and Cauchy’s integral formula we ﬁnd that for t ∈ Pr , n m=0

Rm (f )(t) =

1 2πi

(1 − z)r−1 (1 + z)2n−r

dz z n+1

.

(1.1.77)

Thus we veriﬁed the complex integral representation (1.1.72). We remark that the representation of X(t) in (1.1.72) diﬀers from that of Rn (f )(t) by a factor of (1−z) in the contour integral.

1.1. Bases in Lp

31

Evaluating complex integrals. Next we determine the complex integral (1.1.77). With the following parametrization we replace (1.1.77) by a standard deﬁnite integral whose value is well known. Let z = eiθ . Then (1.1.77) equals 2π dθ 1 (1 − eiθ )r−1 (1 + eiθ )2n−r inθ . (1.1.78) 2π 0 e We simplify the integrand of (1.1.78) using elementary trigonometric identities. e−iθ/2 (1 − eiθ ) = (−2i) sin(θ/2)

and

e−iθ/2 (1 + eiθ ) = 2 cos(θ/2). (1.1.79)

Consequently, e−inθ (1 − eiθ )r−1 (1 + eiθ )2n−r = (−i)r−1 22n−1 e−iθ/2 sinr−1 (θ/2) cos2n−r (θ/2) ! " = (−i)r−1 22n−1 sinr−1 (θ/2) cos2n−r+1 (θ/2) − i sinr (θ/2) cos2n−r (θ/2) . Next replace the integrand in (1.1.78) by the last line of the above equation. Then, by the change of variables ϕ = θ/2 and dθ = 2dϕ, we ﬁnd that (1.1.78) coincides with the integral " (−i)r−1 22n π ! r−1 sin (ϕ) cos2n−r+1 (ϕ) − i sinr (ϕ) cos2n−r (ϕ) dϕ. 2π 0

(1.1.80)

To derive the pointwise estimates (1.1.71) we evaluate the trigonometric integrals of (1.1.80). This requires us to distinguish between the cases where r is odd and where r is even. Observe that cos(ϕ) is an odd(!) function around the point π/2 and that sin(ϕ) is even(!) around π/2. Clearly if r is an even number, then 2n − r + 1 is odd. Hence π/2 π r 2n−r sin (ϕ) cos (ϕ)dϕ = 2 sinr (ϕ) cos2n−r (ϕ)dϕ, 0

0

and

π

sinr−1 (ϕ) cos2n−r+1 (ϕ)dϕ = 0.

0

Similarly if r is odd, then 2n − r is odd, hence π π/2 sinr−1 (ϕ) cos2n−r+1 (ϕ)dϕ = 2 sinr−1 (ϕ) cos2n−r+1 (ϕ)dϕ, 0

and

0

π 0

sinr (ϕ) cos2n−r (ϕ)dϕ = 0.

32

Chapter 1. The Haar System: Basic Facts and Classical Results

Next recall that for x, y > 0 we have π/2 Γ(x)Γ(y) 2 , (1.1.81) sin2x−1 (ϕ) cos2y−1 (ϕ) = Γ(x + y) 0 ∞ √ where Γ(x) = 0 tx−1 e−t dt. In particular Γ(n + 1) = n! and Γ(1/2) = π. In summary it follows from Kiener’s integral representation (1.1.80) and the identity (1.1.81) that for r = 2k and t ∈ P2k , Γ k + 12 Γ n − k + 12 22n |X(t)| = × . (1.1.82) 2π n! Similarly, for r = 2k + 1 and t ∈ P2k+1 , we obtain from (1.1.80) and (1.1.81) that Γ k + 12 Γ n − k + 12 22n |X(t)| = × . (1.1.83) 2π n! Thus (for diﬀerent reasons) the value of X(t) on the set P2k coincides with its value on P2k+1 . We replace the expressions involving Gamma functions by more elementary formulas. Recall the so-called duplication formula for the Gamma function. It states that for x > 0, √ π Γ(2x) 1 . Γ(x + ) = 2x−1 2 2 Γ(x) Apply the duplication formula with x = k and write 1/2 = k/(2k). This gives, √ 1 π (2k)! . Γ(k + ) = 2k 2 2 (k)! Apply it to x = n − k and use 1/2 = (n − k)/(2n − 2k) to obtain that √ 1 π (2n − 2k)! Γ(n − k + ) = 2n−2k . 2 2 (n − k)! Substitute into (1.1.82) and (1.1.83), then simplify to obtain (1.1.71) by arithmetic. The lower bound for X p . Now we turn to the last part of the proof where we 1 apply (1.1.71) to obtain a lower bound for the integral 0 |X(t)|p dt. Since X(t) is constant on the sets Pr we denote Xr = X(t) for t ∈ Pr . Note that the disjoint sets Pr , 0 ≤ r ≤ 2n are deﬁned so that their measure is given by |Pr | = 2−2n 2n r . Hence (1.1.71) gives 1 2n 2n p p 2n 2 |X(t)| dt = |Xr | r 0 r=0 (1.1.84)

n−1 2n 2n p ≥ |X2k | + . 2k 2k + 1 k=0

1.1. Bases in Lp

33

The formula for evaluating consecutive binomial coeﬃcients gives 2n 2n 2n + 1 + = . 2k 2k + 1 2k + 1 2n Note that 2n+1 2k+1 ≥ 2k . Inserting the values (1.1.71) for |X2k | we obtain the identities p p −(p−1) 2n n 2n p 2n |2X2k | . = 2k n k 2k Then by (1.1.84) we obtain that 2n+p

1

p n−1 np 2n−(p−1) 2n |X(t)| dt ≥ × . n k 2k p

2

0

(1.1.85)

k=0

Recall H¨older’s inequality in the form n−1

−(p−1) apk bk

≥

k=0

n−1

ak

p n−1

k=0

−(p−1) bk

.

(1.1.86)

k=0

Apply it with ak = nk and bk = 2n 2k . We continue by giving closed formulas for the two factors appearing on the right-hand side of (1.1.86). n−1 k=0

n k

= 2 − 1 and n

n−1 k=0

2n 2k

≤ 22n .

In summary we obtained a lower bound for the right-hand side of (1.1.85). It follows that p 1 2n p |X(t)| dt ≥ × 2−np × 2−p−1 . n 0 √ The remaining calculations on Stirling’s formula, n! ∼ 2πn · nn · e−n . 2n are based It allows us to compute n ∼ 22n n−1/2 . Inserting this value in the previous lower bound for the norm of X in Lp we ﬁnd 1 |X(t)|p dt ≥ Cn−p/2 2np . 0

Next recall that we put f = 22n 1[0,2−2n [ . Hence f p dt = 22n(p−1) . Consequently the following lower estimate holds true. p −p/2 −np+2n 2 (1.1.87) f p dt. |X(t)| dt ≥ Cn

34

Chapter 1. The Haar System: Basic Facts and Classical Results

Recall that Rn denotes the orthogonal projection onto the span of Walsh functions with multiplicity n. We pointed out that the integral representations (1.1.72) for X and (1.1.76) for Rn (f ) are almost identical. With this observation it is easy to see that the proof of Theorem 1.1.10 gives a lower estimate for the operator norm of Rn on Lp . Theorem 1.1.9, on the other hand, provides an upper estimate. In summary the following estimates hold, √ c2cn(1−2/p) ≤ Rn Lp ≤ (C p)n , for p > 2. (1.1.88) Comparing the right and left hand side in (1.1.88) gives rise to the following problem. Problem 1.1.11. Narrow the gap between the lower and upper estimates for the operator norm of Rn on Lp . Notice that the test function (1.1.67) in the proof of Theorem 1.1.10 is very special. Hence it is likely that the lower estimate in (1.1.88) can be improved. Since the appearance of Paley’s work the Walsh system has been continuously investigated from the point of view of harmonic analysis and approximation theory. Moreover the Walsh system plays a remarkable role in the analysis of classical Banach spaces where it is used to exhibit spaces and operators with extremal (and often unexpected) properties. P. Enﬂo’s original construction of a Banach space without the approximation property is one of the most outstanding examples representing this line of investigation (see the notes for references). Through the work of B. Roider and K. Kiener in 1974 the methods of P. Enﬂo found an early application to a concrete approximation problem for Walsh functions in Lp . In 1992 J. Bourgain [25] extended B. Roider’s theorem: Denoting by Xnp the subspace of Lp spanned by the Walsh functions {wA : A ⊆ {1, . . . , 2n}, |A| ≤ n}, J. Bourgain [25] proves a trace estimate for any operator T on Lp whose range is contained in Xnp , (1.1.89) tr(T) ≤ B22n(1−γ) T Lp , where B = B(p) and γ = γ(p) > 0 when p = 2. The trace of the orthogonal projection onto Xnp is larger than c22n , hence (1.1.89) gives a lower estimate for the norm of the projection. Thus (1.1.89) implies Roider’s estimate (1.1.51).

1.2

Dyadic H 1

In this book we will work intensively with the following dyadic versions of H 1 and BMO. The space BMO consists of all square integrable functions, having mean zero, bJ hJ h= J∈D

1.2. Dyadic H 1

35

for which

⎛

||h||BMO

⎞1/2 1 = sup ⎝ b2J |J|⎠ < ∞. |I| I∈D J⊆I

The space H 1 consists of integrable functions f with integrable square functions. For 1 f= f dt + aJ h J 0

J∈D

the square function is given by

S(f )(x) =

1/2 a2J 1J (x)

,

J∈D

and H 1 is the space of integrable functions for which 1 1 1 ||f ||H = f dt + S(f )(x) dx < ∞. 0

0

Feﬀerman’s inequality Next we identify BMO as the dual space of H 1 . We begin by showing that each h ∈ BMO induces a bounded linear functional on H 1 . This follows from the next theorem, the content of which is known as Feﬀerman’s inequality. In this section we present two proofs of Feﬀerman’s inequality. Theorem 1.2.1. (Feﬀerman’s inequality) Let f be a ﬁnite linear combination of Haar functions, and let h ∈ BMO, then √ f h ≤ 2 2||f ||H 1 · ||h||BMO . Proof. Let h =

bJ hJ be the Haar expansion of h ∈ BMO. Since f hdt = f, hJ bJ ,

we may assume that {J : bJ = 0} is contained in {J : f, hJ = 0}. In particular we assume that h is a ﬁnite linear combination of Haar functions. Let I be a ﬁxed dyadic interval. We deﬁne S(h | I), the square function localized to I as ⎛ ⎞1/2 S(h | I)(x) = ⎝ b2J 1J (x)⎠ . J⊆I

36

Chapter 1. The Haar System: Basic Facts and Classical Results

Now ﬁx a point x in the unit interval. Note that S(h | I)(x) < ∞ for every dyadic interval I. Next deﬁne I(x) to be the largest dyadic interval containing x such that 1 2 S 2 (h | I(x))(x) ≤ 2 sup bL |L|. (1.2.1) I x |I| L⊆I

Note that, by deﬁnition, we have the following upper bound for the square function localized to the interval I(x), √ S(h | I(x))(x) ≤ 2||h||BM O , x ∈ [0, 1]. (1.2.2) Thus Feﬀerman’s inequality is reduced to showing that the following estimate holds true, f hdx ≤ 2 S(f )(x)S(h | I(x))(x)dx. (1.2.3) To prove the integral estimate (1.2.3) we verify ﬁrst that for every dyadic interval J, the following measure estimate holds, |{x ∈ J : I(x) ⊇ J}| ≥ |J|/2.

(1.2.4)

To this end we ﬁx J ∈ D and consider the set A = {x ∈ J : I(x) is strictly contained in J}. Then using (1.2.1), we estimate S 2 (h | J)(x) dx ≥ S 2 (h | J)(x) dx J A 1 2 sup bL |L| dx >2 A x∈I |I| L⊆I |A| ≥2 S 2 (h | J)(x) dx. |J| J

(1.2.5)

The last inequality in (1.2.5) used the following pointwise bound that holds for any x ∈ J, 1 2 1 bL |L| ≥ S 2 (h | J)(x) dx. sup |J| J I x |I| L⊆I Finally in (1.2.5) we cancel the factor J S 2 (h | J)(x) dx and obtain that |A| ≤ |J|/2. Taking complements we obtain the measure estimate (1.2.4), that is |{x ∈ J : I(x) ⊇ J}| ≥

1 |J|. 2

We rewrite the estimate (1.2.4), in the following integral form, using indicator functions, 1 1 1J (x) · 1{y:I(y)⊇J} (x) dx ≥ |J|. 2 0

1.2. Dyadic H 1 Now ﬁx f = follows.

37

aJ hJ in H 1 . With the preceding analysis of h, we estimate as f h dx ≤ |aJ bJ ||J| ≤2 |aJ bJ | 1J (x)1{y:I(y)⊇J} (x) dx =2 |bJ aJ |1J (x) dx ≤2

I(x)⊇J

S(f )(x)S(h | I(x)) dx.

Thus we veriﬁed that (1.2.3) holds. Recalling the pointwise upper bound (1.2.2) completes the proof of Feﬀerman’s inequality. Let h be square integrable, with Haar expansion 1 h= hdx + bJ hJ . 0

J∈D

At a given point x ∈ [0, 1] the sharp function of h is deﬁned to be ⎛

⎞1/2 1 h (x) = ⎝sup b2J |J|⎠ , I |I| J⊆I

where the supremum is extended over all dyadic intervals which contain the given point x ∈ [0, 1]. Looking back at the proof of Theorem 1.2.1 we observe that the sharp function played an important role in the proof of Feﬀerman’s inequality. It appeared on the right-hand side of the estimate (1.2.1). Even more, the sharp function appears implicitly in the deﬁnition of BMO. Indeed, by inspection, the space BMO consists of all functions with uniformly bounded sharp functions, and ||h||BMO = sup h (x).

(1.2.6)

x∈[0,1]

Observe that combining (1.2.1) and the integral estimate (1.2.3) we obtain the following inequality, that immediately gives Feﬀerman’s inequality, 1 √ 1 f hdx ≤ 2 2 S(f )h dx. (1.2.7) 0

0

It is worth pointing out that the above proof focusses exclusively on the BMO function h. The fact that f ∈ H 1 , enters only at the √ very end 1 when we observe that the right hand side of (1.2.7) is bounded by 2 2 h ∞ 0 S(f )dx. As an application of Khintchine’s inequality we next prove a converse to Feﬀerman’s inequality. In that way we are showing that Theorem 1.2.1 is sharp.

38

Chapter 1. The Haar System: Basic Facts and Classical Results

Theorem 1.2.2. For each f ∈ H 1 , with

f = 0, there exists h ∈ BMO such that

1 f, h ≥ √ ||f ||H 1 · ||h||BMO . 2 2 Proof. For the purpose of applying Khintchine’s inequality to the Haar coeﬃcients of a given function f , it is more convenient to ﬁrst re-index the sequence of Rademacher functions using dyadic intervals. Let {rI : I ∈ D} be any enumeration of the Rademacher functions. Let f ∈ H 1 , and let f= aI h I be its Haar expansion. For y ∈ [0, 1], we deﬁne fy by its Haar expansion as fy = aI hI rI (y). Note that fy ∈ H 1 . Applying Khintchine’s inequality we obtain that 1 0

1 0

1 |fy |dy dx ≥ √ 2

a2I h2I (x)

1/2

1 dx = √ ||f ||H 1 . 2

Changing the order of integration one ﬁnds at least one point y ∈ [0, 1] such that 1 ||fy ||L1 ≥ √ ||f ||H 1 . 2 Now we choose g ∈ L∞ such that g, fy = ||g||L∞ ·||fy ||L1 . Next we expand g − g in its Haar series, g− g= cI h I , and deﬁne h=

I cI hI ,

where I = rI (y). By the biorthogonality of the Haar system we have that f, h = fy , g, and by the deﬁnition of the BMO norm we have also, ||h||BMO = ||g − g||BMO ≤ 2||g||L∞ . Combining these observations gives f, h = fy , g = ||fy ||L1 ||g||∞ 1 ≥ √ ||f ||H 1 ||h||BMO . 2 2

1.2. Dyadic H 1

39

By Feﬀerman’s inequality every h ∈ BMO deﬁnes a continuous linear functional on the space H 1 , and moreover the unit ball of BMO norms H 1 by the converse to Feﬀerman’s inequality. We summarize Theorems 1.2.1 and 1.2.2 as follows. Let f ∈ H 1 with f = 0, then √ 1 √ ||f ||H 1 ≤ sup{ f h dx : ||h||BMO ≤ 1} ≤ 2 2||f ||H 1 . (1.2.8) 2 2 We now show that every continuous linear functional on H 1 , determines a function in BMO. For n ∈ N let fn : [0, 1] → R be integrable. Recall that the sequence 1 2 (fn )∞ n=1 belongs to the space L ( ) if ||(fn )∞ n=1 ||L1 (2 )

1

= 0

∞

1/2 dx < ∞.

fn2

n=1

The dual Banach space of L1 (2 ) is canonically identiﬁed with L∞ (2 ), consisting of sequences (hn )∞ n=1 so that

||(hn )∞ n=1 ||L∞ (2 )

= sup x∈[0,1]

∞

1/2 h2n (x)

< ∞,

n=1

where “supx∈[0,1] ” denotes the essential supremum over the unit interval, deﬁning the norm in L∞ . Theorem 1.2.3. For every continuous linear functional L : H 1 → R there exists h ∈ BMO so that (1.2.9) ||h||BMO ≤ ||L||, and so that for every ﬁnite linear combination of Haar functions g,

1

L(g) =

hgdt.

(1.2.10)

0

Proof. The idea of the proof comes with the useful observation that H 1 is canonically represented as a closed subspace of L1 (2 ). Hence by the Hahn–Banach theorem every continuous linear functional extends to an element in the dual space 1 2 ∞ 2 1 of L ( ), which is the space L ( ). The details are as follows. Let f ∈ H , with f = 0. For n ∈ N let hI fn = f, hI . (1.2.11) |I| −n−1 {I:|I|=2

}

Then the square function of f satisﬁes S(f ) =

fn2

1/2 .

40

Chapter 1. The Haar System: Basic Facts and Classical Results

Thus the linear map f → (fn ) provides the embedding of H 1 to a closed subspace of L1 (2 ), and ||f ||H 1 = ||(fn )∞ n=1 ||L1 (2 ) . With the Hahn Banach theorem extend the linear functional L to a functional on ˜ : L1 (2 ) → R, be an extension satisfying L1 (2 ), with the same norm. Let L ˜ ||L|| = ||L||, ˜ n )), L(f ) = L((f for f ∈ H 1 , and (fn )∞ n=1 deﬁned by (1.2.11). Next recall that the dual space of L1 (2 ) is identiﬁed as L∞ (2 ), in the following sense. There exists a sequence (hn ) such that 1/2

2 ˜ = hn (x) , ||L|| sup x∈[0,1]

and for every (xn ) ∈ L1 (2 ), ˜ n )) = L((x

∞

1

xn hn dt.

(1.2.12)

0

n=1

Next we deﬁne h, using the sequence (hn ), by putting h=

∞

hn ,

n=1 {I:|I|=2−n−1 }

hI hI . |I|

We test the equality (1.2.12) on the sequence (fn ) deﬁned by (1.2.11). This gives ˜ n )) = L((f

∞

f,

n=1 {I:|I|=2−n−1 }

hI hI , hn |I|

= f, h. ˜ is given by integration against h, Thus when restricted to (fn ) the action of L

1

˜ n )) = L((f

f hdt. 0

This proves (1.2.10). Now we show that h ∈ BMO, and that

||h||BMO ≤

sup x∈[0,1]

1/2 h2n (x)

.

1.2. Dyadic H 1

41

Evaluate the BMO norm of h by ﬁxing a dyadic interval J with |J| = 2−m−1 . Let (h)J denote the mean value of h over the interval J. By Bessel’s inequality we obtain that, hI |h − (h)J |2 dt = hn , 2 |I| |I| J −n−1 n≥m {I⊆J:|I|=2

≤

∞

J n=1

}

h2n dt.

Dividing by |J| and then taking the supremum over all dyadic intervals J, we ﬁnd that

1/2 ∞

h BMO ≤ sup h2n (x) , x∈[0,1] n=1

and consequently, ˜ ||h||BMO ≤ ||L||

which proves (1.2.9).

We will now present a diﬀerent approach to Feﬀerman’s inequality. In this approach the proof is carried out by analyzing the functions in H 1 . We will show that every f ∈ H 1 with small enough norm can be decomposed into a convex combination of dyadic atoms. A function a is a dyadic atom if there exists a dyadic interval I such that supp a ⊆ I, 1 a2 (t)dt ≤ |I|−1 , 0

and

1

a(t)dt = 0. 0

We will also say that the constant function 1[0,1] is a dyadic atom. For a given f ∈ H 1 with Haar expansion f= xI h I , I∈D

we deﬁne the Haar support of f to be the following collection of dyadic intervals, {I ∈ D : xI = 0}. In the proof of Theorem 1.2.4 we obtain the decomposition of f into atoms by ﬁrst decomposing the Haar support of f into pairwise disjoint collections of dyadic intervals. At this point we use, for the ﬁrst time in this book, a stopping time decomposition of the Haar support. Many more applications of this fundamental technique will be given in later chapters.

42

Chapter 1. The Haar System: Basic Facts and Classical Results

Theorem 1.2.4. For f ∈ H 1 , there exists a sequence of dyadic atoms, (ai ) and a sequence of scalars (ci ), such that 1 ∞ f− f= ci a i , 0

and

∞

i=1

|ci | ≤ 16

1

S(f ). 0

i=1

Conversely, every dyadic atom a belongs to H 1 and satisﬁes 1 S(a)dt ≤ 1. 0

Before we present the proof we should point out how Theorem 1.2.1 (Feﬀerman’s inequality) can be deduced from the atomic decomposition of H 1 . Theorem 1.2.4 may be paraphrased by saying that it suﬃces to prove Feﬀerman’s inequality only for dyadic atoms a ∈ H 1 . Now ﬁx a dyadic atom a and a dyadic interval I such that 1 supp a ⊆ I, and 0 a2 (t)dt ≤ |I|−1 . Next ﬁx h ∈ BMO. Let (h)I denote the mean 1 value of h over the interval I. Then, using 0 adt = 0 and the Cauchy–Schwarz inequality, we estimate as follows, 1 = a(h − (h)I )dt ahdt 0

I

≤ a L2 1I (h − (h)I ) L2 . Finally, observe that the biorthogonality of the Haar system gives that

1I (h − (h)I ) L2 ≤ |I|1/2 h BMO . Below the following notations are repeatedly used. For a collection of dyadic intervals F we denote by max F the maximal intervals of F. As dyadic intervals are nested, max F consists of pairwise disjoint intervals. For a dyadic interval I we denote by Q(I) the collection of all dyadic intervals which are contained in I, thus Q(I) = {J ∈ D : J ⊆ I}. Proof of Theorem 1.2.4. Let f ∈ H 1 , with Haar expansion f= xJ h J . 1 Thus we assume that 0 f = 0. We will decompose f by splitting its Haar support. We start with the unit interval [0, 1]. Let n([0, 1]) be the smallest integer such that Ω([0, 1]) = {t ∈ [0, 1] : S(f )(t) ≥ 2n([0,1]) }

1.2. Dyadic H 1

43

satisﬁes |Ω([0, 1])| ≤

1 . 4

Let F([0, 1]) be the collection of dyadic intervals which are contained in Ω([0, 1]). Then deﬁne C([0, 1]) = Q([0, 1]) \ F([0, 1]). Note that f[0,1] =

xJ h J

J∈C([0,1])

is deﬁned to satisfy the pointwise estimate S(f[0,1] ) ≤ 2n([0,1]) . Next we deﬁne E1 to be the collection of maximal dyadic intervals of F([0, 1]), that is, E1 = max F([0, 1]). We continue the construction inductively. Suppose we have already deﬁned collections of pairwise disjoint dyadic intervals E1 , · · · , Ei . Fix I ∈ Ei . Let n(I) be the smallest integer such that Ω(I) = {t ∈ I : S(f )(t) ≥ 2n(I) } satisﬁes |Ω(I)| ≤

1 |I|. 4

Let F(I) be the collection of dyadic intervals which are contained in Ω(I). Then let C(I) = Q(I) \ F(I). We deﬁne now fI =

xJ h J

J∈C(I)

which satisﬁes the pointwise estimate S(fI ) ≤ 2n(I) . Moreover the support of S(fI ) is contained in the dyadic interval I. Hence S(fI )2 ≤ |I|22n(I) . Now we let E(I) = max F(I),

44

Chapter 1. The Haar System: Basic Facts and Classical Results

and we take the union over I ∈ Ei to deﬁne Ei+1 = {E(I) : I ∈ Ei }. As Ei consists of pairwise disjoint dyadic intervals, it is easy to see that Ei+1 is a collection of pairwise disjoint dyadic intervals. This completes the induction step. # Finally we deﬁne E = Ei . Recall that along with the construction of E we veriﬁed that fI aI = |I|2n(I) is a dyadic atom. This suggests a decomposition of f, f=

fI

I∈E

=

|I|2n(I) aI .

I∈E

It remains to show that the coeﬃcients in front of the atoms are in fact summable. We do this by proving that n(I) |I|2 ≤ 16 S(f ). (1.2.13) I∈E

We start by observing that the stopping time deﬁnition of n(I) implies the measure estimate |I| |{t ∈ I : S(f )(t) ≥ 2n(I)−1 }| ≥ . 4 Then we continue with the proof of the convexity relation (1.2.13),

|I|2n(I) ≤ 4

I∈E

2n(I) |{t ∈ I : S(f ) ≥ 2n(I)−1 }|

I∈E

≤8

2n(I)−1 |{t ∈ I : S(f ) ≥ 2n(I)−1 }|

I∈E

≤8

2k |{S(f ) ≥ 2k }|

k∈N

≤ 16

S(f ).

We ﬁnd an atomic decomposition of f by rewriting the equation f = f=

I∈E

|I|2n(I)

fI . |I|2n(I)

I∈E

fI as

1.2. Dyadic H 1

45

The converse is actually very easy. We ﬁx an atom a, supported in a dyadic interval I such that a2 ≤ |I|−1 . As a = 0, the dyadic square function S(a) has its support in I as well. Hence by the Cauchy–Schwarz inequality we estimate 1/2 S(a) ≤ |I| ( S 2 (a))1/2 . The hypothesis that a is an atom implies of course ( S(a)2 )1/2 ≤ |I|−1/2 . Thus we showed that S(a) ≤ 1. In the course of the above proof we decomposed f by splitting the Haar support into pairwise disjoint collections of dyadic intervals {C(I) : I ∈ E}. Being the result of a stopping time decomposition, the collections C(I) satisfy a very important connectedness property: First, the collection C(I) contains exactly one maximal interval, namely I. Second, the following holds true. Let J ∈ C(I) and let K be contained in the Haar support of f, then J ⊆ K ⊆ I implies K ∈ C(I). The collection C(I) is called a block of dyadic intervals in the Haar support of f. Next the index set E and its construction deserves further # analysis, that provides an important link to BMO. Recall that we deﬁned E = Ei . Reviewing the proof we gave of Theorem 1.2.4 shows that the collections Ei are linearly ordered and shrinking uniformly at a geometric rate. Precisely, the following holds. 1. If I ∈ Ei , J ∈ Ej and I ⊆ J, then i ≥ j. 2. If J ∈ Ej and i ≥ j

|I| ≤ 2−i+j |J|.

{I∈Ei : I⊆J}

As a result we obtain that E satisﬁes the Carleson packing condition: For any J ∈ E, |I| ≤ 2|J|. (1.2.14) {I∈E:I⊆J}

Summing up: The proof of Theorem 1.2.4 produced a decomposition of the Haar support into blocks of dyadic intervals {C(I) : I ∈ E}, such that the index set E satisﬁes the Carleson packing condition (1.2.14).

Sharp maximal functions in Lp Renorming Lp with the square function ||S(f )||p and then passing to the limit p → 1 gives H 1 , which is thus a natural endpoint of the Lp scale. We will show that in the opposite direction BMO appears as endpoint of the Lp scale when we renorm

46

Chapter 1. The Haar System: Basic Facts and Classical Results

with the sharp function and then pass to the limit p → ∞. Let f ∈ L2 ([0, 1]), with Haar expansion f= f+ bI hI . For x ∈ [0, 1], the sharp function is deﬁned as ⎛

⎞1/2 1 f (x) = ⎝sup b2J |J|⎠ , x∈I |I| J⊆I

where the supremum is taken over all dyadic intervals containing x. We observed in (1.2.6), by comparing the deﬁnitions of BMO and f , that f ∈ BMO if f ∈ L∞ . Now we show that the sharp function provides an equivalent renorming of Lp when p > 2. Theorem 1.2.5. Let p > 2. If f ∈ Lp ([0, 1]) with f = 0, then (8(p − 1))

−1

||f ||p ≤ ||f ||p ≤ 2

p p−2

1/p ||f ||p .

Proof. Let q = p/(p − 1) be the H¨older conjugate index of p. As p > 2 we have that q < 2. Choose g ∈ Lq with ||g||q ≤ 2 and g = 1, so that

1

||f ||p =

f gdt. 0

Now recall the integral estimate (1.2.3) and the pointwise upper bound (1.2.1) established in the course of the proof of Feﬀerman’s inequality. They give that

1 0

√ f gdt ≤ 2 2

1

f S(g)dt.

(1.2.15)

0

Now we apply H¨ older’s inequality and Theorem 1.1.6, the square function characterization of Lq , when q < 2.

1 0

f S(g)dt ≤ ||f ||p ||S(g)||q √ ≤ ||f ||p 2 2(p − 1)||g||q √ ≤ 2 2(p − 1)||f ||p .

This shows the left-hand side inequality of Theorem 1.2.5, that is, ||f ||p ≤ 8(p − 1)||f ||p .

1.2. Dyadic H 1

47

We obtain the right-hand side estimate by comparing the sharp function f to the following version of the Hardy–Littlewood maximal function. For x ∈ [0, 1] deﬁne M2 (f )(x) = sup x∈I

1 |I|

1/2

f

2

,

I

where we take the supremum of all dyadic intervals which contain the point x. A moments reﬂection shows that pointwise inequalities hold between the sharp function and the maximal function of Hardy and Littlewood, f (x) ≤ M2 (f )(x).

(1.2.16)

For p > 2, the Hardy–Littlewood maximal theorem gives an upper bound for the Lp norm of M2 (f ), in terms of the Lp norm of f. More precisely by Theorem 1.2.6 below, the following holds for p > 2, ||M2 (f )||p ≤ 2

p p−2

1/p ||f ||p .

Assuming the validity of Theorem 1.2.6, we obtain from (1.2.16) the norm estimate for the sharp function, ||f ||p ≤ ||M2 (f )||p ≤ 2

p p−2

1/p ||f ||p .

Completing the proof that the sharp function provides an equivalent norm on Lp we show the Lp estimate for the Hardy–Littlewood maximal function. Besides M2 we will encounter diﬀerent versions of the Hardy–Littlewood maximal function. Hence we will prove the required estimate in a more general form. Theorem 1.2.6. Fix p, q, such that 1 ≤ q < p. Let f ∈ Lp ([0, 1]), then 1/q 1 q Mq (f )(x) = sup |f | x∈I |I| I satisﬁes the norm estimate ||Mq (f )||p ≤ 2

p p−q

1/p ||f ||p .

Proof. Fix f ∈ Lp ([0, 1]). Without loss of generality we assume that f ≥ 0. Let t > 0 and deﬁne the following function h by truncating f at the places where f (x) ≤ t/2, f (x) if f (x) > t/2, h(x) = 0 otherwise.

48

Chapter 1. The Haar System: Basic Facts and Classical Results

Put g = f −h. Then g(x) ≤ t/2, hence f (x) ≤ h(x)+t/2, and Mq (f ) ≤ Mq (h)+t/2. Consequently {Mq (f ) > t} ⊆ {Mq (h) > t/2}. (1.2.17) By Lemma 1.2.7 below, the maximal function Mq satisﬁes a weak type q − q estimate. We apply Lemma 1.2.7 to h and recall that the support of h is contained in the set E = {x : f (x) > t/2}. This gives tq |{Mq (h) > t/2}| ≤ 2q ||h||qq ≤ 2q f q dx.

(1.2.18)

E

The pieces of information obtained so far are the inclusion relation (1.2.17) and the weak type estimate (1.2.18) Now we combine them and reach an upper bound for the distribution function 2q |{Mq (f ) > t}| ≤ q (1.2.19) f q (x)1{f >t/2} (x)dx. t Finally we obtain estimates for ||Mq (f )||p by integration from (1.2.19) in the following way, ∞ ||Mq (f )||pp = p tp−1 |{Mq (f ) > t}|dt 0 ∞ q p−1 −q ≤2 p t t f q (x)1{f >t/2} (x)dxdt 0

= 2q p

1

0

= 2q

p p−q

(1.2.20)

2f (x)

f q (x)

tp−q−1 dtdx 0

1

f q (x)(2f )p−q (x)dx. 0

It remains to collect terms and to take the p-th root to obtain that ||Mq (f )||p ≤ 2

p p−q

1/p

f p .

Now we prove the fact that the maximal function Mq satisﬁes a weak type q − q estimate. Lemma 1.2.7. For h ∈ Lq , and t > 0 deﬁne Ωt = {x : Mq (h)(x) > t}. Then tq |Ωt | ≤ ||h||qq .

1.2. Dyadic H 1

49

Proof. For x ∈ Ωt there exists a uniquely determined maximal dyadic interval Ix such that |h|q > tq |Ix |. Ix

Let M = {Ix : x ∈ Ωt }. Note that M consists of pairwise disjoint dyadic intervals which cover the set Ωt . Hence |I| = |Ωt |. I∈M

For I ∈ M we have that

|h|q > tq |I|. I

We sum this estimate over all I ∈ M and use that M consists of pairwise disjoint dyadic intervals. This gives the weak type q − q estimate as follows, |h|q ||h||qq ≥ I

I∈M

≥ =

tq |I|

I∈M tq |Ωt |.

For p > 1 we use H p to denote Lp equipped with the square function norm

g H p

=

0

1

1/p 1 gdt + ( a2I h2I )p/2 0

where aI = g,

I

hI . |I|

Assume that q > 2 and that f : [0, 1] → R satisﬁes ||f ||Lq < ∞. Assume that g is a ﬁnite linear combination of Haar functions. The inequality

1

√ f gdt ≤ 2 2

0

1

f S(g)dt

0

implies that f induces a bounded linear functional on H p where 1/p + 1/q = 1. The next theorem states that the converse holds as well: Every bounded linear functional on H p induces f : [0, 1] → R such that ||f ||Lq < ∞. Theorem 1.2.8. Let 1 ≤ p < 2, and let q be the H¨ older conjugate exponent to p satisfying 1/p + 1/q = 1. For every continuous linear functional L : H p → R there exists f : [0, 1] → R so that f ∈ Lq , with ||f ||Lq ≤

√

2

q q−2

1/q ||L||,

50

Chapter 1. The Haar System: Basic Facts and Classical Results

and so that for every ﬁnite linear combination of Haar functions g, L(g) =

1

f gdt. 0

Proof. Let g be a ﬁnite linear combination of Haar functions. For n ∈ N denote by gn the orthogonal projection of g onto the span{hI : |I| = 2−n }. That is,

gn =

g,

{I:|I|=2−n−1 }

hI hI . |I|

(1.2.21)

The following identity holds for the square function of g, S(g) = ( gn2 )1/2 . We represent H p as a closed subspace of Lp (2 ). Let g˜ = (gn )∞ n=1 . Then ||g||H p = ||˜ g ||Lp (2 ) . By the Hahn–Banach theorem we extend the linear functional L to a functional ˜ : Lp (2 ) → R, be an extension satisfying on Lp (2 ), with the same norm. Let L ˜ and ||L|| = ||L||, ˜ g ) for g ∈ H p . L(g) = L(˜ Next recall that the dual space of Lp (2 ) is identiﬁed as Lq (2 ), in the following sense. There exists a sequence (fn ) such that ˜ = ||( (1.2.22) ||L|| fn2 )1/2 ||Lq , and for every (hn ) ∈ Lp (2 ), ˜ n )) = L((h

∞

fn hn .

(1.2.23)

n=1

Next use the sequence (fn ), to deﬁne f. We put f=

∞

fn ,

n=1 {I:|I|=2−n−1 }

hI hI . |I|

Testing the identity (1.2.23) on the sequence (gn ) deﬁned by (1.2.21) we obtain that ∞ hI ˜ n )) = L((g g, hI , fn |I| −n−1 n=1 {I:|I|=2

= g, f .

}

1.2. Dyadic H 1

51

˜ is given by integration as Thus restricted to (gn ) the action of L

1

˜ n )) = L((g

gf dt. 0

Now we show that f ∈ Lq , and that ||f ||q ≤

√ 2

q q−2

1/q ||(

fn2 )1/2 ||q .

At a given point x ∈ [0, 1], we evaluate the sharp function of f by ﬁxing a dyadic interval J such that x ∈ J. Choose m ∈ N such that |J| = 2−m−1 . By Bessel’s inequality, we obtain hI |f − (f )J |2 dx = fn , 2 |I| |I| J −n−1 n≥m {I⊆J:|I|=2

≤

∞

J n=1

}

fn2 dx.

Divide by |J| and take the supremum over all dyadic intervals containing the ﬁxed point x ∈ [0, 1]. Thus we restate the last estimate as a pointwise inequality between the sharp function and the Hardy–Littlewood maximal function M1 . For every x ∈ [0, 1],

1/2 ∞ 2 f (x) ≤ M1 ( fn ) (x). (1.2.24) n=1

By our assumption q > 2, hence by Theorem 1.2.6 the maximal function M1 acts boundedly on the space Lq/2 . Moreover the following norm estimate holds true,

M1 (h) q/2 ≤ 2

q q−2

2/q

h q/2 .

This allows us to ﬁnish the proof as follows. By (1.2.24) and by Theorem 1.2.6, ||f ||q ≤ M1 ( √ ≤ 2

n

q q−2

Invoking (1.2.22) gives that ||f ||q ≤

1/2

fn2 ) q/2 1/q

||(

fn2 )1/2 ||q .

n

√ q 1/q 2 q−2

L , as claimed.

52

1.3

Chapter 1. The Haar System: Basic Facts and Classical Results

Bounded square functions and large deviation inequalities

We return to square functions in Lp , 1 < p < ∞. In Theorem 1.1.6 we combined the inequalities of Burkholder, and Khintchine to prove the square function characterization of Lp asserting that for f ∈ Lp , with f = 0, √ √ (2 2)−1 (p∗ − 1)−1 ||f ||p ≤ ||S(f )||p ≤ 2(p∗ − 1)||f ||p . In this section we show that for 2 ≤ p < ∞ the following equivalence holds, √ p 1 √ ||f ||p , ||f || ≤ ||S(f )|| ≤ √ p p 16 p 2 √ when f = 0. The constants of the above inequality which are of order p, are known to represent the correct asymptotic behavior as p → ∞. Along with the proof we introduce important tools of general interest in real analysis. These include a detailed study of the exponential function, 1 exp{f − S 2 (f )}, 2 large deviation inequalities, stopping time arguments, and the use of so-called good λ inequalities. We apply these methods to study Schechtman’s sign-embedding of high-dimensional Euclidean subspaces into 1n . We also show how to obtain non-trivial multipliers into SL∞ (the Banach space consisting of functions with bounded square functions).

Square functions in Lp (2 < p < ∞) We prove now a strong upper bound for the expectation of the exponentials exp{f − 12 S 2 (f )}. The following theorem has wide ranging applications. These include the law of the iterated logarithms, eigenvalue estimates for the Laplacian, embedding theorems for fractional Sobolev spaces, the existence of high-dimensional Euclidean sections in convex bodies and the square function inequality, √ ||f ||p ≤ 16 p||S(f )||p , for p ≥ 2, where f ∈ Lp and f = 0. The notes at the end of the chapter provide the references to the applications of our next theorem. 1 Theorem 1.3.1. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. Then for every µ > 0,

1

exp{µf − 0

µ2 2 S (f )} ≤ 1. 2

1.3. Bounded square functions and large deviation inequalities Proof. As

1 0

53

f dt = 0 we expand f in its Haar series as f=

aI h I .

We ﬁx a dyadic interval I. The following estimate is basic and elementary. It controls the mean values of exp{aI hI } over I. |I| (exp(aI ) + exp(−aI )) exp{aI hI } = 2 I 2 (1.3.1) aI ≤ |I| exp . 2 Now we split the Haar series of f according to the levels {J : |J| = 2−k }. Write dk = aJ h J . {J:|J|=2−k }

We restate the estimate (1.3.1) in the following way. For every dyadic interval I of length |I| = 2−k , exp{dk − I

d2k a2 } = exp{aI hI − I h2I } 2 2 I ≤ |I|.

For future reference it is convenient to rewrite the last estimate using conditional expectations. Recall that Ek denotes the conditional expectation with respect to the σ-algebra generated by the dyadic intervals of length = 2−k . We showed so far that d2 (1.3.2) Ek (exp{dk − k }) ≤ 1. 2 Next let n ∈ N. For k ≤ n − 1, the conditional expectation En reproduces the functions dk and d2k . Therefore we have the following straightforward identities,

1

exp 0

n k=0

d2 dk − k 2

$

1

dt =

exp 0

k=0

1

=

exp 0

n−1 n−1 k=0

d2 dk − k 2 d2 dk − k 2

$ (exp{dn − $

d2n })dt 2

En (exp{dn −

The basic inequality (1.3.2) and the above identities imply that $ $ n n−1 1 1 d2k d2k dt ≤ dt. exp dk − exp dk − 2 2 0 0 k=0

k=0

d2n })dt. 2

(1.3.3)

54

Chapter 1. The Haar System: Basic Facts and Classical Results

The net eﬀect of (1.3.3) is that we increased the value of the integral while we reduced the upper limit of summation from n to n − 1. Now we repeat this procedure to reduce the upper limit of summation from n − 1 to n − 2, and we continue in this fashion until we reach the level k = 0. This gives in summary the inequality n $ 1 1 d2k d2 exp dk − exp{d0 − 0 }dt dt ≤ 2 2 (1.3.4) 0 0 k=0

≤ 1. By the lemma of Fatou, in the estimate (1.3.4) we may pass to the limit as n → ∞. As $ ∞ S 2 (f ) d2k exp f − dk − = exp 2 2 k=0 ! " we proved that exp f − 12 S 2 (f ) ≤ 1. Finally we ﬁx µ > 0, and replace f by (µf ). This gives that 1 µ2 2 S (f )} ≤ 1. exp{µf − 2 0 The space SL∞ is the class of functions for which the square function is uniformly bounded. Thus f ∈ SL∞ iﬀ S(f ) ∈ L∞ . Our ﬁrst corollary to Theorem 1.3.1 derives a strong upper bound for the distribution of functions f ∈ SL∞ . We show a square-exponential estimate for the Hardy–Littlewood maximal function M1 (f ). To prove this statement we combine Theorem 1.3.1, with the weak type 1 : 1 estimate of Lemma 1.2.7 and Jensen’s inequality. 1 Corollary 1.3.2. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. If S(f ) ∈ L∞ , then t2 |{x : M1 (f )(x) > t}| ≤ 2 exp − . 2||S(f )||2∞ Proof. We assume that ||S(f )||∞ = 1. Write (|f |)I for the mean value of |f | over the interval I. Analogously we let (exp{|f |})I denote the mean value of exp{|f |} over I. By Jensen’s inequality the mean values are related as follows, exp{(|f |)I } ≤ (exp{|f |})I .

(1.3.5)

The inequality (1.3.5) translates into a pointwise estimate for the Hardy–Littlewood maximal function M1 , exp{M1 (f )} ≤ M1 (exp{|f |}). By Lemma 1.2.7, the weak type 1 : 1 estimate for the maximal function M1 we have that exp{µ|f |} , (1.3.6) |{M1 (exp{µ|f |}) ≥ exp α}| ≤ exp α

1.3. Bounded square functions and large deviation inequalities

55

for every µ, α > 0. Theorem 1.3.1 and the hypothesis that ||S(f )||∞ = 1 imply that exp{µ|f |} ≤ exp{µf } + exp{−µf } (1.3.7) µ2 ≤ 2 exp{ }. 2 Inserting the integral estimate (1.3.7) into the right-hand side of the measure estimate (1.3.6) gives that 2 exp{µ|f |} µ |{M1 (exp{µ|f |}) ≥ exp α}| ≤ ≤ 2 exp −α . exp α 2 Summing up we ﬁnd the following upper bound for the distribution function of M1 (f ). Let α > 0, then |{µM1 (f ) > α}| = |{exp{µM1 (f )} > exp α}| ≤ |{M1 (exp(µ|f |) > exp α}| 2 µ ≤ 2 exp −α . 2 √ Specializing this estimate to µ = α = t, gives that 2 t |{M1 (f ) > t}| ≤ 2 exp − , 2 provided that ||S(f )||∞ = 1. We get rid of the restriction ||S(f )||∞ = 1 by applying the above special case to f /||S(f )||∞ . We will next discuss rescaled versions of Corollary 1.3.2. Fix a dyadic interval Q . We now introduce the Hardy–Littlewood maximal function localized to the interval Q. Here we take the supremum only over those intervals which are contained in Q. For x ∈ Q we put, 1 M1,Q (f )(x) = sup |f |. I⊆Q, x∈I |I| I Let f ∈ L2 . We denote the mean value of f over the interval Q by fQ , that is, 1 fQ = f dx. |Q| Q Then by rescaling Corollary 1.3.2 from the unit interval to the interval Q, we obtain the following renormalized distributional estimate: If S(f ) ∈ L∞ , then t2 |{x ∈ Q : M1,Q (f − fQ )(x) > t}| ≤ 2|Q| exp − . (1.3.8) 2||S(f )||2∞

56

Chapter 1. The Haar System: Basic Facts and Classical Results Our aim is still the square function estimate ||f ||p ≤ 16p1/2 ||S(f )||p .

(1.3.9)

The essential step in the proof is the next proposition which gives an estimate for the distribution of M1 (f ) over the set where S(f ) satisﬁes a pointwise upper bound. The distributional estimates of Proposition 1.3.3 are traditionally called good λ inequalities, or more descriptively, relative distributional estimates. We 1/2 in the square function estimate remark that the appearance of the constant " ! 116p on the right hand side of the good (1.3.9) can be traced to the factor exp − 4 "2 ! λ inequality, below. The factor exp − 412 has in turn its origin in the integral estimate of the exponential 1 exp{f − S 2 (f )} ≤ 1. 2 1 Proposition 1.3.3. Let f ∈ L2 ([0, 1]) with 0 f dt = 0. Then for every 0 < ≤ √ 1 − 1/ 2, and t > 0 the following distributional inequality holds, 1 |{M1 (f ) > 2t, S(f ) < t}| ≤ 2 exp − 2 |{M1 (f ) > t}|. 4 Proof. Fix t > 0, and deﬁne Ωt = {x : M1 (f ) > t}. For every x ∈ Ωt , there exists a maximal dyadic interval Ix such that 1 |f | > t. |Ix | Ix Let M = {Ix : x ∈ Ωt }. The collection M covers the set Ωt . By the maximality condition, M consists of pairwise disjoint dyadic intervals, hence, |Q| = |{M1 (f ) > t}|. {Q∈M}

Maximality gives also the following implication: If Q ∈ M, and if Q∗ denotes the dyadic predecessor of Q, then 1 f ≤ t. |Q∗ | Q∗ Let Q ∈ M and assume that there exists x ∈ Q such that M1 (f )(x) > 2t. Observe that then the relevant contribution to the value of the maximal function M1 (f ) at the point x necessarily comes from the intervals which are contained in Q (and not from those containing Q). Precisely, if Q ∈ M, and M1 (f )(x) > 2t, then the following equality holds at the point x, M1 f (x) = M1,Q f (x).

(1.3.10)

1.3. Bounded square functions and large deviation inequalities

57

Expand f in its Haar series, f=

aI h I .

Deﬁne L to be the collection of dyadic intervals I such that the following estimate holds true, a2J h2J (x) ≥ 2 t2 , J⊇I

for some (or equivalently for all) x ∈ I. Let L be the set of points which is covered by the collection L, thus L= I. I∈L

Deﬁne S = D \ L, and put ft = J∈S aJ hJ . The reason for our choice of L and S is that now, by deﬁnition, ft has uniformly bounded square function and satisﬁes ||S(ft )||∞ ≤ t.

(1.3.11)

ft (x) = f (x),

(1.3.12)

Observe that if x ∈ / L, then and furthermore, if J is a dyadic interval containing x we have that ft = f. J

(1.3.13)

J

So far we deﬁned two important collections of dyadic intervals, M and L and we obtained several consequences encoded in their very deﬁnitions. Now we analyze the interplay between M and L. We ﬁx Q ∈ M, and we assume that there exists x ∈ Q such that x ∈ / L. Then we claim that the following integral estimate holds, 1 f ≤ (1 + )t. (1.3.14) |Q| Q To verify this claim we consider the dyadic predecessor of Q and call it Q∗ . Note that the coeﬃcient aQ∗ appearing in the Haar expansion of f is obtained from the mean values of f over Q and over Q∗ as follows, 1 1 f = |aQ∗ |. (1.3.15) |Q∗ | ∗ f − |Q| Q Q Recall that for Q ∈ M, the mean value of |f | over Q∗ is bounded by t, hence 1 1 f≤ ∗ |f | ≤ t. (1.3.16) |Q∗ | Q∗ |Q | Q∗ Furthermore if x ∈ Q, and x ∈ / L, then |aQ∗ | ≤ t.

(1.3.17)

58

Chapter 1. The Haar System: Basic Facts and Classical Results

Inserting the estimates (1.3.17) and (1.3.16) into the identity (1.3.15) gives the upper bound 1 f ≤ (1 + )t, |Q| Q hence inequality (1.3.14) holds true. Next we claim that for Q ∈ M, the following estimate holds, 1 |{x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t}| ≤ 2|Q| exp − 2 . 4

(1.3.18)

We start proving this claim by ﬁrst rewriting the set whose measure we wish to estimate. Using ﬁrst (1.3.10) then (1.3.13) we obtain that, {x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t} = {x ∈ Q : M1,Q (f )(x) > 2t, x ∈ / L} = {x ∈ Q : M1,Q (ft )(x) > 2t, x ∈ / L}. We write (ft )Q for the mean value of ft over Q. By (1.3.13), for x ∈ Q and x ∈ /L the mean values (ft )Q and (f )Q coincide. Invoking (1.3.14) gives that |(ft )Q | ≤ (1 + )t. By inserting (ft )Q we obtain the following inclusion of sets, which incidentally is a one line summary of the proof given so far. / L} ⊆ {x ∈ Q : M1,Q (ft − (ft )Q ) (x) > (1 − )t} . {x ∈ Q : M1,Q (ft )(x) > 2t, x ∈ By deﬁnition ft has uniformly bounded square function with ||S(ft )||∞ ≤ t. Hence (1.3.8) gives the measure estimate (1 − )2 t2 . |{x ∈ Q : M1,Q (ft − (ft )Q ) (x) > (1 − )t}| ≤ 2|Q| exp − 22 t2 Now observe that inside the exponential the factors t2 cancel and that for 0 < ≤ √ 1−1/ 2, we have clearly (1−)2 /2 ≥ 1/4. Thus we veriﬁed the claimed inequality (1.3.18). Finally recall that M is a collection of pairwise disjoint dyadic intervals, covering the set {M1 (f ) > t}. Now we sum over Q ∈ M, thereby ﬁnishing the proof of Proposition 1.3.3: |{M1 (f ) > 2t, S(f ) < t}| = |{x ∈ Q : M1 (f )(x) > 2t, S(f )(x) < t}| Q∈M

1 |Q| ≤ 2 exp − 2 4 Q∈M 1 = 2 exp − 2 |{M1 (f ) > t}|. 4

1.3. Bounded square functions and large deviation inequalities

59

We now prove a square function characterization of Lp with the correct asymptotic behavior of the constants as p → ∞. 1 Theorem 1.3.4. Let p ≥ 2, and let f ∈ Lp ([0, 1]) with 0 f dt = 0. Then √ ||f ||p ≤ ||M1 (f )||p ≤ 16 p||S(f )||p . Proof. We wish to evaluate ||M1 (f )||pp . Fix t > 0 and > 0. We start the proof with the triangle inequality, |{M1 (f ) > 2t}| ≤ |{M1 (f ) > 2t, S(f ) < t}| + |{S(f ) ≥ t}|.

(1.3.19)

The eﬀect of the second summand |{S(f ) ≥ t}| is easy to evaluate. We have ∞ p tp−1 |{S(f ) ≥ t}|dt = −p ||S(f )||pp . (1.3.20) 0

The relative distributional estimate of Proposition 1.3.3 controls the contribution of the ﬁrst summand. It gives 1 |{M1 (f ) > 2t, S(f ) < t}| ≤ 2 exp − 2 |{M1 (f ) > t}|. (1.3.21) 4 We point out that it is the presence of factor exp{−(1/42 )} that allows us to compensate the eﬀect caused by −p appearing in (1.3.20). Combining the estimates (1.3.19) and (1.3.20) with the relative distributional estimate (1.3.21) gives the following bound for ||M1 (f )||pp . 1 2p

1

M1 (f )p (x) = p 0

∞

tp−1 |{M1 (f ) > 2t}|dt

0 ∞

tp−1 |{M1 (f ) > 2t, S(f ) < t}|dt + −p ||S(f )||pp ∞ 1 ≤ 2p exp{− 2 } tp−1 |{M1 (f ) > t}|dt + −p ||S(f )||pp 4 0 1 = 2 exp{− 2 }||M1 (f )||pp + −p ||S(f )||pp . 4 ≤p

0

Now subtract exp{−(1/2 )}||M1 (f )||pp on both sides. This gives 1 1 − 2 exp{− 2 } ||M1 (f )||pp ≤ −p ||S(f )||pp . 2p

(1.3.22)

Note that (1.3.22) can become a meaningful estimate only if is chosen so small that the factor in front of ||M1 (f )||pp is a positive number. For p ≥ 2 deﬁne =

1 (4 ln 2)1/2 (p

+ 2)1/2

.

60

Chapter 1. The Haar System: Basic Facts and Classical Results p

Then exp{−(1/42 )} ≤ 2−p−1 , and −p ≤ (8 ln 2)p/2 p 2 . This gives p

||M1 (f )||pp ≤ 16p p 2 ||S(f )||pp . It remains to raise both sides to the power 1/p. For f ∈ Lp with f = 0 we established the square function estimate √ ||f ||p ≤ 16 p||S(f )||p

for

2 ≤ p < ∞.

Now we turn to proving the converse inequality. In the proof below we present a method due to A. M. Garsia [75]. Theorem 1.3.5. Let p > 2 and f ∈ Lp . Then, √ p ||S(f )||p ≤ √ ||f ||p . 2 Proof. To start the proof we expand f in its Haar series, f = f + aI h I . Then we introduce the following notation to denote partial sum approximations to S 2 (f ), 2 S−1 = 0,

and

Sn2 (t) =

a2I h2I (t) for n ∈ N0 .

{I:|I|≥2−n }

We also abbreviate the notation for the square function of f, by writing S = S(f ). Below we state two crucial identities, and an inequality which uses that for p > 2, the function r p/2−1 is increasing on the positive real axis. 2 p S (f )(t) = p = ≤

S 2 (t)

r p/2−1 dr

0 2 ∞ Sn (t) n=0 ∞

r p/2−1 dr.

2 (t) Sn−1

(1.3.23)

! " 2 Snp−2 (t) Sn2 (t) − Sn−1 (t) .

n=0

For brevity it is convenient to introduce more notation. Let n ∈ N0 , then put p−2 . mn = Snp−2 − Sn−1

Using the summation by parts formula we next rewrite the last line of (1.3.23), ∞ n=0

∞ ! " 2 2 Snp−2 (t) Sn2 (t) − Sn−1 (t) = mn {S 2 (t) − Sn−1 (t)}. n=0

(1.3.24)

1.3. Bounded square functions and large deviation inequalities

61

Next observe that En , the conditional expectation with respect to the σ-algebra generated by the dyadic intervals of length = 2−n reproduces mn . That is En (mn ) = mn . Integrating the estimate (1.3.23) over the unit interval and taking into account the identity (1.3.24) we ﬁnd ∞ 1 2 1 p 2 S (f )(t)dt ≤ mn En (S 2 (t) − Sn−1 (t))dt. (1.3.25) p 0 0 n=0 2 )(t) satisfy the pointwise Observe that the conditional expectations En (S 2 − Sn−1 estimate, 2 ) ≤ En (f 2 ). (1.3.26) En (S 2 − Sn−1

Inserting the estimate (1.3.26) into the right-hand side of (1.3.25) and using again that En (mn ) = mn , allows us to continue the right-hand side of (1.3.25) as follows, ∞ 1 ∞ 1 2 2 mn En (S − Sn−1 )dt ≤ mn f 2 dt. (1.3.27) n=0

0

0

n=0

Finally we change the order of integration and summation, and sum the resulting p−2 telescoping series ∞ . This gives that k=0 mk = S 1 ∞ 1 mn f 2 dt = S p−2 f 2 dt. (1.3.28) n=0

0

0

Thus starting from (1.3.23) we showed with (1.3.25), (1.3.27) and (1.3.28), this crucial inequality, 1 2 1 p S (f )(t)dt ≤ S p−2 f 2 dt. p 0 0 Now we use H¨older’s inequality with a = p/(p − 2) and b = p/2. This gives 1 1/a 1 1/b 1 S p−2 f 2 dt ≤ S (p−2)a f 2b 0

0

(p−2)/p

1

Sp

= 0

0 1

2/p |f p | .

0

Summing up we have that 1 (p−2)/p 1 2/p 2 1 p S ≤ Sp |f p | . p 0 0 0 1 Cancelling the factor ( 0 S p )(p−2)/p on both sides gives that 2 ||S(f )||2p ≤ ||f ||2p . p

62

Chapter 1. The Haar System: Basic Facts and Classical Results

In the following paragraphs we will discuss three diﬀerent applications of the square function characterization of Lp and the SL∞ estimates that formed the basis of its proof. First, we compare sharp functions with square functions in Lp and let p → ∞, second we use SL∞ estimates to produce Euclidean subspaces in √ 1n , and third we exploit the p dependence in Theorem 1.3.5 to prove that the square function of a bounded function is square-exponentially integrable. Endpoints of the Lp scale. Let f ∈ Lp with f = 0. Combining Theorem 1.3.4 and Theorem 1.3.5 gives that for p ≥ 2, √ √ √ (1.3.29) ( 2/ p) S(f ) p ≤ f p ≤ 16 p S(f ) p . Recall that a function is in SL∞ if its square function is uniformly bounded. Thus SL∞ appears as an endpoint of the scale of Lp spaces for p → ∞. Compare this to the renorming obtained in the previous section. For p ≥ 2, (p − 2)/(2p) f p ≤ f p ≤ 8(p − 1) f p .

(1.3.30)

This gives a diﬀerent endpoint to the scale of Lp spaces for p → ∞, namely BMO. A glance at the deﬁnitions of square functions and sharp maximal functions shows that SL∞ is contained in BMO. However much more is true. A result of J. Garnett and P. W. Jones [73] displays the relation between the three spaces L∞ , BMO and SL∞ that appeared as endpoints of the Lp scale. It asserts that for any f ∈ SL∞ with S(f ) ≤ 1 and > 0 there exists g ∈ L∞ satisfying

f − g BMO ≤ and

g ∞ ≤ A−1 ,

(1.3.31)

where A > 0 is a universal constant. In particular the closure of L∞ in BMO contains SL∞ . Applying the Garnett–Jones result, J. Bourgain [18] obtains a remarkable estimate for the Rademacher projection on H 1 . Recall that {rm } denotes the Rademacher system consisting of {+1, −1} valued, independent random variables. J. Bourgain’s inequality states that ∞

h, rm 2 ≤ C h L1 h H 1 ,

for h ∈ H 1 .

(1.3.32)

m=1

The interesting aspect of J. Bourgain’s inequality is the appearance of the L1 norm on the right-hand side. Replacing h L1 h H 1 by h 2H 1 would turn it into a trivial estimate. On the other hand replacing h L1 h H 1 by the smaller number

h 2L1 would lead to a false statement. To prove (1.3.32) put am = h, rm (

h, rn 2 )−1/2 ,

1.3. Bounded square functions and large deviation inequalities and let f = deﬁne

am rm . Since

63

a2m = 1, we have f ∈ SL∞ with S(f ) = 1. Next −1/2

1/2

= h L1 h H 1 . By the Garnett–Jones result there exists g ∈ L∞ so that

f − g BMO ≤ and

g ∞ ≤ A−1 .

(1.3.33)

With the inequalities of Feﬀerman and H¨ older we obtain that 1/2 h, rm 2 = h, f = h, f − g + h, g √ ≤ 2 2 h H 1 f − g BMO + h L1 g ∞ .

(1.3.34)

Use the estimates (1.3.33) in the last line of (1.3.34). Then our choice of > 0 gives J. Bourgain’s inequality (1.3.32). Schechtman’s sign-embedding. In this section we show how SL∞ estimates lead to Euclidean subspaces of proportional dimension in 1n . The resulting embedding — Schechtman’s sign-embedding — turns out to be quite explicit. We write 1n to denote Rn equipped with the norm of the sequence space 1 . Likewise we denote by 2k the space Rk equipped with the norm of the Hilbert space 2 . A straightforward exercise with Khintchine’s inequality yields a subspace X ⊆ 1n and a linear bijection i : 2k → X so that k = [log2 n]

and

i · i−1 ≤ 16.

It is a much harder problem to ﬁnd such an embedding i : 2k → X under the hypothesis that the algebraic dimension of the subspace X is proportional to the dimension of the ambient space 1n . That is, under the hypothesis that k = [δn] . Here, of course, we demand that the factor of proportionality n/k ∼ δ −1 is an absolute constant, not depending on n. In the seminal paper [186] G. Schechtman develops an approach to these problems based integral estimates for the exponentials exp{f − 12 S 2 (f )}. Specifically, G. Schechtman’s construction of the embedding is based on Khintchine’s inequality and Corollary 1.3.2 asserting that |f −

1 0

t2 f | > t ≤ 2 exp − . 2||S(f )||2

(1.3.35)

∞

Subsequently, martingale methods based on (1.3.35) found numerous applications to related problems in the local theory of Banach spaces. See the notes for references that give an overview of this development.

64

Chapter 1. The Haar System: Basic Facts and Classical Results

Next we turn to deﬁning G. Schechtman’s embedding. Let {ei : i ≤ n} denote the unit vector basis in 1n . Thus for coeﬃcients bi ∈ R we get n n bi ei = |bi |. 1 i=1

i=1

n

Recall that {rl } denotes the Rademacher system. Fix k ≤ n and deﬁne 1 rik+j (t)ei , n i=1 n

xj (t) =

j≤k

where

and

t ∈ [0, 1].

Thus xj (t) is formed by attaching random signs {±1} to the unit vectors of 1n and averaging. Note that for each t ∈ [0, 1] the vectors xj (t) are of norm 1 in 1n . The next theorem asserts that for some t ∈ [0, 1], they are equivalent to the unit vector basis of 2k provided that k ∼ δn. Theorem 1.3.6. There exists δ > 0 so that the following holds true. For each n ∈ N and k ≤ δn there exist t ∈ [0, 1] so that ⎛ ⎞1/2 k k 1⎝ 2⎠ aj ≤ aj xj (t) 4 j=1 j=1

⎞1/2 ⎛ k 2 ≤ 4⎝ aj ⎠ ,

1n

(1.3.36)

j=1

for every choice of coeﬃcients a = (aj ) ∈ Rk . Proof. Fix coeﬃcients a = (aj ) ∈ Rk , and assume that kj=1 a2j = 1. The ﬁrst part of the proof shows that with a = (aj ) ﬁxed the estimates (1.3.36) hold for a very large set of t ∈ [0, 1]. The purpose of the second part is to exploit this fact and extend (1.3.36) from one set of coeﬃcients to the entire unit sphere of 2k . Part 1. For t ∈ [0, 1], we ﬁrst deﬁne the function f (t) and then unwind the deﬁnition of the random vectors xj (t). Thus, k f (t) = aj xj (t) j=1 1 n n k 1 = aj rik+j (t) . n i=1 j=1

(1.3.37)

Next we estimate the Haar coeﬃcients of f, its square function and its mean value over the interval [0, 1]. Starting with the Haar coeﬃcients we ﬁx i ≤ n and

1.3. Bounded square functions and large deviation inequalities

65

m ≤ k. Let I be a dyadic interval. The statistical independence of the Rademacher functions gives the following implication: k If hI , rik+m = 0, then aj rik+j (t) hI (t)dt ≤ |I| · |am |. (1.3.38) j=1 The remaining Haar coeﬃcients vanish. By (1.3.38) the square function of f satisﬁes the following upper bound. S 2 (f )(t) ≤

n k 1 2 a n2 i=1 m=1 m

(1.3.39)

1 2 ≤ a . n j=1 m k

Using Khintchine’s inequality (for the left-hand side of (1.3.40)) and the biorthogonality of the Rademacher functions (for the right-hand side of (1.3.40)) we obtain that ⎛ ⎛ ⎞1/2 ⎞1/2 1 k k k √ −1 a2j ⎠ ≤ aj rik+j (t) dt ≤ ⎝ a2j ⎠ . (1.3.40) ( 2) ⎝ 0 j=1 j=1 j=1 Recall now that we imposed that √ ( 2)−1 ≤

a2j = 1. In summary we obtained that

1

f (t)dt ≤ 1 and

S 2 (f )(t) ≤

0

1 . n

(1.3.41)

It follows from (1.3.41) and (1.3.35) that this deviation inequality holds, |f −

1 0

2 s n f | > s ≤ 2 exp − . 2

(1.3.42)

√ √ 1 By (1.3.41) we get {f ≥ 2 or f ≤ (2 2)−1 } ⊆ {|f − 0 f | > (2 2)−1 }. Taking √ −1 complements and using (1.3.42) with s = (2 2) gives that % √ % n& & . (1.3.43) (2 2)−1 ≤ f ≤ 2 ≥ 1 − 2 exp − 16 For obvious reasons, (1.3.43) is frequently called the concentration of measure inequality for f. Summing up the ﬁrst part of the proof, the function f, formed by a single set of coeﬃcients a = (aj ) is bounded (from above and below) on an extremely large subset of the unit interval.

66

Chapter 1. The Haar System: Basic Facts and Classical Results

Part 2. Next we pass from a ﬁxed set of coeﬃcients a = (aj ) to a suﬃciently large collection of coeﬃcients in the unit sphere of 2k . It is at this point of the argument that we will impose a restriction on the size of k. Let 0 > 0. Denote by S k−1 the unit sphere of 2k . Choose a discrete set M ⊆ S k−1 — with minimal cardinality — so that for each b ∈ S k−1 there exists a ∈ M satisfying |a − b| ≤ 0 . Verify, by counting, that the cardinality of M is bounded by exp{+2k/0 }.

(1.3.44)

(See, for instance, Proposition 10.II.E in [213].) Now choose 0 > 0 small enough so that this implication holds: If k √ −1 (2 2) ≤ aj xj (t) (1.3.45) ≤ 2 for all (aj ) ∈ M, j=1 1 n

then

k 1 ≤ aj xj (t) 4 j=1

≤ 4 for all (aj ) ∈ S k−1 .

(1.3.46)

1n

An elementary iteration argument allows one to choose 0 > 0 satisfying the above property, and 0 does not depend on the value of k. (The details of the iteration are worked out in [148].) Next apply the ﬁrst part of the proof to each a = (aj ) ∈ M, and deﬁne ⎧ ⎫ ⎪ k ⎪ ⎨ ⎬ + √ −1 and E = Ea = t : (2 2) ≤ aj xj (t) ≤ 2 Ea . ⎪ ⎪ ⎩ ⎭ j=1

1n

a∈M

Inequality (1.3.43) states that |Ea | > 1 − 2 exp{−n/16}. By (1.3.44), the measure of the intersection E is therefore larger than −n +2k . 1 − 2 exp exp 16 0 Hence, for k ≤ (n/32 − 2)0 say , the set E is non-empty. For each t ∈ E, by its deﬁnition, the estimates (1.3.45) hold true. Passing from (1.3.45) to (1.3.46) completes the proof. We emphasize that (1.3.44) and (1.3.43) are the central and competing inequalities of the preceding proof. √ The square function of bounded functions. We exploit the constants p appearing in Theorem 1.3.5 to prove that the square function of a bounded function is square-exponentially integrable.

1.3. Bounded square functions and large deviation inequalities

67

Corollary 1.3.7. Let f ∈ L∞ . Then for 0 < α < 1/e,

1

exp{α 0

S 2 (f ) }dt < ∞. ||f ||2L∞

Proof. We use the normalization ||f ||L∞ = 1. Expanding the exponential in its Taylor series gives ∞ αn exp{αS 2 (f )} = S(f )2n . n! n=0 Apply Theorem 1.3.5, with p = 2n to f. This gives

1

S(f )2n dt ≤ nn ||f ||2n 2n . 0

Note that nn /n! ≤ en , since nn /n! is just one of the summands in the Taylor expansion of en . Then we have

1

∞ αn 1 exp{αS (f )}dt = S(f )2n dt n! 0 n=0 2

0

≤ ≤

∞ αn n n n! n=0 ∞

αn en .

n=0

For α < 1/e the geometric series converges.

Quadratic exponential integrability is the best estimate available for S(f ) when f is uniformly bounded. To see this consider the following example. Let In = [0, 4−n ). The sequence In is decreasing, In ⊃ In+1 . Next deﬁne the following sequence of pairwise disjoint intervals , 3 Kn = [ 4−n , 4−n ), 4 and let, f=

∞

1Kn .

(1.3.47)

n=0

As the intervals Kn are pairwise disjoint, f is bounded by 1. Now we compute the Haar coeﬃcients. We have −|In | ; hIn f dt = 6

68

Chapter 1. The Haar System: Basic Facts and Classical Results

the other coeﬃcients in the Haar expansion of f vanish. Hence, S 2 (f )(t) = It follows that S 2 (f )(t) =

n 36

∞ 1 1I . 36 n n=0

for t ∈ In \ In+1 .

This shows that S(f ) is unbounded and the distribution function of S(f ) satisﬁes the lower estimate |{S(f ) > t}| ≥ A0 exp{−a0 t2 }. Clearly this shows that quadratic exponential integrability is the best one can get for the square function of f, even when f is uniformly bounded. Observe however that for the example (1.3.47) and m(t) = t the product mf is non-trivial in the sense that mf > 0 and mf ∈ SL∞ . In the next subsection we will ﬁnd such a multiplier for every bounded nonnegative function f.

Multipliers into SL∞ To prove the good-λ inequality in Proposition 1.3.3 we produced functions in SL∞ by deleting certain coeﬃcients in the Haar expansion of a given function. Thereby we contracted the Haar support and lost any information about its pointwise support. In this subsection we present a method to produce elements in SL∞ by contracting simultaneously the Haar coeﬃcients and the pointwise support of a given function. Let f ∈ L∞ ([0, 1]) be non-negative. Let its Haar expansion be

1

f+

f=

aI h I .

0

In the next theorem we deﬁne a multiplier m : [0, 1] → [0, 1], such that h = m · f, is non-trivial and has uniformly bounded square function. We determine m so that the operation f −→ mf is a pointwise contraction on f that also contracts each individual Haar coeﬃcient of f to the extent that h has a bounded square function. The product mf is non-trivial in the sense that mf ≥ exp(− f ∞ ) f. Replacing f with f / f ∞ allows us to restrict ourselves to the case where f is normalized in L∞ ([0, 1]).

1.3. Bounded square functions and large deviation inequalities

69

Theorem 1.3.8. Let f ∈ L∞ ([0, 1]), and assume that 0 ≤ f ≤ 1. Let fI be the mean value of f over the interval I. Then there are constants mI ∈ [1/6, 6] so that the multiplier m : [0, 1] → [0, 1], deﬁned by 1 1 m = exp(f − f ∞ ) exp − a2I mI ( + )h2I 2 fI satisﬁes the following conditions: The product h = m · f is non-trivial, and 1 1 h ≥ exp(− f ∞ ) f. 0

0

The Haar coeﬃcients of h = m · f satisfy the upper bound ⎞ ⎛ |h, hI | ≤ A|I| exp ⎝− a2J ⎠ |aI |. J⊇I

Consequently the dyadic square function of h is uniformly bounded, , hI - 2 h2I ≤ CA2 . h, |I| I

2

Proof. Clearly e−S (f ) is the ﬁrst candidate for a multiplier to satisfy the conclusions of our theorem. However there are serious obstacles that prevent one 2 from getting satisfactory estimates for the Haar coeﬃcients f e−S (f ) , hI . The construction of the multiplier ϕ below is in direct response to these diﬃculties; indeed the combinatorial identity (1.3.48) combined with the estimates (1.3.51) are crucial in proving the main inequality (1.3.52) for the Haar coeﬃcients of f eϕ . Overview. Given 0 ≤ f ≤ 1, we write down its Haar expansion, 1 f+ aI h I . f= 0

First we show that for every dyadic interval there exists a constant mI satisfying 1 ≤ mI ≤ 6, 6 and such that the following identity holds, 1 1 + f exp aI hI − a2I mI h2I dt = f. 2 fI I I Once the sequence {mI } satisfying (1.3.48) is determined we put 1 1 ϕ = f − f ∞ − + a2I mI h2I , 2 fI I

(1.3.48)

70

Chapter 1. The Haar System: Basic Facts and Classical Results

and deﬁne the multiplier as m = eϕ . We will verify below that the function h = mf satisﬁes the conclusions of Theorem 1.3.8. Veriﬁcation of (1.3.48). Let I be a ﬁxed dyadic interval. We begin by evaluating the integral f eaI hI dt. I

Deﬁne R(t) to be the remainder term of order 3 in the Taylor expansion of the exponential exp(aI hI (t)), that is 1 R(t) = exp(aI hI (t)) − 1 − aI hI (t) − a2I h2I (t). 2

Deﬁne also Q=

f (t)R(t)dt. I

Then, |Q| ≤ |a3I | I f (t)dt. Next we rewrite , 1 2 2 aI hI (t) f (t)e dt = f (t) 1 + aI hI (t) + aI hI (t) + R(t) dt 2 I I 2 a = f (t)dt + a2I |I| + I f (t)dt + Q 2 I I Q a2I a2I |I| + + . = f (t)dt 1 + 2 f dt f dt I I I Now recall that we write fI for the mean value of f over the interval I. Then, clearly, we have the identity 1 Q 1 a2I a2I |I| Q 2 + + + = 1 + aI . 1+ + 2 2 f |I|f f dt f dt I I I I Next we determine constants mI so that the following equation holds. 1 1 1 1 Q 2 2 1 + aI + + = exp aI + mI . 2 fI |I|fI 2 fI

(1.3.49)

As |Q| ≤ |a3I ||I|fI and |aI | ≤ fI , the solutions mI of (1.3.49) satisfy the estimates 1 < mI < 6. 6 Summing up we showed that 1 1 + f eaI hI dt = f dt exp a2I mI . 2 fI I I

(1.3.50)

1.3. Bounded square functions and large deviation inequalities

71

Dividing (1.3.50) by exp(a2I ( 21 + f1I )mI ) gives the identity (1.3.48). Thus we chose a constant mI ∈ [1/6, 6] so that the functions f and f exp(aI hI −a2I ( 21 + f1I )mI h2I ), have identical mean value over the interval I. Estimates for f eaI hI , hI . Next we show that the Haar coeﬃcient f eaI hI , hI , admits the upper bound f eaI hI hI ≤ 3|aI | |I|. (1.3.51) I

To verify this we start again with the Taylor expansion of the exponential, exp(aI hI (t)) = 1 + aI hI (t) + E(t). Then |E(t)| ≤ a2I . In the integral (1.3.51) we replace eaI hI (t) by 1 + aI hI (t) + E(t) and estimate as follows, using |E(t)| ≤ a2I . f eaI hI hI = f (1 + aI hI + E) hI I I 2 ≤ f hI + |aI | f + aI f I

I

I

≤ 2|aI | · |I| + a2I |I|. As 2|aI | · |I| + a2I |I| ≤ 3|aI | |I|, we have (1.3.51). The main inequality. We are now ready to prove a crucial estimate for the Haar coeﬃcient of eϕ f, ⎞ ⎛ 1 1 eϕ f hI dt ≤ 3|aI | |I| exp ⎝− (1.3.52) a2J ( + )mJ ⎠ . 2 fJ J⊇I

To this end we split ϕ into three pieces. For x ∈ I we write 1 2 1 2 ϕ(x) = vI + aI hI (x) − aI ( + )mI hI (x) + zI (x), 2 fI where vI is deﬁned to be the number 1 1 1 2 vI = − f ∞ + + f dt + aJ h J − aJ m J h2J . 2 fJ 0 J⊃I

(1.3.53)

J⊃I

Recall that by writing J ⊃ I we denote the summation over all dyadic intervals J that are strictly containing I. Note that zI is the function given by 1 1 + aJ h J − a2J mJ zI = h2J , 2 fJ J⊂I

J⊂I

72

Chapter 1. The Haar System: Basic Facts and Classical Results

here J ⊂ I denotes all dyadic intervals that are strictly contained in I. Now let I1 be the left half of I and let I2 be the right half of I. Inductively applying (1.3.48) we obtain the identities zI e f= f, I1 I1 (1.3.54) zI e f= f. I2

I2

The identities (1.3.54) are crucial in reducing estimates for the Haar coeﬃcient ϕ f, hI , to the estimate for f eaI hI , hI in (1.3.51). Indeed as eaI hI is a constant function on both intervals I1 and I2 , we observe that with (1.3.54), aI hI zI aI zI e e f hI dt = e e f dt − e−aI ezI f dt I1 I2 aI e f dt − e−aI f dt = (1.3.55) I1 I2 = eaI hI f hI dt. By (1.3.53) we observe that ⎛ evI ≤ exp ⎝−

J⊇I

⎞ 1 1 a2J ( + )mJ ⎠ . 2 fJ

(1.3.56)

The identity (1.3.55) and the estimates (1.3.51) and (1.3.56) imply the following upper bound for the coeﬃcient eϕ f, hI , eϕ f hI dt = evI exp −a2I ( 1 + 1 )mI eaI hI ezI hI f dt 2 fI ⎛ ⎞ 1 1 = exp ⎝− a2J ( + )mJ ⎠ eaI hI f hI dt (1.3.57) 2 fJ J⊇I ⎛ ⎞ 1 1 ≤ 3|I| exp ⎝− a2J ( + )mJ ⎠ |aI |. 2 fJ J⊇I

Upper bounds for the square function of eϕ f . Next we shorten notation and abbreviate, ⎞ ⎛ a2 J⎠ . pI = exp ⎝− 12 J⊇I

Note that pI is clearly bounded by 1. The sum {I:x∈I} pI a2I is a lower Riemann ∞ s )ds, where the partition is given by the partial sums of the sum for 0 exp(− 12

1.4. Martingales and biorthogonal systems series

{I:x∈I}

73

a2I . Thus for every x ∈ [0, 1],

pI a2I

{I:x∈I}

≤

∞

exp(− 0

s )ds = 12. 12

It is now quite easy to estimate the square function of eϕ f. Expand cI h I eϕ f = eϕ f + in its Haar series. Then by (1.3.57) and (1.3.56), |cI | ≤ pI aI . At each point x ∈ [0, 1] we have therefore the uniform bound, S 2 (eϕ f )(x) ≤ 9

pI a2I

{I:x∈I}

≤ 108. The mean value of eϕ f . Next we observe that by iteratively applying equation (1.3.48) we arrive at the identity

1

e f dt = exp(−f + ϕ

0

0

1

f )

1

f dt,

∞

0

that implies the lower bound

1

eϕ f dt ≥ exp(−f ) ∞ 0

1

f dt. 0

Finally, as ϕ ≤ 0 the multiplier m = eϕ , satisﬁes 0 ≤ m ≤ 1.

1.4

Martingales and biorthogonal systems

The Haar basis is at the same time the simplest martingale diﬀerence sequence and the simplest biorthogonal system. In this section we extend the content of Burkholder’s Theorem 1.1.5 to all martingale diﬀerence sequences and to a large class of biorthogonal systems, including the Franklin system and the wavelet family. The results of this section illustrate the use of the Haar system in the solution of problems arising in probability and analysis.

74

Chapter 1. The Haar System: Basic Facts and Classical Results

The martingale inequalities of D. Burkholder and E. M. Stein We deﬁne ﬁrst conditional expectations and recall the notions of martingales and martingale diﬀerence sequences. The following summary is based on the appendix in [61] by R. Durrett. We refer to this book for the proofs. Let (Ω, F, | . |) be a probability space. Let G ⊆ F be a sub σ-algebra of F. The theorem of Radon and Nikodym implies that for every f ∈ L1 (Ω, F, | . |) there exists an (essentially unique) g : Ω → R which is measurable with respect to G and satisﬁes f= g for A ∈ G. A

A

The function g is then called the conditional expectation of f with respect to G, for which we use the notation E(f |G) = g. Taking conditional expectations deﬁnes a contraction on Lp (Ω) (1 ≤ p ≤ ∞). Let {Fn } be a sequence of increasing σ-algebras contained in F. Assume that F coincides with the σ-algebra generated by F0 ∪ · · · ∪ Fn ∪ . . . . Let {fn } be a sequence in L1 (Ω, F), such that fn = E(fn+1 |Fn ). Then {fn } is called a martingale and dn = fn − fn−1 is called a martingale diﬀerence sequence. Examples of martingales are most easily obtained by ﬁxing f ∈ L1 (Ω, F) and forming the conditional expectations, fn = E(f |Fn ). The partial sum operators of the Haar basis {hi }∞ i=0 are given by conditional expectation with respect to an increasing sequence of σ-algebras. See Proposition 1.1.1 and itsproof. The same holds for the partial sums of the Rademacher assertion ∞ a2i < ∞. series i=1 ai ri , when Burkholder’s martingale inequality For f ∈ Lp (Ω) the conditional expectations fn = E(f |Fn ) form a convergent sequence in Lp (Ω) (1 ≤ p < ∞). Indeed we have

f − fn Lp (Ω) → 0 for n → ∞. Hence with dn = fn − fn−1 the martingale diﬀerence series ∞ n=1 dn is convergent in Lp (Ω). The next result extends Burkholder’s theorem to martingales. It asserts that in Lp (Ω), 1 < p < ∞, the decomposition of f in its martingale diﬀerences f = E(f |F0 ) +

∞ n=1

dn

1.4. Martingales and biorthogonal systems

75

is unconditionally converging. We present the proof of B. Maurey [139], which consists of reducing estimates for general martingale diﬀerence sequences to the case of Haar functions. Theorem 1.4.1. Let f ∈ Lp (Ω, F), 1 < p < ∞. Let {Fn } be a sequence of increasing σ-algebras contained in F. Let dn = E(f |Fn ) − E(f |Fn−1 ). Then for every choice of n ∈ {+1, −1}, the following estimate holds, ∞ n dn ≤ (p∗ − 1) f Lp (Ω) . E(f |F0 ) + p n=1

(1.4.1)

L (Ω)

Proof. In this proof, we ﬁnd it convenient to use systematically the following notation. For a collection of pairwise disjoint dyadic intervals C we denote Q(C) = {J ∈ D : J ⊆ K}. K∈C

Thus Q(C) is the collection of intervals which are contained in one of the intervals of C. For a single dyadic interval I we write Q(I) = {J ∈ D : J ⊆ I}. Overview. Given > 0 we will construct a sequence of functions hn : [0, 1] → R with the following three properties. First, their Haar support, given by Hn = {I : hn , hI = 0}, forms a sequence of pairwise disjoint collections of dyadic intervals, second, ∞ hn ≤ (1 + ) f Lp (Ω) , (1.4.2) h0 + p n=1

L ([0,1])

and third, for every choice of n ∈ {+1, −1} the following estimate holds, ∞ ∞ n dn ≤ (1 + ) h0 + n hn . (1.4.3) E(f |F0 ) + p p n=1

n=1

L (Ω)

L ([0,1])

Once this sequence is obtained we get (1.4.1) from Burkholder’s inequality. Indeed, using (1.4.3), Theorem 1.1.5 and (1.4.2) gives that ∞ ∞ n dn ≤ (1 + ) h0 + n hn E(f |F0 ) + p p n=1 n=1 L (Ω) L ([0,1]) ∞ ∗ ≤ (p − 1)(1 + ) h0 + hn p n=1

∗

≤ (p − 1)(1 + ) f Lp (Ω) . 2

L ([0,1])

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Chapter 1. The Haar System: Basic Facts and Classical Results

The basic observation. Below we will construct the sequence {hn } satisfying (1.4.2) and (1.4.3) by iterating an elementary observation. Let C be # a collection of pairwise disjoint dyadic intervals covering the unit interval. Thus I∈C I = [0, 1[. Assume that C is decomposed into pairwise disjoint collections C1 , . . . , Cm . Let Ci be the point-set covered by the collection Ci . The sets Ci are pairwise disjoint and cover the unit interval. Choose now scalars ai ∈ R so that m ai |Ci | = 0, (1.4.4) i=1

and form the function h : [0, 1] → R by putting h=

m

ai 1Ci .

(1.4.5)

i=1

Thus we deﬁned the function h to be measurable with respect to the σ-algebra generated by the collection C. By (1.4.4) we have h = 0. Next put H = Q([0, 1]) \ Q(C). Since h is measurable with respect to the σ-algebra generated by C and the Haar support of h is contained in H. Thus hJ h= hJ . h, |J|

(1.4.6) h = 0,

(1.4.7)

J∈H

Summing up for a function h, given by (1.4.5) and (1.4.4), the Haar support is contained in H, deﬁned by (1.4.6). Perturbation, preparation. Now, let f ∈ Lp (Ω, F), 1 < p < ∞. Let {Fn } be a given sequence of increasing σ-algebras contained in F. We assume for simplicity that F0 = {Ω, ∅}. Let dn = E(f |Fn ) − E(f |Fn−1 ). Furthermore we assume that each of the martingale diﬀerences dn is a step function. Measured in Lp , the error of approximating by a step function can be made arbitrarily small. Consequently we may assume that each of the σ-algebras {Fn } is generated by a ﬁnite set of atoms. Let An denote the atoms of {Fn }. The collections {An } are linearly ordered in the following sense. If A ∈ An and # B ∈ An+1 so that A ∩ B = ∅, then B ⊆ A. Taking the union and forming A = An , gives a collection of sets which satisﬁes the following condition. If A, B ∈ A, and A ∩ B = ∅, then A ⊆ B or B ⊆ A.

(1.4.8)

Next we will reproduce the intersection pattern of A, by using collections of dyadic intervals. Precisely we claim that for each A ∈ A there #exists a collection of pairwise disjoint dyadic intervals CA so that with CA = I∈CA I the following conditions hold.

1.4. Martingales and biorthogonal systems

77

1. CΩ = [0, 1], and |CB | = |B| for B ∈ A. 2. If A, B ∈ A, and CA ∩ CB = ∅, then CA ⊆ CB or CB ⊆ CA . 3. CB ⊆ CA iﬀ B ⊆ A. 4. If B ⊆ A, and I ∈ CA , then |B| |I ∩ CB | = . |I| |A|

(1.4.9)

5. If I ∈ CB , J ∈ CA and I ⊆ J then B ⊆ A. Collections {CA : A ∈ A} with the above properties are constructed inductively in the course of the proof of Proposition 2.1.4 in Chapter 2. Before we continue we should comment on condition (1.4.9): Let A ∈ An and I ∈ CA . Then by (1.4.9), conditioned to I the measures of the sets {I ∩ CB : B ∈ An+1 , B ⊆ A} are the same as the measures of {B ∈ An+1 : B ⊆ A} conditioned to A. # The induction argument. Deﬁne Cn = B∈An CB . Note that Cn is a collection of pairwise disjoint dyadic intervals. We are going to construct now a sequence of functions {hn } with Haar support contained in Hn = Q(Cn−1 ) \ Q(Cn ) so that the joint distribution of {hn } is the same as the joint distribution of the martingale diﬀerence sequence {dn }. Recall that F0 = {Ω, ∅}. Hence E(f |F0 ) is a constant function such that f. We let h0 : [0, 1] → R be the constant function deﬁned by h0 (t) = E(f |F 0) = f. Now we deﬁne h1 : [0, 1] → R. For d1 : Ω → R there exist coeﬃcients d(B), B ∈ A1 so that d1 = d(B)1B and d(B)|B| = 0. B∈A1

B∈A1

Next we modify d1 replacing the indicator function 1B : Ω → {0, 1} by 1CB : [0, 1] → {0, 1}. By doing this we obtain a copy of d1 with the same distribution function. We put d(B)1CB . h1 = B∈A1

Note that h1 is measurable with respect to the σ-algebra generated by the collection C1 , and h1 = 0. Consequently with H1 = Q([0, 1]) \ Q(C1 ),

78

Chapter 1. The Haar System: Basic Facts and Classical Results

the Haar expansion of h1 assumes the form h1 =

h1 ,

I∈H1

hI hI . |I|

By construction the martingale diﬀerences {d0 , d1 } and the newly constructed functions {h0 , h1 } are identically distributed. Suppose now that {h0 , h1 , . . . hn } are deﬁned so that for i ≤ n, hi =

I∈Hi

hi ,

hI hI , |I|

where

Hi = Q(Ci−1 ) \ Q(Ci ),

and so that the joint distribution of {d0 , d1 , . . . , dn } and {h0 , h1 , . . . , hn } are identical. Next we determine hn+1 . Let I ∈ Cn . There exists a uniquely determined A ∈ An such that I ∈ CA . Observe that restricting h0 , h1 , . . . , hn to the interval I gives a constant function. Similarly the restriction of d0 , d1 , . . . , dn to A is a constant. Next ﬁx B ∈ An+1 such that B ⊆ A. Then form the collection of dyadic intervals CB,I = {J ⊆ I : J ∈ CB }, and let CB,I denote the point set covered by CB,I . Next we consider the restriction of dn+1 to the atom A ∈ An . For B ∈ An+1 , with B ⊆ A, there exist coeﬃcients d(B) such that 1A dn+1 = d(B)1B . B∈An+1 ,B⊆A

Note that E(dn+1 |Fn ) = 0 implies that A dn+1 = 0. Next we replace the indicator functions 1B : A → {0, 1} by 1CB,I : I → {0, 1}. By (1.4.9), we obtain a copy of 1A dn+1 , by deﬁning hn+1,I = d(B)1CB,I . B∈An+1 ,B⊆A

Observe that hn+1,I is measurable with respect to the σ-algebra generated by {J ∈ Cn+1 : J ⊆ I}. Moreover I hn+1,I = 0. It follows that the Haar support of hn+1,I is contained in Hn+1,I = Q(I) \ Q({J ∈ Cn+1 : J ⊆ I}). Finally deﬁne hn+1 by summing over I ∈ Cn . Thus, hn+1,I . hn+1 = I∈Cn

1.4. Martingales and biorthogonal systems

79

With this deﬁnition of hn+1 the functions {d0 , . . . , dn+1 } and {h0 , . . . , hn+1 } are identically distributed. Finally for Hn+1 = Q(Cn ) \ Q(Cn+1 ), the Haar expansion of hn+1 is given by hI hn+1 , hI . hn+1 = |I| I∈Hn+1

This completes the inductive construction of the sequence {hn }.

Consequences. Combining Theorem 1.4.1 with Khintchine’s inequality gives a large class of square function characterizations for Lp (Ω, F), 1 < p < ∞. We let {Fn } be a sequence of increasing σ-algebras contained in F. Assume that F is the # σ-algebra generated by Fn . For f ∈ Lp (Ω, F) deﬁne d0 = E(f |F0 ),

and

dn = E(f |Fn ) − E(f |Fn−1 ),

for n ∈ N.

2 1/2 the martingale square function of f. By Theorem 1.4.1 We call (d20 + ∞ n=1 dn ) the decomposition of f into martingale diﬀerences f = d0 +

∞

dn

n=1

converges unconditionally in Lp , 1 < p < ∞, and by Khintchine’s inequality we have the following characterization of Lp (Ω) by martingale-square-functions 1/2 ∞ 2 2 cp f Lp (Ω) ≤ d0 + dn ≤ Cp f Lp (Ω) . (1.4.10) p n=1 L (Ω)

For 1 < p < ∞ we denote by H p [(Fn )] the space of all f ∈ Lp (Ω, F) for which 1/2 ∞ 2 2 E(f |F ) + (E(f |F ) − E(f |F )) < ∞.

f H p [(Fn )] = 0 n n−1 p n=1 L (Ω)

By (1.4.10) the identity operator Id : Lp (Ω) → H p [(Fn )] is an isomorphism. The importance of this result lies in its generality: One is completely free to choose the σ-algebras {Fn } to decompose a given f ∈ Lp (Ω). Stein’s martingale inequality A very surprising application of Burkholder’s martingale inequality is J. Bourgain’s proof of E. M. Stein’ s martingale inequality.

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Chapter 1. The Haar System: Basic Facts and Classical Results

Theorem 1.4.2. Let (Ω, F, | · |) be a probability space. Let F1 ⊆ · · · ⊆ Fm ⊆ F be σ algebras in Ω. Then, for any choice of f1 , . . . , fm ∈ Lp (Ω) the following estimate holds true, p p 1 1 m m ∗ p rn (t)E(fn |Fn ) dt ≤ (p − 1) rn (t)fn dt, (1.4.11) p 0 0 p n=1

n=1

L (Ω)

L (Ω)

∗

where p = max{p, p/(p − 1)}, and where r1 , . . . , rm are Rademacher functions. Proof. For t ∈ [0, 1], and x ∈ Ω deﬁne F (t, x) =

m

rn (t)fn (x).

n=1

On the product space [0, 1] × Ω deﬁne the increasing σ-algebras Gn = σ{r1 , . . . , rn } ⊗ Fn ,

n ≤ m.

Observe that taking the conditional expectation with respect to Gn corresponds to forming partial sums as follows, ri (t)E(fi |Fn )(x). (1.4.12) E(F | Gn )(t, x) = i≤n

Let F0 = 0, put Fn = E(F | Gn ), for 1 ≤ n ≤ m, and dn (fi ) = E(fi | Fn ) − E(fi | Fn−1 ), for 2 ≤ n ≤ m. Note that by (1.4.12), we get Fn − Fn−1 = rn E(fn |Fn ) + ri dn (fi ), for 1 ≤ n ≤ m.

i
(In the above formula we read i<1 = 0, which is consistent with the standard convention that the value of a sum taken over the empty set equals 0.) Hence, m

Fn − Fn−1 =

n=1

m n=1

rn E(fn |Fn ) +

m

ri dn (fi ).

n=1 i
Next apply Theorem 1.4.1 to the martingale diﬀerences Fn − Fn−1 deﬁned on the probability space [0, 1] × Ω. Put Cp = max{p, p/(p − 1)} − 1. Then for every s ∈ [0, 1], the following estimate holds. 1 Cpp |F (t, x)|p dtdx Ω 0 p 1 m ≥ rn (s)(Fn − Fn−1 )(t, x) dtdx Ω 0 n=1 p 1 m m = rn (s)rn (t)E(fn |Fn )(x) + ri (s)ri (t)dn (fi )(x) dtdx. Ω 0 n=1

n=1 i
(1.4.13)

1.4. Martingales and biorthogonal systems

81

For s ∈ [0, 1] and x ∈ Ω, we next rewrite the integrand in the last line of the above estimate. Observe that (by multiplication with rn (s)) the following identity holds. p 1 m m rn (s)rn (t)E(fn |Fn )(x) + ri (s)ri (t)dn (fi )(x) dt 0 n=1 n=1 i
n=1 i
Next combine (1.4.14) with (1.4.13) and apply the triangle inequality. Then we 1 exploit that for i < n, 0 ri (s)rn (s)ds = 0. Thus we obtain the following estimates.

1

|F (t, x)|p dtdx Ω 0 p 1 m m rn (t)E(f |Fn )(x) + ri (s)rn (s)ri (t)dn (fi )(x) dtdx ≥ sup s∈[0,1] Ω 0 n=1 n=1 i
Cpp

E.M. Stein in [193] proved that for fn ∈ Lp (Ω), the following martingale inequality holds true, 1/2 1/2 m m 2 2 E(fn |Fn ) ≤ Ap fn , (1.4.15) p n=1 p n=1 L (Ω)

L (Ω)

where Ap ≤ Cp2 /(p − 1). By Khintchine’s inequality we obtain (1.4.11) — with diﬀerent constants — from (1.4.15), and conversely (1.4.11) implies that E. M. Stein’s martingale inequality holds true. The original proof in [193] applies just to scalar valued Lp -functions fn : Ω → R. Above we presented J. Bourgain’s proof since it is solely based on Burkholder’s theorem and the triangle inequality in the space Lp . Consequently the estimate (1.4.11) carries over to Lp functions fn : Ω → E taking values in a Banach space E with the UMD property.

T. Figiel’s approach to biorthogonal systems Next we turn to biorthogonal systems. We introduce a powerful method of T. Figiel [65] showing that a large class of biorthogonal systems is equivalent to the Haar basis in Lp ([0, 1]). As a corollary we obtain the theorem of S.V. Boˇckarev [14] that

82

Chapter 1. The Haar System: Basic Facts and Classical Results

the Franklin functions form an unconditional basis in Lp ([0, 1]). We ﬁx a family of functions fI : [0, 1] → R, indexed by dyadic intervals I ∈ D. In this section we determine conditions on the family {fI } which imply that the linear extension of the map T : hI → fI , deﬁnes a bounded operator on the Lp spaces when 1 < p < ∞. First we impose a set of three conditions on the mean values and on the Haar coeﬃcients of {fI } which imply that the map T : hI → fI , extends to a bounded operator on Lp . Later we give conditions on the pointwise behavior of {fI } to the same eﬀect. We assume that 1

fI dt = 0.

(1.4.16)

0

Using mean values and Haar coeﬃcients we deﬁne what it means for {fI } to be dominated by the Haar system. Let I be a dyadic interval with |I| = 2−i . Let m ∈ Z such that I + m|I| ⊆ [0, 1]. Let J be a dyadic interval such that, J ⊆ I + m|I|, |J| = 2−l |I|, where l ∈ N. Assume that there exist A > 0 and α > 0 independent of I or J so that the mean values and the Haar coeﬃcients of fI satisfy the estimates fI (y)dy ≤ A|I|{1 + |m|}−1−α , (1.4.17) I+m|I| and

0

1

fI (y)hJ (y)dy ≤ A|J|2−l {1 + |m|}−1−α .

(1.4.18)

We say that the family of functions {fI : I ∈ D} is dominated by the Haar system if the three conditions (1.4.16), (1.4.17) and (1.4.18) hold true for every dyadic interval I ∈ D. Theorem 1.4.3. Let 1 < p < ∞. Let {fI : I ∈ D} be a family of functions that is dominated by the Haar system, that is, it satisﬁes (1.4.16), (1.4.17) and (1.4.18). Let (xI ) be sequence of scalars. Then the following estimate holds, ≤ Cp (A, α) , xI fI p xI h I p L ([0,1])

L ([0,1])

where Cp = C(A, α) max{p, p/(p − 1)}. Proof. The idea of the proof is encoded in Figiel’s expansion (1.4.28). Its derivation is contained in the ﬁrst three steps of the proof. In Steps 4 and 5 we show how (1.4.28) implies the conclusion of the theorem.

1.4. Martingales and biorthogonal systems

83

Step 1. The starting point is the following non-standard expansion of fI . First cut fI using the indicator functions of the shifted intervals I + m|I|. fI (x) = 1I+m|I| (x)fI (x). (1.4.19) {m:I+m|I|⊆[0,1]}

Fix m such that I+m|I| ⊆ [0, 1]. Then we expand further each summand appearing in (1.4.19). We would like to use Haar series in such a way that only those Haar functions hJ appear for which J ⊆ I +m|I|. To make this possible we ﬁrst take out the mean value of fI over I + m|I|. Thus we derived an identity with important consequences, 1 fI (y)dy1I+m|I| (x) 1I+m|I| (x)fI (x) = |I| I+m|I| ∞ (1.4.20) hJ + hJ (x). fI , |J| −l l=0 {J⊆I+m|I|, |J|=2

|I|}

The mean values, |I|−1 I+m|I| fI (y)dy, and the remaining series of Haar functions in (1.4.20) will be treated quite diﬀerently below. Step 2. We exploit now that the integral of fI over the unit interval vanishes. Observe that fI = 0 can be written as . / fI (y)dy = 0. (1.4.21) {m:I+m|I|⊆[0,1]}

I+m|I|

Next we take the sum of the mean values appearing in the ﬁrst line of the equation (1.4.20). The identity (1.4.21) is used below to replace the characteristic functions 1I+m|I| by the diﬀerence 1I+m|I| (x) − 1I (x) which is much better behaved. Before we start doing this we introduce a convenient abbreviation. Let 1 fI (y)dy, (1.4.22) bIm = |I| I+m|I| and for I + m|I| ⊆ [0, 1] deﬁne

With (1.4.21) we ﬁnd

Um (hI ) = 1I+m|I| − 1I . I m bm

= 0, hence

bIm 1I+m|I| (x) =

{m:I+m|I|⊆[0,1]}

bIm Um (hI )(x).

(1.4.23)

{m:I+m|I|⊆[0,1]}

The mean value estimate of (1.4.17) implies that for each ﬁxed m and I, sup |bIm | ≤ A {1 + |m|}−1−α . I

(1.4.24)

84

Chapter 1. The Haar System: Basic Facts and Classical Results

Step 3.

Here we analyze the Haar series appearing in (1.4.20). Let

d(m,l) (fI ) =

fI ,

{J⊆I+m|I|, |J|=2−l |I|}

hJ hJ . |J|

(1.4.25)

The Haar support of d(m,l) (fI ) is a collection of pairwise disjoint dyadic intervals. The union of the intervals appearing in the Haar support of d(m,l) (fI ) equals I + m|I|. Hence the square function of d(m,l) (fI ) is supported in the interval I + m|I|. Invoking the coeﬃcient estimates (1.4.18) gives that −1−α −l 2 1I+m|I| . S d(m,l) (fI ) ≤ A {1 + m}

(1.4.26)

Note also that the indicator function 1I+m|I| just coincides with S(Tm (hI )), where Tm is the rearrangement operator deﬁned as the linear extension of the map Tm : hI → hI+m|I| ,

for I + m|I| ⊆ [0, 1].

Moreover, we observe that for each m, l ﬁxed, the set of functions {d(m,l) (fI ) : I ∈ D} is disjointly supported over the Haar system. In summary the following pointwise estimate between the square functions holds true,

−1−α −l xI d(m,l) (fI ) ≤ A2 {1 + m} S Tm ( xI h I ) , (1.4.27) S I

I

where (xI ) is any given sequence of scalars. Step 4. Let us summarize the non-standard expansion of fI developed in the previous three steps. Using that fI = 0 we obtained the formula fI =

bIm Um (hI ) +

m

∞

d(m,l) (fI ).

(1.4.28)

m l=0

By the hypothesis this expansion is convergent. Next we take a ﬁnite linear combination g = I xI fI , where xI ∈ R. We apply the representation (1.4.28) to each of the functions fI . This gives the representation for g as g=

m

xI bIm Um (hI )

+

∞ m l=0

I

xI d(m,l) (fI ).

I

(1.4.29)

Recall that our aim is to estimate the norm of g in Lp by the norm of f = xI h I in Lp . We exploit (1.4.29) by simply applying the triangle inequality. Thus for

g Lp ([0,1]) we obtain an upper bound by the following series, ∞ xI bIm Um (hI ) + xI d(m,l) (fI ) . (1.4.30) p p m m I

L ([0,1])

l=0

I

L ([0,1])

1.4. Martingales and biorthogonal systems

85

In Step 2, we saw that the mean value estimates (1.4.17) imply (1.4.24). Thus using the norm estimate of Haar multipliers (Theorem 1.5.1 below) we ﬁnd that −1−α I xI bm Um (hI ) ≤ Ap {1 + |m|}

Um Lp ([0,1]) f Lp ([0,1]) . p I

L ([0,1])

The analysis of the operators d(m,l) (fI ) is summarized in the estimate (1.4.27) which relates the square function to the basic rearrangement operators Tm . So we get that, −1−α −l/2 xI d(m,l) (fI ) ≤ A {1 + |m|} 2 xI Tm (hI ) p p I

I

L ([0,1])

−1−α

≤ A {1 + |m|}

−l/2

2

L ([0,1])

Tm Lp ([0,1]) f Lp ([0,1]) .

Step 5. In Theorem 3.3.9 below we will prove that the norm of Tm Lp ([0,1]) , and Um Lp ([0,1]) admit an upper bound of the form Cp log2 (4 + |m|) where Cp depends on the value of p only. Hence the estimates for g, established in Step 4, imply that g Lp ([0,1]) ≤ Cp f Lp ([0,1]) . Before we continue we should try to understand one particular aspect about the non-standard expansion that (1.4.28). Namely the role of led to the formula the hypothesis (1.4.16), that is fI = 0. Let fI = aI . In the case where aI = 0 we can, of course, still carry out the expansion (1.4.28). The changes resulting from aI = 0 are displayed in the identity fI = aI 1I +

bIm Um (hI ) +

m

∞

d(m,l) (fI ).

(1.4.31)

m l=0

The question is now if we can also have a result like Theorem 1.4.3 without using that aI = 0, or without making strong assumptions like I∈D |aI | < ∞. The answer to this question is quite surprising and leads to an interesting relation between Lp and BMO : If {fI } satisﬁes (1.4.17) and (1.4.18) and if aI hI ∈ BMO, then the conclusion of Theorem (1.4.3) holds true, that is the map T : hI → fI extends linearly to a bounded operator on Lp , when 1 < p < ∞. How this works and how the BMO condition enters here, will be one of the topics discussed in the next section. Assume that {fI : I ∈ D} is dominated by the Haar system. Incidentally this implies the norm estimates

fI 2 ≤ C|I|1/2 .

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Chapter 1. The Haar System: Basic Facts and Classical Results

Now we want to isolate further conditions on {fI : I ∈ D} so that the map T : hI → fI extends to an operator satisfying two-sided estimates in Lp . That is, we want to know when T : hI → fI extends to an isomorphism between Lp . A minimal requirement is, of course, that the system {fI : I ∈ D} (when augmented with the constant function) is a basis in L2 . So {fI : I ∈ D} has to have a well-deﬁned system of biorthogonal functions attached to it. Call it {gI : I ∈ D}. Let the deﬁning equations for {gI : I ∈ D} be given as follows, 1 1 |I| if I = J, gJ dt = 0 and fI gJ dt = (1.4.32) 0 if I = J. 0 0 Note that then

sup gI 2 |I|−1/2 < ∞.

Our next theorem asserts that if {fI : I ∈ D} and {gI : I ∈ D} are both dominated by the Haar system, then the linear extension of the map T : hI → fI deﬁnes an isomorphism on Lp . Theorem 1.4.4 below gives a very useful criterion for equivalence to the Haar system. It also raises the following signiﬁcant problem: Assume that {fI : I ∈ D} is an unconditional basis in L2 and dominated by the Haar system. Is it true that its biorthogonal system is dominated by the Haar system as well? In many cases the answer is known and positive. Hence Theorem 1.4.4 covers a wide range of examples; it applies, for instance, to the Franklin system, to wavelets and to spline bases. Theorem 1.4.4. Let 1 < p < ∞. Assume that the family {fI : I ∈ D} and its biorthogonal system {gI : I ∈ D} are both dominated by the Haar system, then for every choice of scalars (xI ) the following estimate holds, c xI h I p ≤ xI fI p ≤ C xI h I p , L

L

L

where c = c(A, α)(max{p, p/(p − 1)})−1 , and C = C(A, α) max{p, p/(p − 1)}. Proof. Let 1 < p < ∞. By hypothesis, the family {fI : I ∈ D} is dominated by the Haar system. Hence the right-hand side estimate follows from Theorem 1.4.3. The left-hand side estimate is equivalent to saying that the linear extension of the map S : fI → hI extends to a bounded operator on Lp . However a simple calculation shows that the transposed operator to S is the linear extension of the map S ∗ : hI → gI .

1.4. Martingales and biorthogonal systems

87

By our assumption that {gI : I ∈ D} is dominated by the Haar system it follows that S ∗ is a bounded operator on Lp , by Theorem 1.4.3. It remains to give a set of pointwise estimates on a family {fI : I ∈ D}, that guarantees that it is dominated by the Haar system. This is provided by the next proposition. We will see that the mean value condition (1.4.17) corresponds to a pointwise decay estimate (1.4.34) and that condition (1.4.18) corresponds to a decay estimate for the derivative of fI , expressed in (1.4.35). Proposition 1.4.5. Let A > 0, α > 0. Let {fI : I ∈ D} be a family of functions. Let I ∈ D such that |I| = 2−i . Assume that 1 fI dt = 0 (1.4.33) 0

and for every x ∈ [0, 1],

! "−1−α , |fI (x)| ≤ A 1 + 2i dist(x, I)

(1.4.34)

and every x, y ∈ [0, 1],

! "−1−α , |fI (x) − fI (y)| ≤ A2i |x − y| 1 + 2i dist([x, y], I)

(1.4.35)

where A > 0 and α > 0 are independent of I ∈ D. Then the system {fI : I ∈ D} is dominated by the Haar system. Proof. Fix a dyadic interval I, with |I| = 2−i . Let m ∈ Z. Assume that I + m|I| is contained in the unit interval. By (1.4.34) every x ∈ I + m|I| satisﬁes −1−α

|fI (x)| ≤ A {1 + |m|}

.

Integrating over x ∈ I + m|I| gives the claimed estimate of the mean value of fI over the shifted interval I + m|I|, fI (x)dx ≤ A|I|{1 + |m|}−1−α . I+m|I| Next we turn to estimating the Haar coeﬃcients of fI . Let J be a dyadic interval such that, J ⊆ I + m|I| and |J| = 2−l |I|. If x, y ∈ J, then |x − y| ≤ 2−i−l . Hence the Lipschitz estimate of (1.4.35) implies that for x, y ∈ J and J ⊆ I + m|I| we have −1−α . (1.4.36) |fI (x) − fI (y)| ≤ A2−l {1 + |m|} Let mJ denote the midpoint of the interval J. By (1.4.36), we obtain the following estimate for the Haar coeﬃcients, 1 1 = {fI (y) − fI (mJ )}hJ (y)dy f (y)h (y)dy I J 0

0

≤ |J| sup |fI (y) − fI (mJ )| y∈J

≤ A|J|2−l {1 + |m|}−1−α .

88

Chapter 1. The Haar System: Basic Facts and Classical Results

The books by B. Kashin and A. Saakyan [111], by Y. Meyer and R.R. Coifman [147], or by G. David [58] discuss in detail classes of orthonormal systems for which the hypothesis of Theorem 1.4.4 and Proposition 1.4.5 are satisﬁed.

1.5

Basic operators

We present Figiel’s representation of integral operators using paraproducts, rearrangements and Haar multipliers. We discuss orthogonal (averaging) projections acting on the Haar system, and introduce a basic criterion implying their boundedness on H 1 .

Multipliers and paraproducts First we discuss multiplier operators acting on the Haar system. Let (xI ) be a sequence in R. We deﬁne the multiplier operator X to be the linear extension of the map hI → xI hI . Thus for f with ﬁnite Haar expansion, f= aI h I , I

X(f ) is deﬁned by X(f ) =

xI a I h I .

(1.5.1)

I

So far the multiplier operator X has been deﬁned just algebraically. Now we study its boundedness properties. We show that Burkholder’s inequality implies that for bounded sequences (xI ) the induced multiplier is a bounded operator on Lp , 1 < p < ∞. The proof given below works for any unconditional basis in any Banach space. Theorem 1.5.1. Let 1 < p < ∞. Let (xI ) be a bounded sequence. Then the multiplier deﬁned by (1.5.1) satisﬁes the following norm estimate on Lp , ∗

X(f ) p ≤ (p − 1) sup |xI | f p , I

where p∗ = max{p, p/(p − 1)}. Proof. Fix 1 < p < ∞ and let q denote the H¨ older conjugate index to p. Choose g ∈ Lq satisfying g q = 1, and

X(f ) p = X(f ), g.

(1.5.2)

1.5. Basic operators

89

With the Haar system we expand g = gdt + bI hI . Then by biorthogonality, xI aI bI |I| X(f ), g = I

≤ sup |xI | I

|aI bI | |I|.

(1.5.3)

I

Next choose I ∈ {±1} such that I aI bI = |aI bI |. Again by biorthogonality, I aI bI |I| = I aI hI , g. (1.5.4) I

I

With (1.5.3) and (1.5.4) H¨ older’s inequality gives that X(f ), g ≤ sup |xI | I aI hI g q . I I

(1.5.5)

p

Burkholder’s inequality provides the upper bound

I aI hI p ≤ (p∗ − 1) f p . I

With (1.5.2) and (1.5.5) the last line ﬁnishes the proof.

Next we deﬁne paraproduct operators and study their boundedness on the Lp spaces. Let a ∈ BMO and let f ∈ Lp with 1 < p < ∞. The paraproduct Pa is then deﬁned by, hI hI Pa (f ) = a, f, 1I . (1.5.6) |I| |I| I

The operator Pa is thus deﬁned to be the linear extension of the map hI → a,

hI 1I . |I|

Paraproducts appear quite naturally in connection with Figiel’s method considered in the previous section. There we ﬁxed a family of functions {fI } and asked for conditions which imply that the map T : hI → fI extends to a bounded linear operator on Lp . Following Figiel’s approach we demanded that the functions fI satisfy the mean-value conditions (1.4.17) and estimates for their Haar coeﬃcient expressed in (1.4.18). Recall the following expansion which marked the starting point for Figiel’s work: fI = aI 1I +

m

bIm Um (hI ) +

∞ m l=0

d(m,l) (fI ),

(1.5.7)

90

Chapter 1. The Haar System: Basic Facts and Classical Results

where bIm is deﬁned in (1.4.22), d(m,l) (fI ) is deﬁned in (1.4.25) and where

1

aI =

fI dt. 0

For a ﬁnite linear combination g = I xI fI , with xI ∈ R the representation (1.5.7) allows us to estimate the norm of g as follows,

g Lp ≤ xI aI 1I p I L (1.5.8) ∞ I + xI bm Um (hI ) + xI d(m,l) (fI ) . p m

I

m l=0

L

I

Lp

We saw in the proof of Theorem 1.4.3 that (1.4.17) gives a nice upper bound for the second series appearing on the right-hand side of (1.5.8). The hypothesis and (1.4.18) takes care of the third sum in (1.5.8). So, by reviewing the proof of Theorem 1.4.3 we made the following observation. If the family {fI } satisﬁes (1.4.17) and (1.4.18), then T : hI → fI extends boundedly to Lp , provided that

xI aI 1I Lp ≤ Cp

I

xI hI Lp

(1.5.9)

I

where

1

aI =

fI dt. 0

But clearly the estimate (1.5.9) states that the paraproduct Pa is bounded on on– Lp . Simultaneously, the paraproduct is one of the basic operators of Calder´ Zygmund theory, since it plays a crucial role in the proof of the famous T (1) theorem of G. David and J.L. Journ´e [59] Theorem 1.5.2. Let a ∈ BMO, then Pa deﬁnes a bounded operator on Lp , for 1 < p < ∞, and satisﬁes the estimate ||Pa (f )||p ≤ Cp ||a||BMO ||f ||p , for every f ∈ Lp with 1 < p < ∞. The constant Cp can be chosen as follows Cp =

32(p − 1)−1 4p3/2

if 1 < p ≤ 2, if 2 ≤ p < ∞.

(1.5.10)

1.5. Basic operators

91

Proof. Let f ∈ Lp and let g ∈ Lq , where q denotes the H¨ older conjugate index to p. We will show that |Pa (f ), g| ≤ Cp ||a||BMO ||f ||p ||g||q .

(1.5.11)

The essential step towards the proof is the estimate 1 √ |Pa (f ), g| ≤ 2 2||a||BMO M1 (g)(x)S(f )(x)dx,

(1.5.12)

0

where M1 (g) denotes the Hardy–Littlewood maximal function introduced in Section 1.1 in Theorem 1.2.6. We use the notation gI to denote the mean value of g over the interval I, 1 g. gI = |I| I We start with the identity hI a, hI f, gI |I| I

1 hI = a(x) f, gI hI (x) dx. |I| 0

Pa (f ), g =

(1.5.13)

I

To the integral appearing in the last line of (1.5.13) we apply Feﬀerman’s inequality. This gives that √ hI |Pa (f ), g| ≤ 2 2||a||BMO f, gI hI . (1.5.14) 1 |I| I

H

We estimate the H 1 factor by analyzing its square function pointwise. For x ∈ [0, 1] we have that

f,

I

hI 2 2 2 hI gI hI (x) ≤ sup gI2 · f, 2 h2I (x), |I| |I| I x I

where the supremum is taken over all dyadic intervals containing the point x. 1/2 Note that supI x gI2 ≤ M1 (g)(x), where M1 (g) denotes the Hardy–Littlewood maximal function introduced in Theorem 1.2.6. Thus we obtain that

I

hI f, 2 gI2 h2I (x) |I|

1/2 ≤ M1 (g)(x)S(f )(x).

(1.5.15)

Integrating (1.5.15) gives (1.5.12), and by H¨ older’s inequality we obtain that √ |Pa (f ), g| ≤ 2 2||a||BMO ||M1 (g)||q ||S(f )||p .

(1.5.16)

92

Chapter 1. The Haar System: Basic Facts and Classical Results

The factors on the right-hand side of (1.5.16) are easily estimated with the upper bound for the Hardy–Littlewood maximal function, Theorem 1.2.6, and the square function characterization of Theorem 1.1.6. Thus (1.5.16) implies that |Pa (f ), g| ≤ Cp ||a||BMO ||g||q ||f ||p , ∗

(1.5.17)

∗

where Cp = 8p(p − 1) and p = max{p, p/(p − 1)}. Taking the supremum (1.5.17) over all g ∈ Lq with ||g||q ≤ 1 gives that ||Pa (f )||p ≤ Cp ||a||BMO ||f ||p . A more careful use of square function estimates is possible by distinguishing between the cases 1 < p < 2 and 2 < p < ∞. If we use Theorem 1.1.6 for 1 < p < 2, and Theorem 1.3.5 for 2 < p < ∞, then we obtain improved constants Cp as follows, 32(p − 1)−1 if 1 < p < 2, Cp = (1.5.18) 4p3/2 if 2 < p < ∞.

Rearrangement operators and Calder´ on–Zygmund kernels Next we introduce rearrangement operators. In combination with paraproducts they are an important tool in the study of integral operators. Most notably in the analysis of Calder´ on–Zygmund integral operators. In this section we present Figiel’s representation of integral operators, to illustrate this point. For m ∈ Z, we deﬁne the operators Tm and Um by their action on the Haar basis. Let I be a dyadic interval such that I + m|I| ⊆ [0, 1]. Then we put Tm (hI ) = hI+m|I| , Um (hI ) = 1I+m|I| − 1I . For the remaining intervals we simply put Tm (hI ) = Um (hI ) = 0. Deﬁned that way, the operators Tm and Um are bounded operators on Lp when 1 < p < ∞. We will give the precise estimates later. Now we discuss their connection to integral operators in detail. We start with integral operators for which the kernel, a priorily, belongs to the space L2 ([0, 1] × [0, 1]). To analyze the operator we expand the kernel using the following biorthogonal system in L2 ([0, 1] × [0, 1]). ! " hI+m|I| (x) · hI (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D ∪ ! " 1I+m|I| (x) · hI (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D ∪ ! " hI+m|I| (x) · 1I (y) : I + m|I| ⊆ [0, 1], m ∈ Z, I ∈ D .

1.5. Basic operators

93

This system is complete in L2 ([0, 1] × [0, 1]), provided that we augment it with the constant function 1. Now we ﬁx an integral kernel in K ∈ L2 ([0, 1] × [0, 1]). We will now expand K using the basis introduced above. This is done in three steps corresponding to the three groups of basis functions introduced. We ﬁx dyadic intervals I, J, with |I| = |J|, then we deﬁne the Haar coeﬃcients 1 1 K(x, y)hI (x)hJ (y)dxdy, (K, hJ ⊗ hI ) =

0

0

0

1

1

(K, 1J ⊗ hI ) =

K(x, y)hI (x)1J (y)dxdy,

(1.5.19)

0

1

1

(K, hJ ⊗ 1I ) =

K(x, y)1I (x)hJ (y)dxdy. 0

0

Next we introduce three kernels which also correspond to the three groups of the basis. hI+m|I| (x) hI (y) K1 (x, y) = , K, hI+m|I| ⊗ hI |I| |I| m∈Z {I:I+m|I|⊆[0,1]}

K2 (x, y) =

m∈Z {I:I+m|I|⊆[0,1]}

K3 (x, y) =

m∈Z {I:I+m|I|⊆[0,1]}

1I+m|I| (x) hI (y) K, 1I+m|I| ⊗ hI , |I| |I| hI+m|I| (x) 1I (y) . K, hI+m|I| ⊗ 1I |I| |I|

By deﬁnition, our original kernel K can be recovered from K1 , K2 , K3 , as follows, 1 1 K= K(x, y)dxdy + K1 + K2 + K3 . (1.5.20) 0

0

Following is the deﬁnition of three Haar multipliers. Their deﬁning sequences come from the Haar coeﬃcients (1.5.19). We let m ∈ Z and let I be a dyadic interval, then we deﬁne (K, hI+m|I| ⊗ hI ) . |I| For ﬁxed m the sequence (am (I)) deﬁnes the multiplier hI am (I)f, hI . Am (f ) = |I| am (I) =

(1.5.21)

(1.5.22)

{I:I+m|I|⊆[0,1]}

Theorem 1.5.1 asserts that if (am (I)) is bounded, then the Haar multiplier Am is a bounded operator on Lp and satisﬁes

Am (f ) p ≤ (p∗ − 1) sup |am (I)| f p . I

94

Chapter 1. The Haar System: Basic Facts and Classical Results

The Haar multiplier Am corresponds to the kernel K1 . Now we deﬁne the multiplier corresponding to K2 . Let I be a dyadic interval, then we deﬁne bm (I) =

(K, 1I+m|I| ⊗ hI ) . |I|

For ﬁxed m the sequence (bm (I)) deﬁnes the multiplier

Bm (f ) =

bm (I)f,

{I:I+m|I|⊆[0,1]}

hI hI . |I|

By Theorem 1.5.1, the Haar multiplier Bm satisﬁes the norm estimates

Bm (f ) p ≤ (p∗ − 1) sup |bm (I)| f p . I

It remains to deﬁne the Haar multiplier associated to the kernel K3 . Let I be a dyadic interval, then deﬁne the coeﬃcients cm (I) =

(K, hI ⊗ 1I+m|I| ) , |I|

which in turn deﬁne the multiplier Cm ,

Cm (f ) =

cm (I)f,

{I:I+m|I|⊆[0,1]}

hI hI . |I|

Theorem 1.5.1 gives that

Cm (f ) p ≤ (p∗ − 1) sup |cm (I)| f p . I

The above discussion shows that in Lp the operator norms of the multipliers Am , Bm , Cm , are bounded by the quantity |K|m = (p∗ − 1) sup{|am (I)| + |bm (I)| + |cm (I)|}, I

where the supremum is extended over all dyadic intervals I for which I + m|I| is contained in the unit interval. We are now prepared to formulate Figiel’s representation of integral operators. This is a powerful theorem which expresses a given integral operator as a sum of paraproducts, Haar multipliers and rearrangement operators. The classical T (1) theorem of G. David and J.L Journ´e [59] follows easily from Figiel’s representation, thereby showing the importance of these basic operators.

1.5. Basic operators

95

Theorem 1.5.3. Let K ∈ L2 ([0, 1] × [0, 1]). Let R be the integral operator deﬁned by the kernel K, 1 R(f )(x) = K(x, y)f (y)dy. 0

11 Let a = R(1), b = R (1) and put K0 = 0 0 K(x, y)dxdy. Then R admits the representation Tm (Am (f )) + Um (Bm (f )) + (Um Cm )∗ (f ), R(f ) = K0 + Pa (f ) + Pb∗ (f ) + ∗

m∈Z p

and its norm on L satisﬁes the estimate ||R||p ≤ K0 + 8p(p∗ − 1) (||a||BMO + ||b||BMO ) + Cp

log2 (4 + |m|)|K|m .

m∈Z

Here p∗ = max{p, p/(p − 1)}. We formulated Figiel’s representation theorem for integral operators under the initial hypothesis that K ∈ L2 ([0, 1] × [0, 1]). However the L2 norm of the kernel does not appear in the conclusion of Theorem 1.5.3, that is the norm estimate of the integral operator 1

K(x, y)f (y)dy.

R(f )(x) = 0

It is therefore possible to extend the scope of the representation Theorem 1.5.3 to singular integral operators, using simple limiting procedures. Recall from (1.5.20), that the kernel K appearing in Theorem 1.5.3 is decomposed as 1

1

K(x, y)dxdy + K1 + K2 + K3 .

K= 0

0

For i ∈ {1, 2, 3} let Ri denote the integral operator corresponding to the kernel Ki , 1 Ri (f )(x) = Ki (x, y)f (y)dy. 0

The following three propositions give the respective representations for the integral operators Ri , i ∈ {1, 2, 3}. Put together they give Theorem 1.5.3. We begin with the simplest case which is the operator R1 . Proposition 1.5.4. Let f ∈ Lp . Let Am be the multiplier given by (1.5.21) and (1.5.22). Then R1 (f ) = Tm (Am (f )), (1.5.23) m∈Z

and ||R1 ||p ≤

m∈Z

||Tm ||p Am p ,

96

Chapter 1. The Haar System: Basic Facts and Classical Results

where

Am p ≤ (p∗ − 1) sup |am (I)| . I

Proof. Once we have shown the representation (1.5.23), the norm estimate for R1 follows easily. Indeed by (1.5.22), Theorem 1.5.1 implies that ∗ ||Am ||p ≤ (p − 1) sup |am (I)| . I

We turn to the representation of R1 . Note that the action of Tm on the Haar function hI is given by Tm (hI ) = hI+m|I| . The claimed representation for the integral operator R1 we ﬁnd by unwinding the deﬁnitions involved. The ﬁrst identity below follows from the deﬁnition of the kernel K1 , for the second one we use the deﬁnition of the rearrangement Tm , for the last identity we invoke the deﬁnition of the multiplier Am . In summary we ﬁnd that

1

K1 (x, y)f (y)dy = 0

m

=

m

=

hI hI+m|I| (x) |I| |I|

(K, hI+m|I| ⊗ hI )f,

hI Tm (hI )(x) |I| |I|

I

(K, hI+m|I| ⊗ hI )f,

I

Tm (Am (f ))(x).

m

1

Proposition 1.5.5. Let f ∈ Lp . Let a(x) = 0 K(x, y)dy, and let Pa be the paraproduct deﬁned by a. Then R2 has the representation R2 (f ) = Pa (f ) + Um (Bm (f )). m∈Z

Its operator norm admits an upper bound, ||R2 ||p ≤ 8p(p∗ − 1)||a||BMO +

||Um ||p Bm p ,

m∈Z

and

Bm p ≤ (p − 1) sup |bm (I)| . ∗

I

Proof. Here we start by evaluating the integral operator R2 . The deﬁnition of the kernel K2 gives that

1

K2 (x, y)f (y)dy = 0

hI 1I+m|I| (x) K, 1I+m|I| ⊗ hI f, , |I| |I|

m∈Z I

(1.5.24)

1.5. Basic operators

97

where the inner sum is extended over all dyadic intervals I for which I + m|I| ⊆ [0, 1]. Next we focus on the function 1I+m|I| (x) appearing in the above sum. Using the operator Um we rewrite 1I+m|I| (x) as 1I+m|I| (x) = Um (hI )(x) + 1I (x).

(1.5.25)

Inserting the identity (1.5.25) into (1.5.24) deﬁnes a splitting of the operator R2 into two parts. A ﬁrst part which involves the operators Um , and a remaining part which leads to the appearance of the paraproduct Pa . We isolate ﬁrst the paraproduct. By changing the order of summation we obtain the identity

1I hI f, K, 1I+m|I| ⊗ hI |I| |I| m∈Z {I:I+m|I|⊆[0,1]} ⎞ ⎛ 1I hI ⎝ = K, 1I+m|I| ⊗ hI ⎠ f, . |I| |I| {m:I+m|I|⊆[0,1]}

I

We continue by identifying the sum appearing inside the brackets. We will show 1 that it is just the Haar coeﬃcient a, hI , where a(x) = 0 K(x, y)dy. We begin by evaluating

K, 1I+m|I| ⊗ hI =

{m:I+m|I|⊆[0,1]}

1

1

K(x, y)hI (x)dxdy. 0

(1.5.26)

0

Next observe that the 1double integral on the right-hand side is a Haar coeﬃcient, namely with a(x) = 0 K(x, y)dy,

1

1

K(x, y)hI (x)dxdy = a, hI . 0

(1.5.27)

0

Recall next the deﬁnition of the paraproduct operator Pa . It is given by the equation hI 1I (x) . a, hI f, Pa (f )(x) = |I| |I| I

Combining (1.5.26) with (1.5.27), and inserting the value of a, hI we arrived at the following identity for the paraproduct, Pa (f )(x) =

m∈Z {I:I+m|I|⊆[0,1]}

hI 1I K, 1I+m|I| ⊗ hI f, . |I| |I|

(1.5.28)

98

Chapter 1. The Haar System: Basic Facts and Classical Results Next we unwind the deﬁnition of the multipliers Bm , and thus we ﬁnd that

Um (Bm (f ))

m∈Z

=

Um

I

m∈Z

hI Bm (f ), hI |I|

hI Um (hI ) , K, 1I+m|I| ⊗ hI f, = |I| |I| m∈Z I

where we extended the inner sum over all dyadic intervals I such that I + m|I| ⊆ [0, 1]. It remains to combine the last identity with (1.5.24), (1.5.25) and (1.5.28). This gives that

1

K2 (x, y)f (y)dy = Pa (f )(x) + 0

Um (Bm (f ))(x),

m∈Z

which is the claimed representation for R2 . The resulting upper bound for the norm comes from Theorem 1.5.2 that controls the norm of the paraproduct, ||Pa ||p ≤ 8p(p∗ − 1)||a||BMO , and Theorem 1.5.1 that gives norm estimates of the multiplier Bm . Hence ||Bm ||p ≤ (p − 1) sup |bm (I)| . ∗

I

Following is the representation of R3 . Up to transposition and duality the the proof below is the same as the preceding proof of Proposition 1.5.5. 1 Proposition 1.5.6. Let g ∈ Lp . Let b(y) = 0 K(x, y)dx, and let Pb be the paraproduct deﬁned by b. Then the adjoint operator to R3 has the representation R3∗ (g) = Pb (g) +

Um (Cm (g)),

m∈Z

and ||R3 || ≤ 8p(p∗ − 1)||b||BMO +

||Um ||p Cm p ,

m∈Z

where

||Cm ||p ≤ (p − 1) sup |cm (I)| . ∗

I

1.5. Basic operators

99

Proof. Fix f ∈ Lp , and g ∈ Lq . Then we evaluate R3 (f ), g. We invoke the deﬁnition of the kernel K3 , then we change the order of summation and relabel, 1 1 K3 (x, y)f (y)g(x)dxdy 0

0

hI+m|I| 1I (y) f, K, hI+m|I| ⊗ 1I g, = |I| |I| m I 1J+n|J| hJ = f, . K, hJ ⊗ 1J+n|J| g, |J| |J| n J

Next we look carefully at the last factor in the above sum. That is f, rewrite it using the operator Un as f,

1J+n|J| |J| .

We

1J+n|J| Un (hJ ) 1J = f, + f, . |J| |J| |J|

Now the proof proceeds parallel to that of Proposition 1.5.5. We identify the 1 paraproduct ﬁrst. Recall that we put b(y) = 0 K(x, y)dx. One ﬁnds as in the proof of Proposition 1.5.5 that K, hJ ⊗ 1J+n|J| , b, hJ = n

where we extend the sum over those n ∈ Z for which J + n|J| ⊆ [0, 1]. The paraproduct Pb (g) is deﬁned by hJ 1J . b, hJ g, Pb (g) = |J| |J| J

Consequently, f, Pb (g) =

hJ 1J K, hJ ⊗ 1J+n|J| g, f, . |J| |J| n J

Next we recall the deﬁnition of the multiplier Cn , and note that f, Un (Cn (g)) n

=

hJ Un (hJ ) K, hJ ⊗ 1J+n|J| g, f, . |J| |J| n J

It follows that f, R3∗ (g) = R3 (f ), g = f, Pb (g) +

f, Un (Cn (g)),

n∈Z

which proves the representation of the operator R3 . The norm estimates are obtained as in the proof of Propositions 1.5.4 and 1.5.5.

100

Chapter 1. The Haar System: Basic Facts and Classical Results

Proof of Theorem 1.5.3. We apply Propositions 1.5.4–1.5.6, and use the following upper bound for the norms of the operators Tm and Um in Lp . ||Tm ||p ≤ Cp log2 (4 + |m|), ||Um ||p ≤ Cp log2 (4 + |m|). These estimates will be proven in Chapter 3, Theorem 3.3.9. With this remark the proof of Theorem 1.5.3 is complete. The Hilbert transform is our ﬁrst example illustrating Theorem 1.5.3. For x, y ∈ [0, 1] and x = y we deﬁne H(x, y) = cot Let > 0 and put

H (x, y) =

H(x, y) 0

(x − y) . π if |x − y| > , otherwise.

The kernel H is a bounded function deﬁned on [0, 1] × [0, 1]. It gives rise to an integral operator 1 R f (x) = H (x, y)f (y)dy. 0

Applying Theorem 1.5.3 we will show that the integral operator R is bounded in Lp (1 0. Corollary 1.5.7. Let 1 < p < ∞. The Hilbert transform deﬁned as the limit R(f ) = lim R (f ) →0

p

is a bounded operator on L . Proof. We will show that for f ∈ Lp ([0, 1]), ||R f ||p ≤ Ap ||f ||p ,

(1.5.29)

p

and that lim→0 R (f ) exists in L . To prepare for the application of Theorem 1.5.3 we recall that the kernel H belongs to L2 ([0, 1] × [0, 1]). Furthermore 1 1 H (x, y)dy = 0 and H (x, y)dx = 0. 0

0

Consequently, the paraproducts in Figiel’s representation of R are vanishing operators. Using integration by parts we easily obtain that there exists A > 0 so that for m ∈ Z, the following estimates hold, |(H , hI ⊗ hI+m|I| )| ≤ A(1 + |m|)−2 |I|, |(H , 1I ⊗ hI+m|I| )| ≤ A(1 + |m|)−2 |I|, |(H , hI ⊗ 1I+m|I| )| ≤ A(1 + |m|)−2 |I|.

1.5. Basic operators

101

The constant A does not depend on I, m or > 0. Hence Theorem 1.5.3 gives (1 + |m|)−2 log2 (4 + |m|) ||R ||p ≤ Cp A m∈Z

≤ Ap . In this upper bound the L2 norm of the kernel does not appear. In particular, the norm of R in Lp is bounded independent of . Next we show convergence of R (f ) when f has bounded derivative. As H (x, y)dy = 0, we may insert the value f (x) in the integral deﬁning R f (x). Hence 1

H (x, y){f (y) − f (x)}dy.

R f (x) = 0

Fix 0 < δ < , and estimate the diﬀerence 1 |Rδ f (x) − R f (x)| = {Hδ (x, y) − H (x, y)}{f (y) − f (x)}dy 0 cot (x − y) · |f (y) − f (x)|dy. ≤ π δ≤|x−y|≤ Recall that f is assumed to have bounded derivative, hence |f (y) − f (x)| ≤ ||f ||∞ |x − y|. Furthermore, the product |(x − y) cot(x − y)| is bounded. It follows that |Rδ f (x) − R f (x)| ≤ C||f ||∞ ( − δ). Summing up, R f is a Cauchy sequence in Lp , when f has bounded derivative. By (1.5.29) and the Banach–Steinhaus theorem, the limit deﬁned by R(f ) = lim R (f ) →0

p

induces a bounded linear operator on L .

Now we apply Theorem 1.5.3 to a wide range of singular kernels, and prove a version of the T (1) theorem of G. David and J. Journ´e [59]. A kernel K(x, y) deﬁned for x = y is called a Calder´on–Zygmund kernel if there exists δ > 0 and C > 0 such that for x = y the following estimates hold, |K(x, y)| ≤ C|x − y|−1 , |∂x K(x, y)| + |∂y K(x, y)| ≤ C|x − y|−1−δ , K(x, y) = −K(y, x). We will now determine when a Calder´ on–Zygmund kernel is associated to a bounded linear operator T : Lp → Lp , in the following sense: 1 1 K(x, y)f (y)g(x)dxdy = T (f ), g, 0

0

102

Chapter 1. The Haar System: Basic Facts and Classical Results

whenever f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅. Fix a Calder´ on–Zygmund kernel K. We introduce now a canonical approximation procedure for K. For a dyadic interval I deﬁne the following collection of dyadic intervals, AI = {J : |J| = |I|, dist(I, J) ≥ |I|}. Fix n ∈ N. We deﬁne the approximating kernel K (n) by setting K (n) (x, y) = K(x, y)1I (x)1J (y). {I:|I|=2−n } J∈AI

Note that the approximation is obtained by deleting a thin strip around the diagonal x = y, hence K (n) ∈ L2 ([0, 1] × [0, 1]). By the deﬁning properties for the Calder´ on–Zygmund kernel K there exists a constant Aδ > 0 such that for every n ∈ N the following estimates hold, (n) K ≤ Aδ |m|−1−δ . m

The L2 kernel K (n) deﬁnes the integral operator R

(n)

1

K (n) (x, y)f (y)dy.

(f )(x) = 0

Let an = R(n) (1), and bn = (R(n) )∗ (1). Theorem 1.5.3 gives the following estimates for the operator norm of R(n) : Lp → Lp . Aδ (log2 (4 + |m|))|m|−1−δ ||R(n) ||p ≤ Cp (||an ||BMO + ||bn ||BMO ) + Cp m∈Z

≤ Cp (||an ||BMO + ||bn ||BMO ) + Ap,δ . We point out that the constants Ap,δ appearing in the norm estimate for R(n) are actually independent of n. Theorem 1.5.8. If supn ||an ||BMO < ∞ and supn ||bn ||BMO < ∞, then there exists a bounded linear operator T : Lp → Lp , 1 < p < ∞ such that 1 1 K(x, y)f (y)g(x)dxdy, T (f ), g = 0

0

whenever f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅. Proof. First observe that we deﬁned the approximating kernels by truncating so that for x = y, K(x, y) = lim K (n) (x, y). n→∞

As supn ||an ||BMO < ∞ there exists a subsequence of (an ) which converges, weak∗ in BMO. Denote it again by (an ). Let f have bounded derivative. Let g ∈ Lq . We

1.5. Basic operators

103

show next that (g, R(n) (f )) is a Cauchy sequence. Let n ∈ N and let m ≥ n. Then g, (R(n) − R(m) )(f ) 1 1 (K (n) (x, y) − K (m) (x, y))f (y)g(x)dxdy =

0

0

1

= 0

1

K (n) (x, y) − K (m) (x, y) (f (y) − f (x))g(x)dxdy

0

+ an − am , (gf ). The deﬁnition of the approximating kernels give that |(K (n) (x, y) − K (m) (x, y))| · |f (y) − f (x)| ≤ C|K (n) (x, y) − K (m) (x, y)| · |x − y| ≤ C. Moreover supp {(K (n) −K (m) )} ⊆ {(x, y) : |x−y| ≤ 2−n }. Thus for each x ∈ [0, 1], 1 |(K (n) (x, y) − K (m) (x, y))(f (y) − f (x))|dy ≤ C2−n . 0

It follows that g, (R(n) − R(m) )(f ) is a Cauchy sequence, and for a Lipschitz function f, the following limit exists in Lp , lim R(n) f.

n→∞

By Theorem 1.5.3 and the hypothesis we ﬁnd sup ||R(n) ||p ≤ sup Cp (||an ||BMO + ||bn ||BMO ) + Ap, δ n

n

< ∞. As Lipschitz functions are dense in Lp , the theorem of Banach and Steinhaus implies the existence of a bounded linear operator T : Lp → Lp , such that T (f ) = lim R(n) (f ), n→∞

p

where the limit exists in L ; moreover, ||T ||p ≤ sup ||R(n) ||p . n

Finally note that if supp (f ) ∩ supp (g) = ∅, then dist( supp (f ), supp (g)) > 0. Hence the identity K(x, y) = lim K (n) (x, y), n→∞

holding for x = y, implies that 1 1 K(x, y)f (y)g(x)dxdy = lim 0

0

n→∞

0

1

1

K (n) (x, y)f (y)g(x)dxdy, 0

104

Chapter 1. The Haar System: Basic Facts and Classical Results

provided that f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅. Summing up we found a bounded linear operator T : Lp → Lp , such that, T (f ), g = lim R(n) (f ), g n→∞ 1 1 = lim K (n) (x, y)f (y)g(x)dxdy n→∞

1

0 1

0

K(x, y)f (y)g(x)dxdy,

= 0

0

whenever f ∈ Lp ([0, 1]), g ∈ Lq ([0, 1]) and supp (f ) ∩ supp (g) = ∅.

Orthogonal projections We introduce now a class of orthogonal projections which plays a very important role in this book. We will use them for three main reasons. First we construct several families of complemented subspaces in H 1 and BMO. Second their boundedness properties will be crucial in ﬁnding isomorphic invariants of H 1 , such as the uniform approximation property, the dimension conjecture or dichotomies for complemented subspaces. Third, orthogonal projections yield canonical examples of operators by which we will test (and refute) several natural conjectures. From now on it is convenient to use systematically the following notation. Let H be a collection of dyadic intervals. Let f = aI hI . We deﬁne S(f |H) by the equation S 2 (f | H) =

a2I h2I ,

I∈H

and call it the square function of f restricted to H. If J is a dyadic interval and if we put Q(J) = {I : I ⊆ J}, then S 2 (f | Q(J))dt = a2I |I|. I⊆J

Hence

f 2BMO

= sup J

S 2 (f | Q(J))

dt . |J|

We let C1 , . . . , CN be disjoint collections of dyadic intervals. We form linear combinations of Haar functions by putting bi = {hI : I ∈ Ci }, i ≤ N, (1.5.30) and we normalize in L2 , ki = bi /||bi ||2 .

(1.5.31)

1.5. Basic operators

105

The orthogonal projection onto span{bi }N i=1 is given by Pf =

N

f, ki ki .

(1.5.32)

i=1

Clearly, by the disjointness of the collections C1 , . . . , CN the operator P is of norm 1 in L2 ([0, 1]). Now we introduce a condition on C1 , . . . , CN which implies that the operator norm of P on BMO is bounded independent of N . We let Ci denote the point-set covered by the collection Ci , that is, we let Ci =

I.

I∈Ci

Disjoint collections of dyadic intervals C1 , . . . , CN are said to satisfy Jones’s compatibility condition (J) if the following conditions (J 1)–(J 4) are satisﬁed. (J 1) Each Ci is a collection of pairwise disjoint dyadic intervals. (J 2) There exists a constant B, so that for each J ∈ Cj and each i ≥ j the following estimate holds, |I| |Cj ∩ Ci | ≤B . |J| |Cj | I∈Ci ,I⊆J

(J 3) The collections {Ci : 1 ≤ i ≤ N } are ordered in the following sense: If there exists I ∈ Ci and J ∈ Cj , such that I ⊆ J, then i ≥ j. Conversely, if i ≥ j, if I ∈ Ci and J ∈ Cj , such that I ∩ J = ∅, then I ⊆ J. (J 4) If j ≤ i and if Cj ∩ Ci = ∅, then Ci ⊆ Cj . In particular the sets {Ci : 1 ≤ i ≤ N } are nested; if Cj ∩ Ci = ∅, then either Ci ⊆ Cj or Cj ⊆ Ci . If the compatibility condition (J) is satisﬁed we will show that the BMO norm of the orthogonal projection deﬁned by (1.5.32) is bounded independent of N. √ Precisely we prove below that the norm of P in BMO is bounded by B, where B is the constant appearing in condition (J 2). Following is a basic criterion for the boundedness of orthogonal projections in BMO. Theorem 1.5.9. If C1 , . . . , CN are pairwise disjoint collections of dyadic intervals satisfying Jones’s compatibility condition (J), then, on BMO, the orthogonal projection N f, ki ki Pf = has norm bounded by

√

i=1

B.

106

Chapter 1. The Haar System: Basic Facts and Classical Results

Proof. Let f ∈ BMO with Haar expansion f = cI hI . We expand P f in its Haar series Pf = αI hI , #N where αI = f, ki |Ci |−1/2 , for I ∈ Ci , and where αI = 0 for I ∈ i=1 Ci . Now we ﬁx k ≤ N and a dyadic interval J0 ∈ Ck . Then the following identity holds, 1 1 2 αI |I|. S 2 (P f | Q(J0 )) = |J0 | |J0 | I⊆J0

We will now show that the following estimates holds, 1 2 αI |I| ≤ B||f ||2BMO . |J0 | I⊆J0

By condition (J 3), if I ⊆ J0 and if I ∈ Ci , then i ≥ k. Therefore by changing the order of summation we have the identity I⊆J0

αI2 |I| =

N

f, ki 2

i=k

I∈Ci ,I⊆J0

|I| . |Ci |

(1.5.33)

We will now estimate the right-hand side of (1.5.33) using the conditions (J 1)–(J 4). Recall that condition (J 2) states that for J ∈ Ck and i ≥ k, the following estimate holds, |I| |Ci ∩ Ck | ≤B . (1.5.34) |J0 | |Ck | I∈Ci ,I⊆J0

Thus conditions (J 2) and (J 3) give the following upper bound for E, 1 |J0 |

|Ci ∩ Ck | B . f, ki 2 |Ck | |Ci | N

S 2 (P f | Q(J0 )) ≤

(1.5.35)

i=k

Now we use condition (J 4) to continue. It follows from (J 4) that if Ci ∩ Ck = ∅ and i ≥ k, then Ci ⊆ Ck , and |Ci ∩ Ck | = |Ci |. Moreover if i ≥ k and Ci is not contained in Ck , then Ci ∩ Ck = ∅. Hence the estimate (1.5.35) implies 1 B S 2 (P f | Q(J0 )) ≤ f, ki 2 . (1.5.36) |J0 | |Ck | {i:Ci ⊆Ck }

1.5. Basic operators

107

Next we use the converse part of condition (J 3), Bessel’s inequality and the deﬁnition of BMO to obtain f, ki 2 ≤ c2K |K| {i:Ci ⊆Ck }

J∈Ck K⊆J

≤ ||f ||2BMO

|J|.

(1.5.37)

J∈Ck

Finally recall condition (J 1) that Ck is a collection of pairwise disjoint dyadic intervals. Thus |J| = |Ck |. (1.5.38) J∈Ck

Inserting identity (1.5.38) and the estimate (1.5.37) into (1.5.36) gives 1 S 2 (P f | Q(J0 )) ≤ B||f ||2BMO , |J0 |

as claimed.

By duality the orthogonal projection P is bounded on H 1 if and only if it is bounded on BMO. Theorem 1.5.10. Let C1 , . . . , CN be collections of dyadic intervals satisfying Jones’s compatibility condition (J). Let ki be deﬁned by (1.5.31). Then the orthogonal projection N Pf = f, ki ki i=1

√ deﬁnes a bounded operator on H with norm ≤ 8 B. 1

Proof. We use Feﬀerman’s duality theorem, the orthogonality of P and Theorem 1.5.9. This gives the following bound for the norm of P, √ ||P f ||H 1 ≤ 2 2 sup{P f, g : ||g||BMO ≤ 1} √ = 2 2 sup{f, P g : ||g||BMO ≤ 1} √ √ ≤ 2 2 B sup{f, h : ||h||BMO ≤ 1} √ ≤ 8 B||f ||H 1 . The glueing process. Below we need to be able to reduce the rank of a given orthogonal projection. For this purpose we glue together families of orthogonal projections acting locally, into one large orthogonal projection acting globally. We describe now the glueing process in detail, and study boundedness properties of orthogonal projections obtained by this method. We recall ﬁrst the notion of a block of intervals. It was introduced in Section 1.2. Let K be a collection of dyadic intervals, and let B ⊆ K. We say that B is a block of intervals in K, if the following conditions are satisﬁed.

108

Chapter 1. The Haar System: Basic Facts and Classical Results

1. B contains exactly one maximal interval, say I. 2. For every J ∈ B and K ∈ K the following “connectedness” condition holds true: If J ⊆ K ⊆ I, then K ∈ B. Given a block of intervals B we will systematically use the notation B = B(I) to indicate that I is the unique maximal interval of the block B. Assume now that {B(I) : I ∈ A} is a collection of disjoint blocks in K. Fix N, B ∈ N. Assume that for each I ∈ A the block B(I) contains collections of dyadic intervals, Ci (I), i ≤ N so that C1 (I), . . . , CN (I) satisfy Jones’s compatibility condition (J).

(1.5.39)

Let the constant appearing in the compatibility condition be the ﬁxed number B ∈ N. For i ≤ N deﬁne {hK : K ∈ Ci (I)}, (1.5.40) bi = I∈A

and let P denote the orthogonal projection onto span{bi }, that is, P (f ) =

N

f,

i=1

bi bi . ||bi ||2 ||bi ||2

(1.5.41)

Applying Theorem√1.5.9 to the projection deﬁned by (1.5.41) gives an upper bound for P BMO by B × (the cardinality of A). For the applications to follow we now isolate a further structural condition ensuring that the norm of P is bounded by a function depending only on B and [[A]], where 1 |I| I∈A

[[A]] = sup

|J|.

J∈A, J⊆I

This is the purpose of Theorem 1.5.11. For m ≤ N and J ∈ A let Cm (J) = K. K∈Cm (J)

Loosely speaking the hypothesis of Theorem 1.5.11 states that for I, J ∈ A the collections {Cm (J) : m ≤ N } and {Cm (I) : m ≤ N } are rescaled versions of each other. Precisely, assume that {Cm (J) : m ≤ N } and {Cm (I) : m ≤ N } are linked through the following conditions: For I, J ∈ A and m, n ≤ N,

and

Cm (I) ⊆ Cn (I) if Cm (J) ⊆ Cn (J),

(1.5.42)

|Cm (I)| |Cm (J)| ≤2 . |Cn (I)| |Cn (J)|

(1.5.43)

1.5. Basic operators

109

Remarks. 1. Note that condition (1.5.42) states that for I, J ∈ A, the intersection pattern of {Cm (J) : m ≤ N } coincides with that of {Cm (I) : m ≤ N }. 2. Below we will apply also the following averaged version of condition (1.5.43). Multiply both sides of (1.5.43) by |Cn (J)|, and sum the resulting estimate over J ∈ A. This gives that . / |Cm (I)| ≤2 |Cn (J)| |Cm (J)|. |Cn (I)| J∈A

Dividing further by For every I ∈ A,

J∈A

J∈A

|Cn (J)| yields the averaged version of condition (1.5.43): |Cm (J)| |Cm (I)| ≤ 2 J∈A . |Cn (I)| J∈A |Cn (J)|

Theorem 1.5.11. Assume that (1.5.39) and the rescaling conditions (1.5.42) and (1.5.43) are satisﬁed. Then for the projection P, deﬁned by (1.5.41) the following estimates hold: Let I0 ∈ A, and J0 ∈ B(I0 ), then (1.5.44) S 2 (P (f )|L) ≤ 2B f 2BMO |J0 |, where L = {J ∈ B(I0 ) : J ⊆ J0 }. Consequently, 1

||P ||BMO ≤ (2B[[A]]) 2 .

(1.5.45)

Proof. The proof is divided into two separate parts. First we show how the norm estimate for P follows from (1.5.44). The second part consists in proving (1.5.44). The norm estimate (1.5.45) follows from (1.5.44). Let f ∈ BMO. Let H be the Haar support of P (f ). For J0 ∈ H there exists I0 ∈ A, such that J0 ∈ B(I0 ). Next observe that the following inclusion holds, {J ∈ H : J ⊆ J0 } ⊆ {J ∈ B(I0 ) : J ⊆ J0 } ∪ B(I). I∈A, I⊂I0

Introducing the following abbreviations simpliﬁes considerably our notation. J0 ∩ H = {J ∈ H : J ⊆ J0 }, J0 ∩ B(I0 ) = {J ∈ B(I0 ) : J ⊆ J0 }. Now we rewrite the square function S 2 (P (f )|J0 ∩ H) as S 2 (P (f )|J0 ∩ H) = S 2 (P (f )|J0 ∩ B(I0 )) +

I∈A, I⊂I0

S 2 (P (f )|B(I)).

110

Chapter 1. The Haar System: Basic Facts and Classical Results

We conclude that the norm estimate (1.5.45) for P follows from the inequalities S 2 (P (f )|J0 ∩ B(I0 )) ≤ 2||f ||2BMO B|J0 |, S 2 (P (f )|B(I)) ≤ 2||f ||2BMO B|I|. In order to prove (1.5.44), it suﬃces to verify the ﬁrst of the above estimates; once this is done we specialize to I0 = J0 and observe that I0 was an arbitrary interval from A. Veriﬁcation of (1.5.44). Recall that H is the Haar support of P (f ). Restricted to the block B(I0 ) the Haar support of P (f ) can be written as B(I0 ) ∩ H = C1 (I0 ) ∪ · · · ∪ CN (I0 ). Hence there exists m0 ≤ N such that J0 ∈ Cm0 (I0 ). Now deﬁne the set of integers Um0 = {m : Cm (I0 ) ⊆ Cm0 (I0 )}. By the hypothesis (1.5.42) the set Um0 does not depend on the choice of the interval I0 . Indeed, if for m0 and m we have that Cm (I0 ) ⊆ Cm0 (I0 ), then for any other I ∈ A the corresponding inclusion holds, namely, Cm (I) ⊆ Cm0 (I). (For Um0 to be independent of I0 , we used the hypothesis that the intersection pattern of {Cm (I) : m ≤ N } is independent of I.) Now we rewrite, using the index set Um0 , S 2 (P (f )|J0 ∩ B(I0 )) =

m∈Um0

f

bm 2 S 2 (bm |J0 ∩ B(I0 )) . ||bm ||2 ||bm ||22

(1.5.46)

Fix m ∈ Um0 , and focus on the factor S 2 (bm |J0 ∩ B(I0 )). We claim that for m ∈ Um0 , the following uniform estimate holds, 2 ||bm ||22 S (bm |J0 ∩ B(I0 )) ≤ 2B . (1.5.47) |J0 | ||bm0 ||22 Observe that the right-hand side of (1.5.47) depends on m0 and m only. In particular it is independent of I0 ∈ A and J0 ∈ B(I0 ). We turn to verifying (1.5.47). Observe that (1.5.40), the deﬁnition of bm , gives |K|. (1.5.48) S 2 (bm |J0 ∩ B(I0 )) = K∈J0 ∩Cm (I0 )

Next we exploit that Jones’s compatibility condition holds for {Cm (I0 ) : m ≤ N }. Condition (J2) states that K∈J0 ∩Cm (I0 )

|Cm (I0 )| |K| ≤B . |J0 | |Cm0 (I0 )|

(1.5.49)

1.5. Basic operators

111

Note that the upper bound on the right-hand side of (1.5.49) is (now) independent of J0 . Next we remove also the dependence on the particular interval I0 . By the rescaling hypothesis (1.5.43) the value of the ratios |Cm (I)|/|Cm0 (I)| is independent of I ∈ A. Hence by averaging we obtain the inequality |Cm (I)| |Cm (I0 )| ≤ 2 I∈A . (1.5.50) |Cm0 (I0 )| I∈A |Cm0 (I)| The identity (1.5.48) followed by the estimates (1.5.49) and (1.5.50) proves the claim made in (1.5.47) since, |Cm (I)|. ||bm ||22 = I∈A

In summary, using locally the compatibility assumption (J) together with the rescaling hypotheses (1.5.42) and (1.5.43) we passed from a small scale quantity to a large scale estimate; we showed that integrating (1.5.46) and invoking (1.5.47) gives that 2 S (P (f )|J0 ∩ B(I0 )) 2B bm 2 ≤ f, . (1.5.51) 2 |J0 | ||bm0 ||2 ||bm ||2 m∈Um0

Finally we use Bessel’s inequality, and the fact that index set Um0 is actually independent of I0 to conclude that bm 2 f, ≤ |Cm0 (I)| · ||f ||2BMO ||bm ||2 (1.5.52) m∈Um0 I∈A = ||bm0 ||22 ||f ||2BMO . Combining (1.5.51) and (1.5.52) implies (1.5.44).

The pigeon hole principle. Often we use a very simple pigeon hole principle to realize the hypothesis of Theorem 1.5.11. As before we are given disjoint blocks of dyadic intervals, {B(I) : I ∈ A}. Let B(I) contain collections of dyadic intervals C1 (I), . . . , CN (I), which satisfy Jones’s compatibility condition (J). Moreover, we assume that the number of families N, and the constant B appearing in the compatibility condition remain ﬁxed, as I ranges over A. We let Ci (I) = J. J∈Ci (I)

Now we assume that a lower bound holds for the density of Ci (I) in I : There exists δ > 0 such that |Ci (I)| > δ|I|,

for i ≤ N and I ∈ A.

(1.5.53)

112

Chapter 1. The Haar System: Basic Facts and Classical Results

Then, by the pigeon hole principle, there exists K = K(δ) and a decomposition of A, A = A 1 ∪ · · · ∪ AK , so that each Ak satisﬁes the hypothesis of Theorem 1.5.11. That is, if I, J ∈ Ak and m, n ≤ N, then Cm (I) ⊆ Cn (I) if Cm (J) ⊆ Cn (J), and

|Cm (J)| |Cm (I)| ≤2 . |Cn (I)| |Cn (J)|

Let k ≤ K. Deﬁne Pk to be the orthogonal projection onto the span of bi = {hK : K ∈ Ci (I)}, i ≤ N. I∈Ak

Then, by Theorem 1.5.11 each of the projections Pk has norm on BMO bounded by 4(B[[Ak ]])1/2 . The sum satisﬁes the norm estimate K 1 Pk ≤ 4K(B[[A]]) 2 , (1.5.54) k=1

BMO

where A = A1 ∪ · · · ∪ AK , and where K = K(δ).

1.6

Notes

The work of A. Haar [79] and J. Schauder [185] is represented by Proposition 1.1.1 and Theorem 1.1.2. See [76] by C. Goﬀman and G. Pedrick for a presentation of the basic properties of Schauder bases, in particular for a proof of St. Banach’s theorem mentioned in the text. Theorem 1.1.3 is the result of S. Szarek [200]. The text presents the method introduced by R. Latala and K. Oleszkiewicz in [124], with simpliﬁcations given in [180]. The proof of Theorem 1.1.4 is taken from the appendix D of [192] by E. M. Stein which also contains the trick by which we transferred Khintchine’s inequality from L4 to L1 . Proofs of Burkholder’s Theorem 1.1.5 are contained in [33] and [35]. Theorem 1.1.6, the square function characterization of Lp , is originally due to R.E.A.C. Paley [167]. The constants of equivalence in Paley’s proof are diﬀerent from the ones obtained in Theorem 1.1.6. Applying the unconditionality of the Haar system in Lp (1 < p < ∞) we obtain estimates for the Walsh system. The proof given for Pisier’s inequality in the scalar case (that is Theorem 1.1.7 referred to as the dyadic version of Poincar´e’s inequality) is due to A. Naor and G. Schechtman [162]. It extends without changes to UMD valued Lp spaces. For a diﬀerent proof see M. Talagrand [201]. Theorem 1.1.8, asserting that the Walsh system in Walsh–Paley order forms a Schauder

1.6. Notes

113

basis for Lp (1 < p < ∞) is due to R. E. A. C. Paley [167]. Theorem 1.1.9, the extension of Khintchine’s inequality to Walsh functions of ﬁnite multiplicity, is obtained by A. Bonami [15] and by K. Kiener [112] independently. Theorem 1.1.10 is a result of B. Roider. The proof in the text is based on K. Kiener’s work in [113] and on additional, unpublished manuscripts of K. Kiener. In [25] J. Bourgain gives an extension of B. Roider’s Theorem 1.1.10. For a systematic treatment of the Walsh system consult the monograph [77] by B. Golubov, A. Eﬁmov and V. Skvortsov. There, in particular, one ﬁnds the identities of Walsh and Paley used in the text. The Walsh system is used as a model providing insight into problems concerning the trigonometric system. The work of R. Hunt in [87] and [88] that analyzes the Walsh model is an excellent preparation for the study of L. Carleson’s treatise [44] on almost everywhere convergence of Fourier series. In [203] Ch. Thiele forms the notion of a quartile operator in the context of the Walsh system. Subsequently quartile operators played a crucial role in the work of M. Lacey and Ch. Thiele [121], [122] on the boundedness of the bilinear Hilbert transform. In Banach space theory Walsh functions are used for the dual purpose of constructing operators (Walsh matrices) and spaces (spanned by subsets of the Walsh system). Typical applications include: (a) The construction of operators with unexpected or extremal properties in the work of P. Enﬂo [62], A. Szankowski [197], [199], A. Pelczy` nski and C. Sch¨ utt [173], C. Sch¨ utt [189], V.I. Gurarii, M. Kadec and V.I. Macaev [78]. (b) The construction of complemented subspaces in Lp (1 < p < ∞) by J. Bourgain, H. Rosenthal and G. Schechtman [29], and subspaces in L1 with extremal properties obtained by J. Bourgain [16]. (c) The proof of the existence of a fundamental, total and bounded biorthogonal sequence in every separable Banach space by R. Ovsepian and A. Pelczy` nski [166], [170]. (d) The study of type, cotype and K-convexity in the work of G. Pisier [177], [178]. The proof of Theorem 1.2.1 (Feﬀerman’s Inequality) is taken from the classic paper [64] by C. Feﬀerman and E. M. Stein. Theorem 1.2.2 is probably a folk result, Theorem 1.2.3 is from A.M. Garsia [75]. The direct approach to the atomic decomposition of H 1 is due to R. Coifman [55]. The proof of Theorem 1.2.4 given in the text is due to S. Janson and P.W. Jones [90]. The sharp function characterization of Theorem 1.2.5 is due to C. Feﬀerman and E. M. Stein [64]. The proof in the text follows A.M. Garsia [75]. Theorem 1.2.6 is the dyadic version of the Hardy–Littlewood maximal theorem, for its proof we used the presentation in [192]. For Theorem 1.2.8 see again A.M. Garsia [75]. Large deviation inequalities as considered in Section 1.3 originate with the work of W. Hoeﬀding [85] and K. Azuma [5]. The proof of Theorem 1.3.1 and

114

Chapter 1. The Haar System: Basic Facts and Classical Results

Corollary 1.3.2 are taken from [49] by S. Y. A. Chang, M. Wilson and Th. Wolﬀ. The proof of Theorem 1.3.4 based on Proposition 1.3.3 was compiled using [6] by R. Banuelos together with [49] by S. Y. A. Chang, M. Wilson and Th. Wolﬀ and [7] by R. Banuelos and C. Moore. A diﬀerent approach to Theorem 1.3.4 was developed by J. Pipher [174]. The proof of Theorem 1.3.5 and Corollary 1.3.7 are from Garsia’s book [75]. Theorem 1.3.6 is due to G. Schechtman [186]. Its proof introduces martingale methods, in particular SL∞ estimates, to obtain Euclidean subspaces of proportional dimension in 1n . For an overview of the development that followed, we refer to the survey articles by G. Schechtman [187] and W. B. Johnson and G. Schechtman [97] as well as to the books by V. Milman and G. Schechtman [148] and G. Pisier [179] . Historically the ﬁrst application of large deviation inequalities to a problem in Banach spaces appears in the note [141] by B. Maurey. T. Figiel, J. Lindenstrauss and V. Milman [68], B. Kashin [110], and G. Schechtman [186] present distinct solutions to the problem of ﬁnding Euclidean subspaces of proportional dimension in 1n . Multipliers into SL∞ as presented in Theorem 1.3.8 are constructed in [104]. The monograph [148] by V. Milman and G. Schechtman contains far-reaching applications to Banach space theory of large deviation inequalities such as Theorem 1.3.1. In a completely diﬀerent direction N. Makarov [133] applies Theorem 1.3.1 to study the boundary behavior of conformal maps, thereby N. Makarov obtains important estimates of harmonic measure. Recently, J. Bourgain, H. Brezis and P. Mironescu [26], obtained surprising, new applications of large deviation in√ equalities and the p dependence in Theorem 1.3.4 to embedding theorems of fractional Sobolev spaces. See also the survey article [30] by H. Brezis. Theorem 1.3.1 provides the basis for an algorithm that determines the rectiﬁable pieces of a given data set in Rn . Thus the work of G. Lerman [126] — that builds on the travelling salesman theorem of P. Jones [103] — gives unexpected applications of SL∞ estimates to geometric measure theory. The theorem of J. Garnett and P. W. Jones [73] is presented with a stochastic proof in Durrett’s book [61]. Both the original proof in [73] and the stochastic proof in [61] are easily specialized to the context of dyadic martingales. The estimate (1.3.32) for the Rademacher projection H 1 is a result of J. Bourgain [18]. J. Bourgain’s estimate (1.3.32), is related (in spirit) to R. E. A. C. Paley’s inequality on lacunary coeﬃcients [168] and to the interpolatory inequalities for Fourier multipliers, appearing in the work of S. Kislyakov [115] and [116]. (The basis of the relationship is the Lusin square function characterization presented in Section 6.1.) The proof of Theorem 1.4.1 is due to B. Maurey [139] (see also [144]). The proof of Stein’s inequality (Theorem 1.4.2) is due to J. Bourgain and applied in 1983/4 [24]. It extends without changes to the case of Lp spaces of functions with values in a UMD space. The only published account of J. Bourgain’s proof seems to be in [69] by T. Figiel and P. Wojtaszczyk. Theorem 1.4.3, Theorem 1.4.4 and Proposition 1.4.5 are results of T. Figiel [65].

1.6. Notes

115

The proof of Theorem 1.5.1 is obtained (by specialization) from [130], Proposition 2.c.17. Theorem 1.5.2 is taken from [147] by Y. Meyer and R. Coifman. The proof in the text is strictly limited to the case of real-valued coeﬃcients and not extending to coeﬃcients in a UMD space. However there exist paraproduct estimates due to J. Bourgain that extend to the UMD case. See [69] for the only published proof. Theorem 1.5.3 and the proof presented in the text are due to T. Figiel [66]. The survey [69] by T. Figiel and P. Wojtaszczyk contains a very informative summary of Figiel’s work on singular integral operators. Our presentation incorporates also lecture notes of talks given by T. Figiel at a Conference on the Geometry of Banach spaces in Strobl (1989), and at the Weizmann Institute in Rehovot (1988). Z. Ciesielski [53] gives a detailed presentation of the Haar system on [0, 1] × [0, 1] indexed by dyadic squares. J. O. Stromberg’s construction [196] of a modiﬁed Franklin system, is based on the same principles. T. Figiel’s use of the square indexed Haar system in the expansion of a kernel on [0, 1] × [0, 1] resulted in the representation of integral operators as paraproducts, (simple) rearrangements and multipliers. Using the standard Haar system on [0, 1] × [0, 1] indexed by dyadic rectangles leads to a diﬀerent representation of integral operators. It was invented by Y. Meyer [146] to give the ﬁrst wavelet proof of the T (1) theorem of G. David and J.L Journ´e [59] . Comparing Y. Meyer’s proof [146] of the T (1) theorem to that of T. Figiel [66] the following attracts attention: The norm estimates for rearrangement operators in [66] are replacing the applications of Cotlar’s lemma in [146]. Having freed the proof of the T (1) theorem from Cotlar’s lemma, T. Figiel [66] extended the T (1) theorem beyond Hilbert spaces to the UMD class (being able to do this provided the main stimulus for T. Figiel’s work). Simultaneously and independent of the work of T. Figiel the advantages of expanding integral kernels along the Haar system indexed by dyadic squares were understood by G. Beylkin, R. Coifman and V. Rokhlin [10] while developing numerical algorithms for the evaluation of integral operators. The boundedness criteria for orthogonal projections as formulated in Theorem 1.5.9–Theorem 1.5.11 are due to P. W. Jones [102]. The use of orthogonal projections has led to deep results in function spaces other than H 1 and BMO. For instance, the results on complemented subspaces of rearrangement invariant function spaces in the monograph [93] by W. Johnson, B. Maurey, G. Schechtman, L. Tzafriri.

Chapter 2

Projections, Isomorphisms and Interpolation This chapter reviews three basic concepts of functional analysis. We discuss comnski, plemented subspaces (mostly in H 1 ), the decomposition method of A. Pelczy´ and elements of complex interpolation theory.

2.1

Complemented subspaces

In this section we study Banach spaces which are isomorphic to a complemented subspace of H 1 . To this end we introduce three families of Banach spaces closely related to H 1 . These are atomic H 1 over spaces of homogeneous type, martingale H 1 associated to an increasing sequence of σ algebras, and Banach spaces spanned by nested, three-valued martingale diﬀerence sequences. Furthermore we discuss the Rosenthal space and related Banach spaces spanned by independent random variables. Standard concepts and terminology We recall standard concepts associated with the study of Banach spaces. We deﬁne projections, isomorphisms, well isomorphic sequences of ﬁnite-dimensional spaces, complemented copies, inﬁnite direct sums, and the ﬁnite-dimensional building blocks of the spaces H 1 , Lp , and BMO. Projections. A projection P on a Banach space E is a bounded linear operator P : E → E such that P 2 = P . The range of a projection is called a complemented subspace. Clearly the orthogonal projections discussed in Section 1.5 satisfy P 2 = P.

118

Chapter 2. Projections, Isomorphisms and Interpolation

Isomorphisms. Two Banach spaces E and F are isomorphic if there exists a bijective linear operator T : E → F so that

T · T −1 < ∞.

(2.1.1)

In that case T : E → F is called an isomorphism between E and F. The inﬁmum of (2.1.1) taken over all isomorphisms between E and F is called the Banach–Mazur distance between E and F. If two Banach spaces E and F are isomorphic, then we say that E is a copy of F and we write E ∼ F. A property of a Banach space which is preserved by isomorphisms is called an isomorphic invariant. Well isomorphic sequences. Given two sequences of ﬁnite-dimensional Banach spaces {En } and {Fn }. We say that {En } is well isomorphic to {Fn } if there exist bijective linear operators Tn : En → Fn such that sup Tn · Tn−1 < ∞.

n∈N

Complemented copies. Below we will be concerned with Banach spaces that are isomorphic to a complemented subspace of H 1 . Observe that a Banach space E is isomorphic to a complemented subspace of a Banach space F if there exist linear operators M : E → F and N : F → E so that Id E −→ E M N F

and

M · N < ∞.

(2.1.2)

Indeed, M (E) is a subspace of F that is isomorphic to E, and P = M ◦ N is a projection on F whose range equals M (E). If (2.1.2) holds, then we say that F contains a complemented copy of E. Inﬁnite direct sums. Among the simplest procedures to generate new Banach spaces from old ones is the formation of inﬁnite direct sums. Let En , n ∈ N be a sequence of Banach spaces, let 1 ≤ p ≤ ∞. Then ( En )p is deﬁned as the space of sequences (xn )∞ n=1 , where xn ∈ En , equipped with the norm 1/p ||xn ||pEn when 1 ≤ p < ∞, (2.1.3) when p = ∞. sup ||xn ||En For 1 ≤ p < ∞ let q be the H¨ older conjugate index to p, satisfying 1/p + 1/q = 1. The dual space to ( En )p is naturally identiﬁed as ∗ En = En∗ , p

where

En∗

denotes the dual space to En .

q

2.1. Complemented subspaces

119

Finite-dimensional building blocks. We encounter inﬁnite direct sums most frequently in connection with the following examples. Deﬁne the algebraic vector space Vn = span{hJ : |J| ≥ 2−n }. The algebraic dimension of Vn is equal to 2n+1 − 1. Equipping Vn with the norm of H 1 , Lp or BMO gives the ﬁnite-dimensional building blocks of H 1 , Lp or BMO, Hn1 = (Vn , || · ||H 1 ) ,

Lpn = (Vn , || · ||Lp ) ,

BMOn = (Vn , || · ||BMO ) .

Clearly Hn1 is complemented in H 1 ; it is the range of the norm 1 projection,

P (f ) =

f, hJ

{J:|J|≥2−n }

hJ . |J|

The projection P is also a norm 1 operator on Lp and on BMO. Hence Lpn and BMOn are one-complemented subspaces of Lp respectively BMO. The following proposition is simple and useful. Its proof shows how inﬁnite direct sums are represented as subspaces of H 1 . Proposition 2.1.1. The space ( Hn1 )1 is isomorphic to a complemented subspace of H 1 . Proof. Let (In ) be a sequence of pairwise disjoint dyadic intervals. Let fn : [0, 1] → In be an aﬃne function mapping [0, 1] bijectively onto the interval In . The slope of fn is |In |. Deﬁne the following collections of dyadic intervals contained in In , Cn = {J ⊆ In : |J| ≥ 2−n |In |}. Let Yn be the algebraic vector space Yn = span{hJ : J ∈ Cn }. The composition operator Tn deﬁned by Tn : g → |In |g ◦ fn , is an isometry from (Yn , || · ||H 1 ) onto Hn1 . Note that for yn ∈ Yn we have the identity ||yn ||H 1 . (2.1.4)

yn H 1 = 1 Let Y be the closure in H of span{Yn : n ∈ N}. By the above identity (Y, || · ||H 1 ) is isometric to ( Hn1 )1 . Moreover Y is a complemented subspace of H 1 . Indeed let P be the orthogonal projection deﬁned by

P (f ) =

∞ n=1 J∈Cn

Then ||P ||H1 = 1 and Y is the range of P.

f, hJ

hJ . |J|

120

Chapter 2. Projections, Isomorphisms and Interpolation

Let 2n denote the n-dimensional Euclidean space. With obvious modiﬁcations the proof of Proposition 2.1.1 shows that ( 2n )1 is isomorphicto a com1 plemented subspace of H . By duality the spaces ( BMOn )∞ and ( 2n )∞ are isomorphic to complemented subspaces of BMO. In the next section we will considerably strengthen this observation and prove a theorem of P. Wojtaszczyk that ( BMOn )∞ is isomorphic to BMO. This provides us with a useful alternative representation of the space BMO. Consequences of this isomorphism will be drawn in Chapter 5 where we prove isomorphic invariants of H 1 and BMO. Martingale H 1 spaces Martingale H 1 spaces provide a class of complemented subspaces of H 1 . Let (Ω, F, | · |) be a probability space and let F0 ⊆ · · · ⊆ Fn ⊆ · · · F

# be a sequence of increasing σ-algebras on Ω so that F is generated by Fn . For an integrable function f : Ω → R, the induced square function is deﬁned to be 1/2

|E(f |Fn ) − E(f |Fn−1 )|2 , n

where E(f |Fn ) denotes the conditional expectation of f with respect to Fn . Then H 1 [(Fn )] is the space of integrable functions f : Ω → R, with integrable square function. The norm on H 1 [(Fn )] is deﬁned by 1/2 E(f |F0 )2 + ||f ||H 1 [(Fn )] = |E(f |Fn ) − E(f |Fn−1 )|2 . Ω

n

The space BMO[(Fn )] consists of all functions g ∈ L2 (Ω) so that

g BMO[(Fn )] = sup E(|g − E(g|Fn−1 )|2 Fn ) < ∞. n∈N

The space BMO[(Fn )] is (identiﬁable with) the dual space of H 1 [(Fn )] as follows. Let f, g ∈ L2 (Ω), then √ |E(f · g)| ≤ 2||f ||H 1 [(Fn )] g BMO[(Fn )] . Thus every g ∈ BMO[(Fn )] induces a bounded linear functional on H 1 [(Fn )], by L(f ) = lim E(fn · gn ), n→∞

fn = E(f |Fn ) and gn = E(f |Fn ).

Conversely for every bounded linear functional L : H 1 [(Fn )] → R there exists g ∈ BMO[(Fn )] so that L(f ) = lim E(fn · gn ) n→∞

for f ∈ H 1 [(Fn )].

2.1. Complemented subspaces

121

The book of A. M. Garsia [75] is the standard reference to martingale inequalities, H 1 [(Fn )] and BMO[(Fn )]. Dyadic H 1 is in the class of H 1 [(Fn )] spaces. To see this let Fn be the σalgebra generated by dyadic intervals of length ≥ 2−n . This observation, and many other analogies between general martingale H 1 space and the dyadic H 1 , suggest strong ties between these spaces. On the other hand also the Banach spaces 1 and L1 ([0, 1]) are in this class. To see how L1 ([0, 1]) arises, let F0 = {∅, [0, 1]} and deﬁne F1 = F2 · · · to be the Lebesgue σ-algebra on the unit interval [0, 1]. For this ﬁltration H 1 [(Fn )] coincides with L1 ([0, 1]). The Banach spaces L1 , 1 , and H 1 share very few isomorphic invariants. In particular, by a result of A. Pelczy´ nski, L1 ([0, 1]) is not isomorphic to a subspace of a Banach space with an unconditional basis. We conclude that without putting restrictions on the ﬁltration (Fn ) the ties between the spaces in the class H 1 [(Fn )] and dyadic H 1 are not very strong. However, after imposing the simplest condition ruling out the case of L1 , B. Maurey [144] obtained this truly remarkable theorem: Theorem 2.1.2. If the σ-algebras Fn are purely atomic, then H 1 [(Fn )] is isomorphic to a complemented subspace of H 1 . It takes up almost all of Chapter 4 to prove B. Maurey’s result. In Chapter 4 we present also two applications to Maurey’s theorem. First we use it to obtain the isomorphic classiﬁcation of martingale H 1 spaces when the underlying ﬁltration consists of purely atomic σ-algebras. Separately we apply its proof to show that the span of ﬁnitely many Haar functions in Lp is well isomorphic to pn . Atomic H 1 spaces The next examples of complemented subspaces in H 1 are the atomic H 1 spaces appearing in analysis. On a set X let d(., .) be a quasi-metric, that is, d(x, y) = 0 if and only if x = y, d(x, y) = d(y, x), d(x, z) ≤ K(d(x, y) + d(y, z)) for x, y, z ∈ X. The balls of radius r in X are the following sets, B(x, r) = {y ∈ X : d(x, y) < r}. We let µ be a probability measure on X satisfying the doubling condition, µ(B(x, 2r)) ≤ Aµ(B(x, r)), for some A and all x ∈ X, r > 0. Following R. Coifman and G. Weiss [56] we call the triple (X, d, µ) a space of homogeneous type. An atom for (X, d, µ) is either a constant function on X or a function a : X → R such that adµ = 0,

122 and

Chapter 2. Projections, Isomorphisms and Interpolation

a2 dµ ≤ µ(B)−1 ,

where B is a ball in X such that supp a ⊆ B. We deﬁne the associated atomic H 1 space to be the space of functions f : X → R which admit a decomposition into atoms, f= ci a i , where ai are atoms for (X, d, µ), and where |ci | < ∞. We denote this space by H 1 (X, d, µ). The norm of f ∈ H 1 (X, d, µ) is ||f ||H 1 (X,d,µ) = inf{ |ci |}, where the inﬁmum is taken over all decompositions of f into atoms. The following two examples appear most frequently. 1. Historically the ﬁrst example of a space of homogeneous type is the boundary of the unit disk T = {z ∈ D : |z| = 1}, where the metric is given by the Euclidean distance, and the measure is the Lebesgue measure on T. The 1 . In Chapter 6 we present a thorough space H 1 (T, dt, | · |) is denoted Hat 1 discussion of the space Hat . Section 6.1 covers the basic theorems about 1 , relating it to functions for which the Hilbert transform and the Lusin Hat area function are integrable; Section 6.2 contains L. Carleson’s proof of B. 1 is isomorphic to H 1 . Maurey’s theorem that Hat 2. By Theorem 1.2.4 we identiﬁed the space of homogeneous type (X, d, µ) for which the associated atomic H 1 space coincides with dyadic H 1 . Indeed, let X be the unit interval [0, 1], let µ be Lebesgue measure on [0, 1], and let d(x, y) be the length of the smallest dyadic interval containing both x and y. Then (X, d, µ) is a space of homogeneous type. The following example sheds some light on this metric. We ﬁx numbers n, q ∈ N such that n ≥ q. Consider Iq−1 a dyadic interval of length 2−q+1 . Let m be the midpoint of Iq−1 . For each q ≤ j ≤ n deﬁne Jj = [m − 2−j , m),

and

Ij = [m, m + 2−j ).

Observe that Jj and Ij are adjacent dyadic intervals, and that Jq ∪ Iq = Iq−1 .

(2.1.5)

2.1. Complemented subspaces

123

The interval Iq−1 is then the smallest dyadic interval containing Ij and Jj . Hence for q ≤ j ≤ n, x ∈ Ij and y ∈ Jj we obtain d(x, y) = |Iq−1 |. Given x ∈ [0, 1], the ball B(x, r) of radius r in (X, d, µ) is just the dyadic interval I for which x ∈ I and r/2 ≤ |I| < r. Hence a function a is an atom for (X, d, µ) if there exists a dyadic interval I such that supp a ⊆ I, a2 (t)dt ≤ |I|−1 , and

a(t)dt = 0.

By Theorem 1.2.4 for f ∈ H 1 , there exists a decomposition f− f= ci a i , where ai are atoms for (X, d, µ), and ci are scalars such that |ci | ≤ 16 S(f ). Conversely, any atom a for (X, d, µ) satisﬁes S(a) ≤ 1. In Section 6.5 we will show that atomic H 1 spaces are isomorphic to complemented subspaces in H 1 , for each choice of (X, d, µ). More precisely we will prove the following Theorem [155]. Theorem 2.1.3. For every space of homogeneous type, H 1 (X, d, µ) is isomorphic to a complemented subspace of H 1 . Moreover H 1 (X, d, µ) and H 1 are isomorphic if (and only if ) µ{x ∈ X : µ({x}) = 0} > 0. We close this section by describing the dual space of H 1 (X, d, µ). For x ∈ X and r > 0 let B(x, r) = {y ∈ X : d(x, y) < r}. Let h ∈ L2 (X) and B = B(x, r), then we write mB (h) to denote the mean value of h over B, thus, dµ . mB (h) = h B µ(B)

124

Chapter 2. Projections, Isomorphisms and Interpolation

A function h ∈ L2 (X) belongs to BMO(X, d, µ) if

h BMO(X,d,µ) = sup B

dµ |h − mB (h)| µ(B) B 2

1/2 < ∞,

(2.1.6)

where the supremum is taken over all balls B = B(x, r), x ∈ X, and r > 0. The classical survey article [56] by R. R. Coifman and G. Weiss is our standard reference to spaces of homogeneous type, H 1 (X, d, µ) and BMO(X, d, µ). The duality results described below are proved there in detail. The dual space of H 1 (X, d, µ) is characterized as BMO(X, d, µ). This characterization comprises two statements. First, each h ∈ BMO(X, d, µ) induces a bounded linear functional on H 1 (X, d, µ). For f ∈ H 1 (X, d, µ), with atomic decomposition f= λi ai and |λi | < ∞, the mapping f → lim

N →∞

N i=1

λi

hai dµ ,

(2.1.7)

X

is a well-deﬁned linear functional on H 1 (X, d, µ). Conversely, if x∗ is in the dual space of H 1 (X, d, µ) then there exists a unique h ∈ BMO(X, d, µ) so that

h BMO(X,d,µ) ≤ C x∗ and so that for f ∈ H 1 (X, d, µ) the value of x∗ (f ) is given by the mapping (2.1.7). Three-valued martingale diﬀerences Dyadic intervals are distinguished by the following properties: First for a dyadic interval I there exists n ∈ N such that |I| = 2−n . Second, if I, J are dyadic intervals and I ∩ J = ∅, then either I ⊆ J or J ⊆ I. We now consider arbitrary collections of measurable sets which satisfy the second property but not necessarily the ﬁrst. Let E be a countable collection of measurable subsets in the measure space [0, 1], equipped with Lebesgue measure. We say that E is a nested collection (of measurable sets) if the elements of E satisfy the following condition, if A, B ∈ E and A ∩ B = ∅, then A ⊆ B or B ⊆ A. We let E play the role of dyadic intervals and deﬁne spaces of three-valued martingale diﬀerence sequences. As soon as E is ﬁxed the construction of these spaces is in complete analogy to that of H 1 and BMO. A nested collection E gives rise to spaces X[E] and Y [E] of three-valued martingale diﬀerence sequences. We start with an analog to the Haar system. Let {rA : A ∈ E} be any enumeration of the sequence of Rademacher functions. Then for A ∈ E, put dA (w, t) = 1A (w)rA (t).

2.1. Complemented subspaces

125

Note that the functions {dA } assume only the values {+1, −1, 0}, they form a martingale diﬀerence sequence. The support of dA is the set A × [0, 1]. Hence the support sets of the functions {dA : A ∈ E} form a nested collection of measurable sets. We deﬁne the space X[E] to consist of those f = A∈E aA dA for which ||f ||X[E] =

1/2 a2A 1A

< ∞.

A∈E

basis This norm makes X[E] a Banach space, and {dA : A ∈ E} an unconditional for X[E]. We deﬁne the space Y [E] to consist of all functions h = A∈E bA dA , for which ⎛ ⎞1/2 1 b2B |B|⎠ < ∞. ||h||Y [E] = sup ⎝ |A| A∈E B∈E, B⊆A

Repeating the proof of Theorem 1.2.1 (Feﬀerman’s inequality) gives the integral estimate √ f h dx ≤ 2 2||f ||X[E] ||h||Y [E] (2.1.8) for any ﬁnite linear combination f = aA dA . Moreover Y [E] can be identiﬁed with the dual space of X[E]. By choosing diﬀerent collections E we should be able to produce several diﬀerent spaces in the class X[E]. Below we ﬁnd that the following list of Banach spaces can be obtained by choosing E appropriately, 2n )1 , ( 2n )1 ⊕ 2 , ( 2 )1 , 1 , 2 , 1 ⊕ 2 , ( (2.1.9) Hn1 )1 ⊕ 2 , ( Hn1 )1 ⊕ ( 2 )1 , H 1 . ( Hn1 )1 , ( It was pointed out by P. Wojtaszczyk [211] that the spaces in (2.1.9) are all diﬀerent. By that we mean that no two of them are isomorphic. We will now show how to select nested collections E so that X[E] is isomorphic to a given space from the list (2.1.9). 1. Let C be a collection of dyadic intervals. Then clearly C is a nested collection of measurable sets and for E = C, the unconditional basis for X[E], dI (w, t) = 1I (w)rI (t),

I ∈ C,

is equivalent to the Haar functions {hI : I ∈ C} in H 1 . Thus the class of spaces X[E] contains the class of all subspaces in H 1 which are spanned by subsequences of the Haar system. Clearly H 1 can be obtained in that way. In the course of proving Proposition 2.1.1 we showed how to choose a subsequence of the Haar system so that its closed linear span is isometric to ( Hn1 )1 . Finally we obtain the space 1 by letting C be a collection of pairwise disjoint dyadic intervals.

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2. Here we construct E such that X[E] is isomorphic to 2 . Let I0 = [0, 1] and for n ∈ N let In = [0, 1 − 4−n ]. Then the collection E = {In : n ∈ N0 } is a nested collection and X[E] is isomorphic to 2 . Indeed the unconditional basis for X[E] is dn (w, t) = 1In (w)rn (t). For this system we have

∞ 1/2 ∞ 1 2 an ≤ dn an 2 n=0 n=0

≤

X[E]

∞

1/2 a2n

.

n=0

3. Now we deﬁne a collection E such that X[E] is isomorphic to ( 2 )1 . For n ∈ N, let In be a sequence of pairwise disjoint dyadic intervals. Then for m ≥ 1 choose In,m ⊆ In such that |Im,n | = |In |(1 − 4−m ). Now consider E = {In,m : m, n ∈ N}, and deﬁne the normalized unconditional basis for X[E], dn,m (w, t) = |In,m |−1 1In,m (w)rn,m (t), where rn,m denotes independent Rademacher functions. Let {an,m } be a sequence of scalars. For each n ﬁxed we have an 2 estimate,

∞

∞ 1/2 ∞ 1/2 1 2 2 an,m ≤ am,n dn,m ≤ an,m . 2 m=1

m=1

m=1

X[E]

Next we observe that for n = n and m, m ∈ N, the intervals In,m and In ,m are disjoint. Hence,

∞ 1/2 ∞ ∞ ∞ 1 2 an,m ≤ an,m dn,m 2 n=1

m=1

n=1 n=1

≤

∞ n=1

∞

1/2 a2n,m

X[E]

.

m=1

In other words the unit vectors of ( 2 )1 are equivalent to {dn,m : n, m ∈ N}. 4. In our ﬁnal example we describe an entire class of collections E for which X[E] is isomorphic to 1 . We consider now collections E which satisfy the Carleson packing condition, 1 |B| < ∞. [[E]] = sup A∈E |A| B⊆A, B∈E

We claim that then X[E] is isomorphic to 1 . To see this we consider the unconditional basis, dA (w, t) = 1A (w)rA (t), A ∈ E.

2.1. Complemented subspaces

127

We will show that {|A|−1 dA : A ∈ E} is equivalent to the unit vector basis of 1 . To this end, we ﬁx aA ∈ R. Let f= aA dA (w, t), and deﬁne h=

sign (aA )dA (w, t).

1/2 A |aA ||A| = moment’s reﬂection shows that ||h||Y [E] ≤ [[E]] , and that hf. Applying (2.1.8), that is Feﬀerman’s inequality for nested martingale diﬀerences, we obtain the upper bound √ (2.1.10) |aA ||A| ≤ 2 2[[E]]1/2 ||f ||X[E] . Next we show that the converse estimate holds true as well. Indeed, note that ||dA ||X[E] = |A|, and apply the triangle inequality to f . This gives the corresponding lower bound, |aA ||A| ≥ ||f ||X[E] . (2.1.11) Combining (2.1.10) and (2.1.11) we ﬁnd that {|A|−1 dA : A ∈ E} is equivalent to the unit vector basis of 1 . It is now easy to combine these constructions and to obtain the remaining members of the list (2.1.9). We remarked already that the spaces appearing in (2.1.9) are known to be isomorphically diﬀerent (see [211]). Next we show that each of the spaces X[E] is isomorphic to a complemented subspace of H 1 . Thus we have at once ten diﬀerent examples of complemented subspaces of H 1 . Nested collections can be decomposed naturally in a sequence of generations. This is done as follows. Given A ∈ E we denote by G1 (A, E) ⊆ E the collection of maximal elements of E which are strictly contained in A. This is called the ﬁrst generation of A. As E is nested, G1 (A, E) is a collection of pairwise disjoint sets. Now we deﬁne the following generations by induction, using the ﬁrst step as a model. Suppose that the ﬁrst p − 1 generations, G1 (A, E), . . . , Gp−1 (A, E), are already deﬁned. Then we put H = {B ∈ E : B ⊂ A} \ {G1 (A, E) ∪ · · · ∪ Gp−1 (A, E)}, and deﬁne the p-th generation of A as Gp (A, E) = G1 (A, H). Let G0 (E) denote the collection of maximal elements E. Inductively, for p ∈ N, we deﬁne Gp (E) = G1 (A, E). A∈Gp−1 (E)

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Chapter 2. Projections, Isomorphisms and Interpolation

Proposition 2.1.4. For any nested family of measurable sets E, the space X[E] is isometric to a complemented subspace of H 1 . Proof. To simplify the presentation of the proof we make the convenient but inessential assumption that for each A ∈ E, G1 (A, E) is a ﬁnite set. We will construct collections {CA : A ∈ E} of pairwise disjoint dyadic intervals which provide a model of the collection E, since {CA } will be deﬁned to satisfy the following properties. 1. Let CA be the point-set covered by the collection CA . Then {CA : A ∈ E} is a nested family of subsets of [0, 1] and |A| = |CA |. 2. CB ⊆ CA iﬀ B ⊆ A. 3. If B ⊆ A, and I ∈ CA , then |B| |I ∩ CB | = . |I| |A| 4. If I ∈ CB , J ∈ CA and I ⊆ J, then B ⊆ A. Now we construct these collections by induction. We ﬁx a maximal element in E, say A. As |A| ≤ 1, there exists a collection CA of pairwise disjoint dyadic intervals so that I CA = I∈CA

satisﬁes |CA | = |A|. It is easy to ensure that for A, A ∈ G0 (E), with A = A we have CA ∩CA = ∅. Thus we deﬁne CA for each A in the generation G0 (E). Next we turn to CB for the sets B in the ﬁrst generation of E. Let A ∈ G0 (E). Recall that G1 (A, E) consists of pairwise disjoint elements from E. For each I ∈ CA and each B ∈ G1 (A, E) we ﬁnd easily a collection CB,I of pairwise disjoint dyadic intervals with the following properties: Let CB,I be the point-set covered by the collection CB,I , then CB,I ⊆ I and |B| |CB,I | = . |I| |A| Furthermore we arrange also for the following condition to hold: If B1 , B2 ∈ G1 (A, E) are diﬀerent (and hence disjoint sets), then CB1 ,I ∩ CB2 ,I = ∅. Now we deﬁne CB for B ∈ G1 (A, E) by taking the union, CB = CB,I . I∈CA

Having deﬁned CB for B ∈ G0 (E), . . . Gp (E) we continue as follows. Fix A ∈ Gp (E). Then G1 (A, E) consists of pairwise disjoint elements from E. Hence for each I ∈ CA

2.1. Complemented subspaces

129

and each B ∈ G1 (A, E) there exists a collection of pairwise disjoint dyadic intervals CB,I such that CB,I ⊆ I, and |B| |CB,I | = . |I| |A| Now we obtain CB for B ∈ G1 (A, E) by taking unions, CB,I . CB = I∈CA

The resulting collections {CA : A ∈ E} satisfy the above conditions 1) − 4). Consequently they satisfy Jones’ compatibility condition (J). Now we form the linear combinations {hI : I ∈ CA }, bA = and put kA = bA /||bA ||2 . By Theorem 1.5.9, the orthogonal projection Pf = f, kA kA A∈E

is a bounded operator on H 1 with norm ≤ 1. The range of P is the closed linear span of {bA : A ∈ E}. Thus it remains to prove that in H 1 , the closure of span{bA : A ∈ E} is isomorphic to X[E]. To show this let dA (ω, t) = 1A (ω)rA (t),

A ∈ A,

be the unconditional basis for X[E] and ﬁx a sequence aA ∈ R. Observe that the distribution of A∈E a2A d2A is the same as the distribution of A∈E a2A b2A . Thus it follows that aA dA = aA bA . A∈E

X[E]

A∈E

H1

So far we used the spaces X[E] to produce examples of complemented subspaces in H 1 . They play a second, even more important role in the isomorphic classiﬁcation of martingale H 1 spaces. The link to the martingale H 1 spaces is provided by Maurey’s theorem. Following is a more detailed formulation of Theorem 2.1.2. Theorem 2.1.5. Let (Fn ) be a sequence of increasing and purely atomic σ-algebras. There exists a nested collection E, satisfying A, B ∈ E and A ⊂ B implies |A| ≤ |B|/2, such that H 1 [(Fn )] is isomorphic to X[E]. Consequently H 1 [(Fn )] is a space with an unconditional basis.

130

Chapter 2. Projections, Isomorphisms and Interpolation

Maurey’s result is the main step in the classiﬁcation of martingale H 1 spaces. We will prove Theorem 2.1.5 in Chapter 4 and complement it with the following result. Theorem 2.1.6. Let E be a nested collection of measurable sets satisfying A, B ∈ E and A ⊂ B implies |A| ≤ |B|/2. Then X[E] is isomorphic to one of the spaces 1 , ( Hn1 )1 , or H 1 . Above we found ten diﬀerent Banach spaces in the class X[E]. It is not possible to obtain more spaces by this method, since the isomorphic types of the family X[E] are classiﬁed in [151]: For every choice of E, the space X[E], is isomorphic to one of the ten spaces listed in (2.1.9) above. The Rosenthal space and independent random variables Next we use independent random variables to construct complemented subspaces in H 1 , respectively BMO. We present the Rosenthal space, its predual, the independent sum of BMOn and the intersection of weighted L2 with BMO. The Rosenthal space. We deﬁne the intersection of ∞ with a weighted 2 space. More precisely, let (wi ) be a bounded sequence of positive real numbers such that and lim wi = 0. wi = ∞, Then the sequence space R = R({wi }) consists of all bounded sequences (ai ) for which the following norm is ﬁnite, 1/2 ||(ai )||R = sup a2i + a2i wi . The space R that is now called the Rosenthal space was introduced and studied ﬁrst by H. Rosenthal in [181]. Rosenthal showed in particular that the isomorphic type of R = R({wi }) is determined by the following two conditions on the weights, wi = ∞, and lim wi = 0. Precisely, if {vi } is a sequence of positive real numbers so that vi = ∞ and lim vi = 0, then R = R({vi }) is isomorphic to R = R({wi }). The proof shownski’s ing the existence of an isomorphism between R and R is based on A. Pelczy´ decomposition method. It was observed in [158] that the Rosenthal space is isomorphically diﬀerent from the spaces Y [E] that are generated by nested three-valued martingale diﬀerence sequences. Proposition 2.1.7. The Rosenthal space R is isomorphic to a complemented subspace of BMO.

2.1. Complemented subspaces

131

Proof. The proof is divided into three basic steps. We ﬁrst deﬁne a subspace of BMO, by constructing the sequence (bi ) in (2.1.12). Then we show that in BMO the sequence (bi ) is equivalent to the unit vector basis of the Rosenthal space. In a third step we verify that the orthogonal projection onto the linear span of (bi ) is bounded on BMO. Combined, these three steps imply that the Rosenthal space is isomorphic to a complemented subspace of BMO. Step 1. We are given a sequence (wi ) of positive real numbers ≤ 1. Without loss of generality we assume that the sequence (wi ) is sorted in decreasing order. First we group them according to their value, as follows. For n ≥ 1 deﬁne the index set In = {i : 2−n ≤ wi ≤ 2−n+1 }. Let in denote the cardinality of the set In . Next we select pairwise disjoint intervals of natural numbers Kn , and Mn such that |Kn | = n , |Mn | = in , and so that sup Kn < inf Mn ≤ sup Mn < inf Kn+1 . We use the index set Kn to deﬁne the indicator functions Xn =

1 (1 + rs ). 2

s∈Kn

The supports of the functions Xn form a sequence of independent sets, the measure of the support of Xn equals 2−n . Next we attach the following frequencies to the indicator function Xn . For j ∈ Mn , we form bn,j = Xn rj . Now transform the lexicographic order on the set {(n, j) : n ∈ N, j ∈ Mn } into the linear order on the natural numbers, and accordingly we relabel the sequence {bn,j }, as bi = hI , i ∈ N. (2.1.12) I∈Ci

Let Ci be the point-set covered by the collection Ci . Then by construction the measure of Ci is related to the weights wi as follows, wi /2 ≤ |Ci | ≤ wi .

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Chapter 2. Projections, Isomorphisms and Interpolation

Step 2. We will now show that in BMO the sequence {bi } spans the Rosenthal space. Fix a bounded sequence ai ∈ R and put f=

ai bi .

We verify the equivalence of norms ||f ||2BMO ≤ max a2i +

∞

i∈N

a2i wi ≤ 2||f ||2BMO .

(2.1.13)

i=1

We show ﬁrst the left-hand side estimate in (2.1.13). To this end we expand f in its Haar series and let let αI hI be the Haar expansion of f. Next we ﬁx n ∈ N and I ∈ Cn . Note that for i ≥ n + 1 the sets I and Ci are independent. Hence |bi |2 = |I ∩ Ci | = |I| · |Ci |. I

Therefore we observe the identities

αJ2 |J|

=

|I|a2n

+

J⊆I

= |I|a2n +

∞ i=n+1 ∞

a2i

|bi |2 I

a2i |I||Ci |.

i=n+1

Dividing by |I|, and taking the supremum over all dyadic intervals I gives $ ∞ 1 2 2 2 sup αJ |J| ≤ sup an + ai |Ci | n I |I| i=n+1 J⊆I

≤ sup a2i + i

∞

a2i |Ci |.

i=1

As |Ci | ≤ wi we have veriﬁed the left-hand side of (2.1.13). To prove the reverse estimate, observe that {bi } is disjointly supported over the Haar system. Hence for f = ai bi we have ||f ||BMO ≥ sup |ai |. The norm of f in L2 is bounded by its BMO norm, hence, ||f ||2BMO ≥ ||f ||2L2 ∞ = a2i |Ci |. i=1

Combining this with |Ci | ≥ wi /2 we obtain the right-hand side of (2.1.13).

2.1. Complemented subspaces

133

Step 3. The orthogonal projection onto span{bi } is deﬁned by P (f ) =

∞

f,

i=1

bi bi . ||bi ||2 ||bi ||2

By (2.1.13) boundedness of P on BMO is easy to check. Indeed (2.1.13) gives that ||P (f )||2BMO ≤ sup f, i

∞ bi 2 bi 2 + f, . 2 ||bi ||2 ||bi ||22 i=1

Next observe that ||bi ||22 = |Ci |. Hence by Bessel’s inequality f,

bi 2 ≤ ||f ||2BMO . ||bi ||22

Moreover we have the L2 estimate ∞ i=1

f,

bi 2 ≤ ||f ||22 ||bi ||22 ≤ ||f ||2BMO .

In summary we showed that ||P (f )||2BMO ≤ 2||f ||2BMO , thus proving that the Rosenthal space is isomorphic to a complemented subspace of BMO. The predual of Rosenthal’s space. Let R∗ be the space of all sequences (ai ) for which

∞ 1/2

(ai ) R∗ = min{|ai |, a2i wi−1 } < ∞. i=1

This space can be identiﬁed — canonically — with the predual of Rosenthal’s space. By Proposition 2.1.7 and duality the space R∗ is isomorphic to a complemented subspace of H 1 . A direct proof of this fact, not using duality, follows from work of W. B. Johnson and G. Schechtman [95]. It was veriﬁed in [158] that R∗ is not isomorphic to one of the spaces listed in (2.1.9). The independent sum of BMOn . The next example is very close to Rosenthal’s space. Recall that L2n ⊂ L2 and BMOn ⊂ BMO denote the subspaces spanned by the Haar functions {hJ : |J| ≥ 2−n }. Deﬁne the intersection BMOn ∩ L2n ∞

2

equipped with the norm

||(xn )|| =

max ||xn ||2BMO n∈N

+

∞ n=1

1/2 ||xn ||22

.

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Chapter 2. Projections, Isomorphisms and Interpolation

This space iscommonly called the independent sum of the spaces BMOn , and denoted by ( BMOn )ind . The name comes from the embedding deﬁned below. There we use statistically of BMOn as building blocks of a independent copies subspace isomorphic to ( BMOn )∞ ∩ ( L2n )2 . We point out that implicitly we actually determine a subsequence of the Walsh system that spans a copy of the independent sum ( BMOn )ind . Theorem 2.1.8. The independent sum ( BMOn )ind is isomorphic to a complemented subspace of BMO. Proof. We start by deﬁning a subspace of BMO which is isomorphic to the independent sum ( BMOn )ind . First we ﬁx i ∈ N and let N = {i, . . . , i + n}. Now we deﬁne a copy of BMOn using the Rademacher functions {rj : j ∈ N }. We start by deﬁning b[0,1] = ri . Now we suppose that the functions bI have been constructed for all dyadic intervals of length ≥ 2− . Then ﬁx a dyadic interval J of length 2− , and split J into its left half J1 , and its right half J2 . Then we put bJ1 = ri++1 1{bJ =1} , bJ2 = ri++1 1{bJ =−1} . In this way we deﬁned bI for every dyadic interval I with |I| ≥ 2−n . We let G(N ) = span{bI : |I| ≥ 2−n }. Clearly, when equipped with the BMO norm, the space G(N ) is isometric to BMOn . Incidentally we deﬁned G(N ) in such a way that it coincides with the span of Walsh functions {wA : A ⊆ {i, . . . , i + n}}. Moreover G(N ) is the range of a norm 1 projection. To see this, we renormalize in L2 , and put kI = bI /||bI ||2 . Then by Theorem 1.5.9 the orthogonal projection PN (f ) = f, kI kI is bounded on BMO with norm ≤ 1. The range of PN is G(N ). Next we produce a sequence of independent copies of BMOn . We ﬁx a sequence of disjoint intervals Nn ⊂ N, such that |Nn | = n and sup Nn < inf Nn+1 . Deﬁne Gn = G(Nn ) using the above construction. Note that if xn ∈ Gn and if xm ∈ Gm , then xn and xm are independent functions, for m = n. Repeating the proof of the estimates (2.1.13) shows that for any sequence of functions xn ∈ Gn , the following equivalence of norms holds true, max ||xn ||2BMO +

2 ||xn ||22 ≤ xn

BMO

≤ 2 max ||xn ||2BMO + 2

||xn ||22 . (2.1.14)

2.1. Complemented subspaces

135

Note that the orthogonal projection onto span{Gn , n ∈ N} is given by the operator P = PNn . By the equivalence of norms (2.1.14) one obtains that P is a bounded operator on BMO. Indeed, let f ∈ BMO, then PNn (f ) ∈ Gn , hence

P (f ) 2BMO =

PNn (f ) 2BMO ≤ 2 max PNn (f ) 2BMO + 2

PNn (f ) 22 ≤ 2 f 2BMO + 2 f 22 . As f 2 ≤ f BMO , this shows that P (f ) BMO ≤ 2 f BMO . Summing up we proved our claim that ( BMOn )ind is isomorphic to a complemented subspace of BMO. It is not known whether the Rosenthal space and the independent sum of BMOn are isomorphic Banach spaces. Problem 2.1.9. Prove that R is not isomorphic to ( BMOn )ind . In Chapter 5 we study structural results on complemented subspaces of H 1 . One of them states the following dichotomy: A complemented subspace of H 1 2 either 1contains a copy of , or it is isomorphic to a complemented subspace of ( Hn )1 . To arrive at that result we use that the members of the following family of subspaces are well complemented in BMO. We ﬁx p < q and let S(p, q, M ) = span{hJ : 2−q ≤ |J| < 2−p }. We deﬁne the unit ball in S(p, q, M ) as the intersection of the unit ball of BM O and 1/M times the unit ball of L2 . This is achieved using the norm max{M ||x||L2 , ||x||BMO }. The dual space of S(p, q, M ) consists of T (p, q, M ) = span{hJ : 2−q ≤ |J| < 2−p }, where the norm of x ∈ T (p, q, M ) is deﬁned as |x|T (p,q,M ) = inf{||y||H 1 + ||z||L2 M −1 : x = y + z}. Proposition 2.1.10. If M ≤ 2p/2 , then the space S(p, q, M ) is isomorphic to a complemented subspace of BMO. Embedding and projection have norm bounded by 8. By duality T (p, q, M ) is isomorphic to a complemented subspace of H 1 , again embedding and projection have norm bounded by 8.

136

Chapter 2. Projections, Isomorphisms and Interpolation

Proof. By Theorem 2.1.8 it suﬃces to construct embedding and projection between S(p, q, M ) and ( BMOn )ind . We let A = {J : J dyadic |J| = 2−p }. Then for x ∈ S(p, q, M ) we have the simple and useful identity max{M ||x||L2 , ||x||BMO } = max

⎧ ⎨ ⎩

1/2 M 2 ||x · 1J ||2L2

J∈A

,

max ||x1J ||BMO J∈A

⎫ ⎬ ⎭

.

Recall that we chose M 2 ≤ 2p , and that the support of x · 1J has measure ≤ 2−p . Therefore we may ﬁnd a linear map of slope ≤ M 2 mapping J into [0, 1]. This linear change of variables maps the function x · 1J into an element zJ ∈ BMOn such that, ||zJ ||L2 = M ||x · 1J ||L2 , ||zJ ||BMO = ||x · 1J ||BMO . In this way we obtain an embedding T : S(p, q, M ) →

BMOn

, ind

x → (zJ )J∈A

for which max{M ||x||L2 , ||x||BMO } ≤ ||T x||( BMOn )ind ≤ 2 max{M ||x||L2 , ||x||BMO }. Note that the orthogonal projection onto the range of T is (trivially) bounded on ( BMOn )ind with norm = 1. Thus by Theorem 2.1.8, S(p, q, M ) is isomorphic √ to a complemented subspace of BMO. The norm of the embedding is ≤ 2 2 the norm of the projection is bounded by 2. Sequences of uniformly complemented subspaces. Given a sequence of ﬁnite dimensional Banach spaces {En }, we say that {En } is well isomorphic to uniformly complemented subspaces of F if there exist linear operators An : En → F and Bn : F → En so that Id −→ En En An Bn F

and

sup An · Bn < ∞.

n∈N

Equivalently we say that F contains uniformly complemented copies of {En }. Let {pn }, {qn } and {Mn } be sequences of natural numbers so that pn ≤ qn and Mn ≤ 2pn /2 . Then, by Proposition 2.1.10 the spaces {T (pn , qn , Mn )} are well isomorphic to uniformly complemented subspaces of H 1 . (Or equivalently, H 1 contains uniformly complemented copies of {T (pn , qn , Mn )}.)

2.1. Complemented subspaces

137

Weighted intersections The next class of examples is obtained by intersecting ∞ with a weighted version of BMO. The weight is given by a ﬁxed sequence of scalars, 0 ≤ xI ≤ 1. We deﬁne the space S ∞ to consist of all bounded sequences (aI ) for which the following norm is ﬁnite, ||(aI )||S ∞ = || xI aI hI ||BMO + sup |aI |. The norm in S ∞ depends of course explicitly on the chosen weight. For example, when xI = 1 for all dyadic intervals I, then the norm of S ∞ is equivalent to the norm of BMO. When the weight disappears, that is when xI = 0, for all I, then the norm of S ∞ is equivalent to that of ∞ . When xI ∈ {0, 1}, then S ∞ is the direct sum of ∞ and the span of a subsequence of the Haar basis. Our notation, however, suppresses this dependence. In Chapter 4 we will give the isomorphic classiﬁcation of the spaces S ∞ . It turns out that BMO and ∞ are the only isomorphic types which appear in this class. The ﬁrst step in this direction is the following result. Theorem 2.1.11. The space S ∞ is isomorphic to a complemented subspace of BMO. The norm of embedding and projection are independent of the choice of scalars, 0 ≤ xI ≤ 1. Proof. The proof consists of two steps. It starts by showing that for every choice of scalars 0 ≤ xI ≤ 1, there exists a sequence gI ∈ BMO which spans a subspace of BMO that is isomorphic to S ∞ . Here the construction of the functions gI reﬂects the given sequence of weights {xI }, so that the norm of the isomorphism is independent of the given weight. In the second step we show that the orthogonal projection gI gI P (f ) = f, ||gI ||2 ||gI ||2 I

is bounded in BMO. For convenience we assume that each of the numbers xI is a negative power of 2. Then depending on the weights {xI } we choose a rapidly increasing sequence of natural numbers 1 << m0 << m1 << m2 · · · . Below it will become clear how rapidly increasing this sequence has to be. Let Ai be the dyadic intervals of length 2−mi . First deﬁne the collection C[0,1] = A0 . Then select a collection B[0,1] ⊆ C[0,1] such that x2[0,1] = |B[0,1] |, where B[0,1] is the point-set covered by B[0,1] . We can make this choice, since we are assuming that x[0,1] is a negative power of 2, and that 1 << m0 . Denote by I1 the left half of the unit interval and by I2 the right half of the unit interval. Next ﬁx J ∈ C[0,1] , and select CI1 (J) ⊂ A1 and CI2 (J) ⊂ A1 so that the point-set covered by CIi is contained in J and has measure equal to |J|/2. Moreover we select the intervals in such a way that the relative density of CIi (J) is the same on

138

Chapter 2. Projections, Isomorphisms and Interpolation

the left half of J and on the right half of J, that is, 1 |Jj |, i, j ∈ {1, 2}, 2 where CIi (J) is the point-set covered by CIi (J), and where J1 is the left half of J, and J2 is the right half of J. Next we select BI1 (J) ⊆ CI1 (J), and BI2 (J) ⊆ CI2 (J) such that 1 2 |BI1 (J)| xI 1 = , 2 |J| and 1 2 |BI2 (J)| xI 2 = , 2 |J| where BIi (J) is the point-set covered by the collection BIi (J). Now we take the union over J ∈ C[0,1] . This gives for i ∈ {1, 2}, the collections CIi = CIi (J) and BIi = BIi (J). |CIi (J) ∩ Jj | =

J∈C[0,1]

J∈C[0,1]

We continue by induction following the pattern of the ﬁrst step. Thus for each I we construct two families, BI and CI , consisting of pairwise disjoint dyadic intervals satisfying the following properties: The collection BI is a subcollection of CI . Let CJ be the point-set covered by the collection CJ . Then |CJ | = |J|, and moreover the family {CI } is a nested family of sets, so that CI ⊆ CJ if I ⊆ J. Note that if K ∈ CI and L ∈ CJ and K ⊆ L, then we have CI ⊆ CJ , or equivalently, I ⊆ J. Moreover we observe the following: Fix a dyadic interval J, and let gJ = {hK : K ∈ BJ }. Suppose that I is a dyadic interval which is strictly contained in J . Then for K ∈ BJ , we have the following identities expressing a strong degree of self similarity, 1 1 |I| = x2I . (2.1.15) g 2 = ||gI ||22 |K| K I |J| |J| Having made these observations it is easy to show that the orthogonal projection P deﬁned by gI gI P (f ) = f, ||gI ||2 ||gI ||2 I

is bounded in BMO. To prove boundedness of P, we ﬁx f ∈ BMO and let g = P (f ). Then we ﬁx a dyadic interval J, and we let K ∈ BJ . We calculate using the observation in (2.1.15), 1 gI2 1 gI 2 1 |g − g|2 = f, |K| K |K| K ||gI ||2 |K| K ||gI ||22 I⊆J (2.1.16) gJ 2 1 gI 2 1 . = f, + f, ||gJ ||2 ||gJ ||22 ||gI ||2 |J| I⊂J

2.2. Pelczy´ nski’s decomposition method

139

We continue estimating the ﬁrst summand of the second line in (2.1.16). Note that ||gJ ||22 = |BJ |, and that the Haar support of gJ is the collection BJ . Hence by Bessel’s inequality and Cauchy–Schwarz we estimate as follows. . /2 1 1 gJ 2 1 f, = f, hK 2 ||gJ ||2 ||gJ ||2 |BJ | |BJ | K∈BJ . / . / 1 1 2 −1 |K| · f, hK |K| ≤ |BJ | |BJ | K∈BJ

K∈BJ

≤ C||f ||2BMO . Now we turn to estimating the second summand in the second line of (2.1.16). Recall that for a given interval J the support of the function gJ is contained in the set CJ . Recall also that |CJ | = |J|, and that the collection {CI } is a nested family of sets. Now we estimate using Bessel’s inequality and the deﬁnition of BMO. I⊂J

f,

1 gI 2 1 ≤ f, hK 2 |K|−1 ||gI ||2 |J| |J| K⊆CJ

≤ ||f ||2BMO . Thus we showed that the orthogonal projection P is bounded on BMO. With a similar calculation we evaluate now the norm in BMO of a linear combination g = aI gI . To this end we ﬁx a dyadic interval J and K ∈ BJ . Then we calculate using the observation (2.1.15), 1 1 1 |g − g|2 = a2 g 2 |K| K |K| K |K| K I I I⊆J

= a2J +

a2I x2I

I⊂J

|I| . |J|

From this identity we deduce easily the following equivalence of norms, ||g||BMO ∼ || aI xI hI ||BMO + sup |aI |. Summing up, we showed that the weak *-closed linear span of {gI } is isomorphic to S ∞ , and complemented in BMO.

2.2

Pelczy´ nski’s decomposition method

We now present an extremely ﬂexible and powerful tool to prove that two Banach spaces are isomorphic: the decomposition method of A. Pelczy´ nski. In this monograph we will encounter several examples where the decomposition method

140

Chapter 2. Projections, Isomorphisms and Interpolation

permits us to show that two given Banach spaces are isomorphic, even when the explicit determination of the isomorphism is unknown. Let X be a Banach space and let 1 ≤ p ≤ ∞. The p direct sum of X is the Banach space of all sequences {fi : i ∈ N} with fi ∈ X such that {||fi || : i ∈ N} ∈p . We denote it by ( X)p . The norm of a given sequence {fi : i ∈ N} in ( X)p is deﬁned as follows,

||fi ||pX

1/p

sup ||fi ||X

when

1 ≤ p < ∞,

when

p = ∞.

With these ingredients we can formulate and prove the decomposition method. Theorem 2.2.1. (A. Pelczy´ nski.) 1 ≤ p ≤ ∞. Let X and Y be Banach spaces and suppose that X is isomorphic to its p direct sum. Then the following holds. If X is isomorphic to a complemented subspace of Y , and Y is isomorphic to a complemented subspace of X, then X is isomorphic to Y . Proof. If X is isomorphic to a complemented subspace of Y , then (by deﬁnition) there exists a Banach space E such that Y ∼ E ⊕ X. Now we show that Y is isomorphic to X ⊕ Y . Indeed Y ∼E⊕Y ∼E⊕ X p ∼E⊕X ⊕ X p

∼ Y ⊕ X. Next we show that X is isomorphic to Y ⊕ X. We write X ∼ F ⊕ Y and then, X∼ X p ∼ F ⊕Y p ∼ ⊕Y F ⊕ Y p p ∼ F ⊕Y ⊕Y p

∼ X ⊕ Y.

Inﬁnite direct sums. The development of Banach space theory, as recorded for instance in [130], testiﬁes to the ﬂexibility and extreme power of A. Pelczy´ nski’s decomposition method. In this book we use it frequently to show that a given requires space is isomorphic to H 1 . The hypothesis of the decomposition method then that we know that H 1 is isomorphic to its 1 direct sum, denoted ( H 1 )1 . This motivates our next theorem.

2.2. Pelczy´ nski’s decomposition method

141

Theorem 2.2.2. H 1 is isomorphic to ( H 1 )1 . Proof. Clearly, ( H 1 )1 is isomorphic to its own 1 direct sum, and contains a complemented copy of H 1 . We proceed by constructing a complemented subspace 1 of H isomorphic to ( H 1 )1 . Fix a sequence of pairwise disjoint dyadic intervals {In : n ∈ N}. Then we let Cn = {J ∈ D : J ⊆ In }. The closure of span{hJ : J ∈ Cn } is isomorphic to H 1 . Moreover for fn ∈ span{hJ : J ∈ Cn } we have the identity fn 1 . ||fn ||H 1 = H # Hence the closure of X = span{hJ : J ∈ Cn } in H 1 is isomorphic to ( H 1 )1 . The restriction operator aI h I → aI h I R: n I∈Cn

I∈D

is of norm 1, and provides aprojection onto X. By Pelczy´ nski’s decomposition principle, this implies that ( H 1 )1 is isomorphic to H 1 . The following result of P. Wojtaszczyk provides a surprising representation of BMO. Theorem 2.2.3. BMO is isomorphic to ( BMOn )∞ . Proof. By duality Theorem 2.2.2 implies that BMO is isomorphic to ( BMO)∞ . Hence ( BMOn )∞ is isomorphic to a complemented subspace of BMO. By the decomposition principle of A. Pelczy´ nski it remains to show that BMO is isomorphic to a complemented subspace of ( BMOn )∞ . We start by deﬁning the embedding. Let Dn = {I ∈ D : |I| ≥ 2−n }. For f ∈ BMO, we let hJ hJ . fn = f, |J| J∈Dn

The sequence (fn ) converges in the weak-∗ topology of BMO and the limit is f. Moreover ||f || = supn ||fn ||. Thus the operator E : BMO → ( BMOn )∞ f → (fn ), is bounded with ||E|| = 1. Next we deﬁne an operator R:( BMOn )∞ → BMO (xn ) → R((xn )) satisfying the following property: If (xn ) is a weak-∗ convergent sequence with xn ∈ BMOn , and if xn → x in σ(BMO, H 1 ),

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Chapter 2. Projections, Isomorphisms and Interpolation

then R((xn )) = x. Any operator with this property inverts the action of E. Indeed, for f ∈ BMO, (Ef )n → f in the σ(BMO, H 1 ) topology. Therefore REf = R((Ef )n ) = f. It remains to show the existence of the operator R. For that we use a compactness argument. We let B be the unit ball of ( BMOn )∞ and let C be the unit ball of BMO. Both sets are equipped with their respective weak-∗ topologies, thus they are compact. Now consider the maps Rm : B → C (xn ) → xm . By Tychonov’s theorem the product space C B is compact and therefore the maps Rm have a cluster point in C B . We let R : B → C be such a cluster point. Clearly additivity and homogeneity of the maps Rm transfer to the cluster point Rwhich can therefore be extended linearly to a bounded linear operator R : ( BMOn )∞ → BMO. Moreover as R is a cluster point of the sequence {Rm }, it satisﬁes R((xn )) = x, if xn ∈ BMOn , and

xn → x in σ(BMO, H 1 ).

H 1 with values in 2n . Next we introduce the Banach space of H 1 functions taking values in 2n (the n-dimensional Euclidean space). An integrable f : [0, 1] → 2n with Haar series f (t) = hI (t)xI , xI ∈ 2n belongs to H 1 (2n ) iﬀ its square function 1/2 h2I (t)||xI ||22 is integrable. With the norm 1 0

h2I (t)||xI ||22

1/2 dt,

the space H 1 (2n ) is clearly a Banach space. We will show now that H 1 and H 1 (2n ) are isomorphic in such a way that the norm of the isomorphism and its inverse is bounded independently of n. By contrast H 1 and H 1 (2 ) are two Banach spaces which are not isomorphic. This important theorem of J. Bourgain will be presented in Chapter 5.

2.2. Pelczy´ nski’s decomposition method

143

Theorem 2.2.4. For every n ∈ N, there exists an isomorphism T : H 1 → H 1 (2n ) such that ||T || · ||T −1 || < C, where C is independent of n. Proof. Clearly H 1 is isomorphic to a complemented subspace of H 1 (2n ), and by nski’s decomTheorem 2.2.2, H 1 is isomorphic to its 1 sum. Hence by Pelczy´ position method, the existence of the isomorphism is guaranteed when we show that H 1 (2n ) is isomorphic to a complemented subspace of H 1 , and the norm of embedding and projection are bounded independently of n. We will now present the construction. We ﬁx n ∈ N. Then we pick a sequence sm ∈ N and a sequence of intervals Rm ⊆ N such that the cardinality of Rm equals n, and sm−1 < inf Rm < sup Rm < sm . We ﬁrst use the sequence {sm } and the Rademacher functions {rsm }, to construct a block basis equivalent to the Haar system as follows. Put k[0,1] = rs1 , and suppose that kJ has been deﬁned for all dyadic intervals of length ≥ 2−m , then we take a dyadic interval J of length equal to 2−m . Split J into dyadic intervals J1 , J2 of length 2−m−1 , and deﬁne kJ1 = 1{kJ =1} rsm+1 , kJ2 = 1{kJ =−1} rsm+1 . In this way we construct kJ for each dyadic interval J. Next we attach to kJ a set of Rademacher functions to produce copies of 2n . Fix i ≤ n, and denote the i-th element in the set (of integers) Rm by i(m). Then, for a dyadic interval J of length 2−m we deﬁne x(J, i) = kJ ri(m) . Our ﬁrst claim is that the orthogonal projection onto span{x(J, i) : J ∈ D, i ≤ n} is bounded on H 1 with norm ≤ 8. The orthogonal projection is given explicitly by P (f ) =

n , J∈D i=1

f,

x(J, i) ||x(J, i)||2

-

x(J, i) . ||x(J, i)||2

To prove the norm estimate we only have to observe that the collections CJ,i = {K ∈ D : hK , x(J, i) = 0},

J ∈ D, i ≤ n,

(can be ordered to) satisfy Jones’ compatibility condition (J).

144

Chapter 2. Projections, Isomorphisms and Interpolation

Our second claim is that in H 1 the closure of span{x(J, i) : J ∈ D, i ≤ n} is isomorphic to H 1 (2n ). We ﬁx a sequence of scalars a(J, i), and observe the identity 1/2 1/2 n n 2 2 2 2 x(J, i) a(J, i) = kJ (s) a(J, i) ds J∈D i=1

i=1

J∈D

=

1/2

kJ2 (s)||aJ ||2

ds,

J∈D

where ||aJ ||22 = ni=1 a(J, i)2 . Finally we observe that {kJ : J ∈ D} has the same joint distribution as the Haar system. Hence n x(J, i)a(J, i) = h J aJ J∈D i=1

H1

J∈D

H 1 (2n )

where aJ = (a(J, i), . . . , a(J, n)). This completes the proof that H 1 (2n ) is isomorphic to a complemented subspace of H 1 . The norm of embedding and projection are bounded independent of n.

2.3

Interpolation of operators

In this section we present powerful methods of complex interpolation. We give applications to averaging projections and rearrangement operators. The Riesz convexity theorem is a widely used interpolation result. In the most basic form its statement involves a single operator T, bounded and deﬁned both on L1 and L∞ . With ||T ||Lp = ||T : Lp → Lp || the conclusion of the Riesz convexity theorem is that for 1 ≤ p ≤ ∞ and θ = 1/p, θ ||T ||Lp ≤ ||T ||1−θ L1 ||T ||L∞ .

To illustrate this result we prove it for ﬁnite-dimensional spaces pn , consisting of Rn equipped with the norm

n 1/p |xi |p .

x p = i=1

Let the linear operator T on pn be given by the matrix (tj,k )nj,k=1 . For x ∈ Rn the j-th component of T x is just nk=1 tj,k xk . Hence

T x pp =

n n | tj,k xk |p . j=1 k=1

2.3. Interpolation of operators

145

Rewrite the right-hand side, multiplying and dividing by the factor ( nl=1 |tj,l |)p . Thus we prepare for the application of Minkowski’s inequality. We obtain the identity /p n . n n t p n j,k |tj,l | |

T x p = xk |p . |t | j,l l=1 j=1 l=1

k=1

For each ﬁxed j ≤ n apply Minkowski’s inequality in the form |

n k=1

|tj,k | t n j,k n xk |p ≤ |xk |p . l=1 |tj,l | l=1 |tj,l | n

k=1

We obtain this upper bound,

T x pp

≤

. n n j=1

≤ sup

j≤n

.

/p−1 |tj,l |

l=1 n

n

|tj,k ||xk |p

k=1

/p−1 ⎡

⎣sup

|tj,l |

n

k≤n j=1

l=1

⎤ |tj,k |⎦

n

|xk |p .

k=1

Taking the p-th root on both sides gives that p−1

1

p

T x p ≤ M1 p M∞

x p ,

n n where M∞ = supk≤n j=1 |tj,k | and M1 = supj≤n l=1 |tj,l |. To complete the proof of the Riesz convexity theorem it remains to observe that T : 1n → 1n = ∞ ∞ is replaced by L2 , then the M1 , and T : ∞ n → n = M∞ . If the endpoint L theorem of Riesz asserts that for 1 ≤ p ≤ 2 and θ = 2 − 2/p, θ ||T ||Lp ≤ ||T ||1−θ L1 ||T ||L2 .

In this section we prove interpolation theorems parallel to the Riesz convexity theorem, where we use H 1 in place of L1 . The averaging projections P (f ) = where ki =

f,

ki ki , ||ki ||2 ||ki ||2

{hI : I ∈ Ci },

were studied in Section 1.5 as operators on H 1 . There we showed (Theorem 1.5.9), that P is bounded on H 1 if the deﬁning collections of dyadic intervals Ci satisfy Jones’s compatibility condition. On the other hand being an orthogonal projection, P is bounded on the space L2 with norm ≤ 1, as long as the collections {Ci } are pairwise disjoint. With interpolation theorems for single operators acting between

146

Chapter 2. Projections, Isomorphisms and Interpolation

Hardy spaces we ﬁnd that P is bounded on H p , for 1 < p < 2, satisfying the norm estimate p ||P (f )||H p ≤ C||P ||1−θ H 1 ||f ||H , where θ = 2 − 2/p. (Recall that for f = aI hI , we denote S(f ) = ( a2I h2I )1/2 and f H p =

S(f ) Lp .) The family of rearrangement operators of the Lp normalized Haar system is a basic example where interpolation theorems for a single operator are not quite suﬃcient; more reﬁned tools of complex interpolation are used instead. A rearrangement operator is determined by an injective map τ acting on dyadic intervals. By linear extension it induces a rearrangement of the Lp normalized Haar system, hτ (I) hI . (2.3.1) Tp : 1/p → |I| |τ (I)|1/p On a given Haar function hI the operator Tp acts as follows, Tp (hI ) = hτ (I)

|I| |τ (I)|

1/p .

The deﬁnition of Tp depends on both the underlying rearrangement and the value of p. We use results of complex interpolation to prove that for the operators {Tp : 1 < p < 2} the boundedness on Lp is actually determined by the rearrangement alone. Precisely, we will prove that if there exists 1 < p0 < 2, such that Tp0 is bounded on Lp0 , then for all 1 < p < 2, the operator Tp is already bounded on Lp . The methods presented in this section are based on ideas from two distant areas of real and complex analysis. These include: (a) Stopping time arguments and a procedure for eﬀectively decomposing functions. This idea already appeared in the proof of the atomic decomposition of H 1 . See Theorem 1.2.4. (b) A maximum principle for subharmonic functions valid in unbounded domains such as the vertical strip V = {x + iy : x ∈ (0, 1), y ∈ R}. We summarize now the results of this section. 1. We identify the Calder´ on product of H 1 and H 2 . Let 1 < p ≤ 2 and θ = p 2 − 2/p. For u ∈ H with u = 0 the norm satisﬁes θ x(I)1−θ y(I)θ hI } ≤ C||u||H p , (2.3.2) c||u||H p ≤ inf{||x||1−θ H 1 ||y||2 : u = y(I)hI . In the inequalities (2.3.2) the conwhere x = x(I)hI and y = stants c > 0 and C > 0 are independent of p. By deﬁnition the Calder´ on product of H 1 and H 2 is given by the central expression in (2.3.2).

2.3. Interpolation of operators

147

1 1 2. We present G. Pisier’s norm on H . Let g ∈ H have Haar expansion g = g(I)hI . Let 1 < p < 2 be ﬁxed and let θ = 2 − 2/p. Then the norm of g in H 1 can be evaluated via H p (1 < p < 2) and L2 in the following way. c||g||1−θ g(I)1−θ w(I)θ hI ||H p ≤ ||g||1−θ (2.3.3) H 1 ≤ sup || H1 ,

where the supremum is extended over all w ∈ L2 with ||w||2 ≤ 1, and Haar expansion w = w(I)hI . 3. Let 1 < p < 2. For Tp , deﬁned by (2.3.1), denote ||Tp ||p = ||Tp : H p → H p ||. Then ≤ C||Tp ||p , where θ = 2 − 2/p. c||Tp ||p ≤ ||T1 ||1−θ 1 4. Let 1 < p < 2 and θ = 2 − 2/p. On H p the norm of an analytic family of operators {Jz } satisﬁes the convexity estimate ||Jθ (f ))||H p ≤ CR01−θ R1θ ||f ||H p , where R0 = supt ||Jit ||H 1 , and R1 = supt ||J1+it ||L2 . A crucial link between the renormings of H p (1 < p < 2) and G. Pisier’s norm on H 1 is given by Feﬀerman’s inequality as formulated in Theorem 2.3.1 below. For instance, both, the inequalities (2.3.24) and the norm (2.3.3) express converse estimates to (2.3.4) thereby showing its sharpness. Theorem 2.3.1. Let x ∈ H 1 , v ∈ BMO, and let w, y, ∈ L2 , be ﬁnitely supported over the Haar system. For θ ∈ (0, 1), form the functions u= x(I)1−θ y(I)θ hI and f = v(I)1−θ w(I)θ hI , where x(I) = x, hI /|I|, y(I) = y, hI /|I|, v(I) = v, hI /|I| and w(I) = w, hI /|I|. Then, 1 1−θ √ uf ≤ 2 2||x||H 1 ||v||BMO (||y||L2 ||w||L2 )θ . (2.3.4) 0

Proof. First we apply H¨ older’ s inequality with the parameters p = 1/(1 − θ) and q = 1/θ. Then we use Feﬀerman’s inequality and the Cauchy–Schwarz inequality as follows. 1 1−θ 1−θ θ θ = uf x(I) v(I) w(I) y(I) |I| 0

1−θ θ |x(I)v(I)||I| |w(I)y(I)||I| √ 1−θ θ ≤ 2 2||x||H 1 ||v||BMO (||y||L2 ||w||L2 ) . ≤

148

Chapter 2. Projections, Isomorphisms and Interpolation

Calder´ on’s product of H 1 and H 2 Recall that on Hp (1 ≤ p < ∞) the norm is deﬁned as follows. Given an integrable function f with f = 0, let S(f ) = (

f (I)2 h2I )1/2

where

f (I) = f,

I

and put

1/p

1

f H p =

hI , |I|

S(f )

p

.

0

In this subsection we will identify the Calder´ on product of H 1 and H 2 by proving the estimates stated in (2.3.2). We start with the left-hand side of (2.3.2). Its proof consists of two applications of H¨ older’s inequality. 1 2 Proposition 2.3.2. Let 1 < p≤ 2, and θ = 2 − 2/p. Then for x ∈ H , and y ∈ L , with Haar expansion x = x(I)hI and y = y(I)hI , the following estimate holds, θ x(I)1−θ y(I)θ hI p ≤ ||x||1−θ H 1 ||y||2 . H

Proof. Fix x ∈ H , and let y ∈ L , be ﬁnitely supported over the Haar system. Let u= x(I)1−θ y(I)θ hI . 1

2

First use H¨older’s inequality for sequence spaces with the conjugate exponents 1θ 1 and 1−θ . This shows that the following estimate holds pointwise for the square function of u, (2.3.5) S(u) ≤ S(x)1−θ S(y)θ . Next apply H¨ older’s inequality for integrals to the conjugate exponents 2 . This gives pθ

1 0

p(1−θ)

1

S(x)p(1−θ) S(y)pθ ≤ 0

and

pθ/2

1

S(y)2

S(x)

1 p(1−θ)

.

(2.3.6)

0

Combining the estimates (2.3.6) and (2.3.5) we obtain that θ ||u||H p ≤ ||x||1−θ H 1 ||y||2 .

Next we verify that the right-hand side estimate of (2.3.2) holds true. We will show that for u ∈ H p the Haar coeﬃcient, u(I) = u, hI /|I|, admits a factorization, u(I) = x(I)1−θ y(I)θ , so that the norm of x = x(I)hI and y = y(I)hI satisfy θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

2.3. Interpolation of operators

149

We constructively determine x ∈ H 1 and y ∈ L2 with these properties using the following decomposition theorem that is based on stopping time arguments. Theorem 2.3.3. Let 1 ≤ s < ∞. Let g ∈ H s with g = 0. There exists a sequence of functions {gi } disjointly supported over the Haar system such that g=

∞

gi ,

i=1

and so that with Bi = ||S(gi )||∞ , the series −1+s/2 gi Bi , x= gi Bi−1+s and y = are converging respectively in H 1 , and L2 . They satisfy the norm estimates, √ s/2 ||x||H 1 ≤ 8 · 22s ||g||sH s and ||y||2 ≤ 8 · 2s ||g||H s . Proof. Let 1 ≤ s < ∞, and pick g ∈ H s with g = 0. We will now deﬁne the decomposition of g using a stopping time argument based on the size of the square function S(g). We start with the unit interval [0, 1]. Let n([0, 1]) be the smallest integer such that Ω([0, 1]) = {t ∈ [0, 1] : S(g)(t) ≥ 2n([0,1]) } satisﬁes

1 . 4 Let F([0, 1]) be the collection of dyadic intervals which are contained in Ω([0, 1]). Then deﬁne C([0, 1]) = Q([0, 1]) \ F([0, 1]), |Ω([0, 1])| ≤

and g[0,1] =

J∈C([0,1])

g,

hJ hJ . |J|

Note that the square function of g[0,1] satisﬁes the pointwise estimate S(g[0,1] ) ≤ 2n([0,1]) .

(2.3.7)

Next deﬁne the collection of pairwise disjoint dyadic intervals, E1 = max F([0, 1]). We continue the construction inductively. Suppose we have already deﬁned collections of pairwise disjoint dyadic intervals E1 , . . . , Ei . Fix I ∈ Ei . Let n(I) be the smallest integer such that the set Ω(I) = {t ∈ I : S(g)(t) ≥ 2n(I) }

150

Chapter 2. Projections, Isomorphisms and Interpolation

satisﬁes

1 |I|. 4 Let F(I) be the collection of dyadic intervals which are contained in Ω(I). Then deﬁne C(I) = Q(I) \ F(I). |Ω(I)| ≤

Note that gI =

g,

J∈C(I)

hJ hJ |J|

satisﬁes the pointwise estimate, S(gI ) ≤ 2n(I) ,

(2.3.8)

and that the support of S(gI ) is contained in I. Now we put E(I) = max F(I), and taking unions over I ∈ Ei we deﬁne Ei+1 = {E(I) : I ∈ Ei }. Thus the collection Ei+1 consists of pairwise disjoint dyadic intervals. This completes the construction part of the proof. # We now analyze consequences resulting from this decomposition. We put E = Ei . First notice that for any I ∈ E, 1 (2.3.9) |J| ≤ |I|. 2 J∈E∩I,J=I

We will next verify that the following convexity relation holds, |I|2sn(I) ≤ 22s+3 ||S(g)||ss .

(2.3.10)

I∈E

We start the veriﬁcation of (2.3.10) with this crucial observation: The stopping time deﬁnition of n(I) implies the measure estimate |{t ∈ I : S(g)(t) ≥ 2n(I)−1 }| ≥

|I| . 4

Multiplying the above line with 2sn(I) , and taking the sum over all I ∈ E, gives |I|2sn(I) ≤ 4 2sn(I) |{t ∈ I : S(g) ≥ 2n(I)−1 }| I∈E

I∈E

≤ 4 · 2s

2s(n(I)−1) |{t ∈ I : S(g) ≥ 2n(I)−1 }|

I∈E

≤4·2

s

2ks |{S(g) ≥ 2k }|

k∈N

≤8·2

2s

S s (g).

2.3. Interpolation of operators

151

Hence (2.3.10) is veriﬁed. The pieces {gI : I ∈ E} are disjointly supported over the Haar system. Therefore S 2 (gI ), S 2 (g) = I∈E

and the decomposition g = gI converges unconditionally in H s . We relabel {gI : I ∈ E} as {gi : i ∈ N}, and accordingly we rename {2n(I) : I ∈ E} as {Bi : i ∈ N}. −1+s/2 Next we verify the L2 norm estimate for y = gi Bi . By (2.3.8) we have that S(gi ) ≤ Bi . Hence the convexity relation (2.3.10) gives the estimates −1+s/2 2 || gi Bi ||2 = S 2 (gi )Bi−2+s (2.3.11) ≤ |I|2sn(I) I∈E

≤ 8 · 22s ||S(g)||ss . gi Bi−1+s , we use triangle To obtain the estimate for the H 1 norm of x = inequality, (2.3.8) and the convexity relation (2.3.10), S(gi )Bi−1+s || gi Bi−1+s ||H 1 ≤ (2.3.12) ≤ |I|2sn(I) I∈E

≤ 8 · 22s ||S(g)||ss .

Remark. Let z = gi Bi−1 , then ||z||BMO ≤ 2. This follows from the pointwise estimate S(gi ) ≤ Bi and the fact that E satisﬁes 2-Carleson condition (2.3.9). || gi Bi−1 ||BMO ≤ [[E]]1/2 sup Bi−1 ||gi ||BMO i (2.3.13) ≤ 2. By simple arithmetic we will verify below that the decomposition of Theorem 2.3.3 yields the right-hand side estimate of (2.3.2). Theorem 2.3.4. Let 1 ≤ p ≤ 2 and θ = 2 − 2/p. For every u ∈ H p with u = 0, there exist x ∈ H 1 and y ∈ L2 , such that θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H ,

and u=

x(I)1−θ y(I)θ hI ,

where x(I) and y(I) denote the Haar coeﬃcients of x, respectively y.

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Proof. Let 1 < p < 2 and θ = 2 − 2/p. To u ∈ H p we apply the decomposition Theorem 2.3.3. Thus u can be decomposed as u= ui , and with Bi = ||S(ui )||∞ we form x= ui Bi−1+p

and

y=

−1+p/2

ui Bi

.

(2.3.14)

To prove the theorem we show that the functions u ∈ H p , y ∈ L2 , and x ∈ H 1 are related through their Haar coeﬃcients and by a convexity estimate for their norms. We claim that the following holds, x, hI 1−θ y, hI θ = u, hI and

θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

θ 5 p We start by verifying the convexity estimate ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H . Recall the upper bounds established in Theorem 2.3.3,

||x||H 1 ≤ 22p+3 ||u||pH p

p/2

and ||y||2 ≤ 2p+3/2 ||u||H p .

Multiplying we ﬁnd that p(1−θ)

θ ||x||1−θ H 1 ||y||2 ≤ A||u||H p

pθ/2

||u||H p ,

where A = 2(2p+3)(1−θ) 2(p+3/2)θ . Using that θ = 2 − 2/p we calculate, p(1 − θ) + pθ/2 = 1

and

A ≤ 25 .

Next we check that the Haar coeﬃcients satisfy the identity x, hJ 1−θ y, hJ θ = u, hJ .

(2.3.15)

To this end let J be a dyadic interval in the Haar support of u. The construction of Theorem 2.3.3 deﬁned the functions {ui }, by splitting the Haar support of u. Hence for the given J there exists a unique i such that J is in the Haar support of ui . Then, x, hJ = Bi−1+p u, hJ ,

and

−1+p/2

y, hJ = Bi

u, hJ .

Raise the coeﬃcients to the respective powers 1 − θ and θ, and multiply them. This gives (−1+p)(1−θ) (−1+p/2)θ x, hJ 1−θ y, hJ θ = Bi Bi u, hJ . With our choice of θ = 2 − 2/p we have p (−1 + p)(1 − θ) + (−1 + )θ = 0. 2 Inserting gives (−1+p)(1−θ)

Bi

(−1+p/2)θ

Bi

= 1,

hence the claimed relation (2.3.15) between the Haar coeﬃcients.

The renorming of (2.3.2) follows from Theorem 2.3.4 and Proposition 2.3.2.

2.3. Interpolation of operators

153

Pisier’s norm on H 1 Here we specialize to H 1 the norm invented by G. Pisier in [176]. To this end we carry further the ideas introduced in the course of proving the renorming (2.3.2). Our treatment is guided by the interpretation of G. Pisier’s norm as a uniqueness result for the endpoint of complex interpolation scales of Banach lattices, due to M. Cwikel, P. Nilsson and G. Schechtman [57]. g(I)hI . For 1 < p < 2, Theorem 2.3.5. Let g ∈ H 1 have Haar expansion g = and θ = 2 − 2/p, % & 1−θ θ c||g||1−θ ≤ sup g(I) w(I) h : ||w|| ≤ 1 ≤ ||g||1−θ (2.3.16) 1 I 2 H H1 , p H

−5

where c = 2

.

Proof. The ingredients of the proof are Proposition 2.3.2 for the right-hand side of (2.3.16), and the stopping time decomposition of Theorem 2.3.3 for the left-hand side. g(I)hI , and let 1 < p < 2. We Fix g ∈ H 1 with Haar expansion g = show ﬁrst that the right-hand side inequality holds. Choose w ∈ L2 with Haar expansion w = w(I)hI . Then put u= g(I)1−θ w(I)θ hI . Proposition 2.3.2 gives an upper estimate for the norm of u ∈ H p , θ ||u||H p ≤ ||g||1−θ H 1 ||w||2 .

This proves that the right-hand side of (2.3.16) holds true. To prove the left-hand side of (2.3.16) we will constructively determine w ∈ L2 so that ||w||2 ≤ 25/2 and 1−θ θ ≤ 4 g(I) w(I) h (2.3.17) ||g||1−θ p. 1 I H H

1. Apply Theorem 2.3.3 to g. Let {gi } be the resulting Assume ﬁrst that ||g||H 1 = decomposition so that g = gi . By deﬁnition supp S(gi ) is a dyadic interval, say Ii . By (2.3.8) and (2.3.10) we have (2.3.18)

S(gi ) ∞ |Ii | ≤ 25 ||g||H 1 . Deﬁne 1/2

wi = gi Yi and form w = (2.3.18),

,

where

Yi =

S(gi ) ∞ |Ii | ,

S(gi ) 22

wi . The choice of Yi gives wi 22 = S(gi ) ∞ |Ii |. Hence by

w 22 ≤ 25 ||g||H 1 .

(2.3.19)

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Chapter 2. Projections, Isomorphisms and Interpolation

Let {wi (J)}, respectively {gi (J)}, denote the Haar coeﬃcients of wi , respectively gi . Then the following identities hold, g(J)1−θ w(J)θ hJ = gi (J)1−θ wi (J)θ hJ i

=

J θ/2 Yi gi .

i

The stopping time decomposition of g implies that p pθ/2 g(J)1−θ w(J)θ hJ p . Yi

gi pH p ≤ 4

(2.3.20)

H

i

A detailed veriﬁcation of (2.3.20) is given in Proposition 3.1.5 below. Observe that by H¨ older’s inequality,

1 0

p−1

1

1

≤

S 2 (gi )

S(gi ) 0

0

p−1 S(gi )p S(gi ) p−1 . ∞ |Ii |

(2.3.21)

Indeed, ﬁrst we pull out S(gi ) ∞ from the second factor on the left-hand side of (2.3.21). Thus it is bounded by

p

1

S(gi ) p−1 ∞ .

S(gi ) 0

Next using H¨ older’s inequality and the fact that S(gi ) is supported on the interval Ii one gets p 1 1 S(gi ) ≤ S(gi )p |Ii |p−1 . 0

0

Collecting terms gives (2.3.21). Next recall that pθ/2 = p − 1. Hence we rewrite (2.3.21) as pθ/2

gi H 1 ≤ Yi

p

gi H p .

Taking the sum over i gives that pθ/2

gi H 1 ≤ Yi

gi pH p . i

(2.3.22)

i

Combining the estimates (2.3.22) and (2.3.20) gives (2.3.17) under the assumption that g H 1 = 1. The homogeneities in these estimates and in (2.3.19) imply the general case. Illustrating how interpolation methods are applied we prove a theorem relating norms of rearrangement operators. Recall that a rearrangement operator

2.3. Interpolation of operators

155

is determined by an injective map τ acting on dyadic intervals. We let Tp be the linear extension of the map Tp :

hτ (I) hI → . 1/p |I| |τ (I)|1/p

For the operator norm of Tp on H p we write as before ||Tp ||p = ||Tp : H p → H p ||. Theorem 2.3.6. For 1 < p < 2 let θ = 2 − 2/p. Then c||Tp ||p ≤ ||T1 ||1−θ ≤ C||Tp ||p , 1 where c = 2−5 , and C = 25 . Proof. We start with the right-hand side inequality showing that T1 is bounded on H 1 provided that Tp is bounded on H p . The idea is to apply the renorming of Theorem 2.3.5 to the range of the rearrangement operator. We ﬁx g ∈ H 1 , and w ∈ L2 . Let g(I), w(I) denote their respective Haar coeﬃcients. Next form u=

g(I)1−θ w(I)θ hI .

Recall that 1 − θ + θ/2 = 1/p to observe that the following identity holds between the Haar coeﬃcients of T1 g, T2 w and Tp u, (T1 g)(J)1−θ (T2 w)(J)θ = (Tp u)(J).

(2.3.23)

After these preparatory remarks we start estimating. First we express the norm of T1 g in H 1 utilizing Pisier’s renorming. Let c = 2−5 , then by Theorem 2.3.5, ⎧ ⎫ ⎨ ⎬ c||T1 g||1−θ (T1 g)(J)1−θ w(J)θ hJ H 1 ≤ sup ⎩ : ||w||2 ≤ 1⎭ . J∈τ (D) p H

Next we observe that on L2 the operator T2 is of norm 1, and invertible on its image. Hence the preceding supremum coincides with ⎧ ⎫ ⎨ ⎬ 1−θ θ : ||T2 w||2 ≤ 1 . sup (T g)(J) (T w)(J) h 1 2 J ⎩ ⎭ p J∈τ (D) H

Observe that we know the value of the coeﬃcients, (T1 g)(J)1−θ (T2 w)(J)θ . (They were identiﬁed in (2.3.23) above.) Inserting we rewrite the above supremum as . / $ 1−θ θ sup Tp g(I) w(I) hI : ||w||2 ≤ 1 . I∈D

Hp

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Chapter 2. Projections, Isomorphisms and Interpolation

As Tp is bounded on H p the last line admits the upper estimate $ 1−θ θ ||Tp ||p sup g(I) w(I) hI : ||w||2 ≤ 1 . I∈D

Hp

By Theorem 2.3.5 the second factor on the right-hand side of the preceding inequality is bounded by ||g||1−θ H 1 . Thus we showed that the following estimate holds, ≤ 25 ||Tp ||p . ||T1 ||1−θ 1

Next we prove the converse estimate. Let u ∈ H p with Haar expansion u = u(I)hI . By Theorem 2.3.4, there exist x ∈ H 1 , and y ∈ L2 , such that u(I) = x(I)1−θ y(I)θ

θ 5 p and ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

Note that the Haar coeﬃcients of Tp u satisfy (Tp u)(J) = (T1 x)(J)1−θ (T2 y)(J)θ . Hence we estimate using ﬁrst Proposition 2.3.2, and then ||T2 ||2 = 1. θ ||Tp u||H p ≤ ||T1 x||1−θ H 1 ||T2 y||2 θ ≤ ||T1 ||1−θ ||x||1−θ 1 H 1 ||y||2

≤ 25 ||T1 ||1−θ ||u||H p . 1

Dual estimates Here we show the dual estimates to the renorming (2.3.2) and give a second proof of Theorem 2.3.5. Let 1 < p < 2, θ = 2 − 2/p and u ∈ H p with u = 0. Then 1−θ θ c||u||H p ≤ sup u v(I) w(I) hI dt ≤ C||u||H p , (2.3.24) where the supremum is taken over v = v(I)hI and w = w(I)hI , satisfying ||v||BMO ≤ 1, and ||y||2 ≤ 1. The constants c > 0 and C > 0 in (2.3.24) are independent of p. The following theorem asserts that the left-hand side estimate in (2.3.24) holds true. Theorem 2.3.7. Let 1 ≤ p ≤ 2 and θ = 2 − 2/p. For u ∈ H p with u = 0, there exist v ∈ BMO, w ∈ L2 , with ||v||BMO ≤ 1 and ||w||2 ≤ 1, so that their Haar coeﬃcients v(I), w(I) give 1−θ θ ||u||H p ≤ C u v(I) w(I) hI dt , where C > 0 is independent of p.

2.3. Interpolation of operators

157

The proof of Theorem 2.3.7 uses the stopping time argument of Theorem 2.3.3 applied to the sharp function in place of the square function. Assume ﬁrst that 1 ≤ p < 4/3 and let q denote the H¨ older conjugate index to p. In Section 1.2 we proved that for u ∈ H p there exists f ∈ Lq so that 1 uf dt and f q ≤ 1. ||u||H p ≤ C 0

Next let 4/3 < p ≤ 2. By the square function characterization of Lp there exists f ∈ Lq so that 1 uf dt and S(f ) q ≤ 1. ||u||H p ≤ C 0

In either case f admits a representation as f= v(I)1−θ w(I)θ hI , where v = v(I)hI and w = w(I)hI satisfy ||v||BMO ≤ C and ||w||2 ≤ C. The reader is invited to supply the details by adapting the proof of Theorem 2.3.3. The right-hand side estimate in (2.3.24) is a corollary to Theorem 2.3.4 and to Feﬀerman’s inequality as stated in Theorem 2.3.1. Corollary 2.3.8. Let 1 < p < 2, and θ = 2 − 2/p. For v ∈ BMO, w ∈ L2 , and u ∈ H p , the following estimate holds, 1 √ 1−θ θ θ u( v(I) w(I) hI ) ≤ 26 2||u||H p ||v||1−θ BMO ||w||2 . 0

Proof. By the decomposition Theorem 2.3.4, u ∈ H p can be represented as u= x(I)1−θ y(I)θ hI , where θ 5 p ||x||1−θ H 1 ||y||2 ≤ 2 ||u||H .

Inserting the representation of u and applying Theorem 2.3.1 we obtain that 1 √ 1−θ θ θ u( v(I) w(I) hI ) ≤ 226 ||u||H p ||v||1−θ BMO ||w||2 , 0

as claimed. p

Summing up, the renorming of H (2.3.2) follows from Theorem 2.3.4 and Proposition 2.3.2. Its dual version (2.3.24) we obtain by combining Theorem 2.3.7 and Corollary 2.3.8. In the next sub-section we combine (2.3.2) and (2.3.24) to prove a basic boundedness principle for analytic families of operators.

158

Chapter 2. Projections, Isomorphisms and Interpolation

The duality proof of Theorem 2.3.5. It is instructive to see how G. Pisier’s original proof [176] — via its interpretation by M. Cwikel, P. Nilsson and G. Schechtman [57] — specializes to the context of H 1 . To this end we give a second proof for the left-hand side of (2.3.16) using these sources. Fix g ∈ H 1 , w ∈ L2 and consider u= g(I)1−θ w(I)θ hI . By Corollary 2.3.8 the following lower bound holds, √ 26 2||u||H p ≥ sup u z(I)1−θ y(I)θ hI dt : ||y||2 ≤ 1, ||z||BMO ≤ 1 % & = sup g(I)1−θ z(I)1−θ w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||z||BMO ≤ 1 . Inserting the preceding lower bound gives an expression displaying a high degree of symmetry, % & √ 26 2 sup g(I)1−θ w(I)θ hI p : ||w||2 ≤ 1 H & % 1−θ 1−θ θ ≥ sup g(I) z(I) w(I) y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1, ||z||BMO ≤ 1 . Expressing the self duality of Hilbert spaces we write % & % & w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1 = µ(I)θ |I| : |µ(I)||I| ≤ 1 . Inserting this identity and thereby reintroducing a diﬀerent form of asymmetry, we ﬁnd that & % sup g(I)1−θ z(I)1−θ w(I)θ y(I)θ |I| : ||y||2 ≤ 1, ||w||2 ≤ 1, ||z||BMO ≤ 1 % & = sup |µ(I)||I| ≤ 1, ||z||BMO ≤ 1 . g(I)1−θ z(I)1−θ µ(I)θ |I| : 1

1

Finally we invoke duality between the sequence spaces θ and 1−θ in the following way. For any sequence of scalars a(I), % & 1−θ 1 sup a(I)µ(I)θ |I| : |µ(I)||I| ≤ 1 = |a(I)| 1−θ |I| . With a(I) = g(I)1−θ z(I)1−θ , the preceding identity gives % & sup g(I)1−θ z(I)1−θ µ(I)θ |I| : |µ(I)||I| ≤ 1, ||z||BMO ≤ 1 1−θ = sup |g(I)z(I)||I| : ||z||BMO ≤ 1 = sup

1−θ gz : ||z||BMO ≤ 1 .

2.3. Interpolation of operators

159

In summary we showed the following minorizations, thereby completing the proof of the theorem. % & sup g(I)1−θ w(I)θ hI p : ||w||2 ≤ 1 H 1−θ √ ≥ (26 2)−1 sup gz : ||z||BMO ≤ 1 ≥ 2−8 ||g||1−θ H1 .

Analytic families of operators Now we open a second source of strong estimates. Following is the three lines theorem. This is a maximum principle for analytic functions in the vertical strip V. Its applications form the core results of complex interpolation theory. Theorem 2.3.9. Let F (z) be analytic in V = {x + iy : x ∈ (0, 1), y ∈ R}. Assume that F extends continuously to the boundary of V, and that there exists a polynomial P such that log |F (x + iy)| ≤ P (y), x + iy ∈ V. If supt |F (it)| ≤ R0 and supt |F (1 + it)| ≤ R1 , then for every θ ∈ (0, 1), |F (θ)| ≤ R01−θ R1θ . First we slightly change the appearance of Theorem 2.3.9. Thus the above maximum principle becomes directly applicable to show boundedness of operators. For z ∈ V we let Jz be a linear operator deﬁned a priorily on the set of ﬁnite linear combinations of Haar functions. For f, g ﬁnitely supported over the Haar system, we assume that the map F (z) = Jz (g)f is analytic in the vertical strip V, and extends continuously the boundary. Furthermore we assume that there exists a polynomial P, depending on the choice of f, g, such that log |F (x + iy)| ≤ P (y),

where

x + iy ∈ V.

On the boundary of V we have Jit and J1+it implicitly deﬁned as follows. We put Jz (g)f, Jit (g, f ) = lim z→it J1+it (g, f ) = lim Jz (g)f, z→1+it

thereby deﬁning linear operators Jit , J1+it , which act initially on ﬁnite linear combinations of Haar functions. Under these conditions we call {Jz : z ∈ V } an analytic family of operators.

160

Chapter 2. Projections, Isomorphisms and Interpolation

Throughout this sub-section we assume that for t ∈ R, the operators Jit , and J1+it have bounded extensions to H 1 and L2 respectively, and that they satisfy the norm estimates