Rearrangements with supporting Trees, Isomorphisms and Combinatorics of coloured dyadic Intervals

TL;DR

This paper characterizes a class of rearrangements with supporting trees, demonstrating their boundedness and isomorphic properties on large subspaces of L^p, and explores combinatorial strategies involving colored dyadic intervals.

Contribution

It introduces a new class of rearrangements with supporting trees and analyzes their boundedness and isomorphic behavior on L^p spaces, along with combinatorial game strategies.

Findings

Rearrangements with supporting trees have bounded vector-valued extensions.

Large subspaces of L^p are isomorphic under these rearrangements.

A combinatorial game with colored dyadic intervals is analyzed for strategic outcomes.

Abstract

We determine a class of rearrangements that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension. We show that there exists a large subspace of $L^p$ on which a bounded rearrangement operator acts as an isomorphism. The combinatorial issues of these problems give rise to a two-person game, to be played with colored dyadic intervals. We determine winning strategies for each of the players.

Equations

$$T(h_I)=h_{\tau (I)}.$$

$$A_I\cap A_J\ne \varnothing \text{implies}{A}_I\subseteq A_J\text{or}{A}_J\subseteq A_I.$$

$$|I\cap A_I|\ge \delta |I|\text{and}|\tau (I)\cap A_I|\ge \delta |I|,I\in \mathcal{F}.$$

$$\text{mbox}{\mathrm{span}}_{L_E^p}\left\{h_I{x}_I:x_I\in E,I\in \mathcal{F}\right\},1<p<\infty .$$

$$\left|\bigcup _{J\subseteq I}\tau (J)\right|\le C|I|\text{and}\left|\bigcup _{J\subseteq I}{\tau }^{-1}(J)\right|\le C|I|,$$

$$\tau :{\mathcal{F}}_i\to \tau ({\mathcal{F}}_i)\text{admit a supporting tree?}$$

$${\Vert T\otimes \text{mbox}{\mathrm{Id}}_E\Vert }_{L_E^p}=\infty \text{and}{\Vert T^{-1}\otimes \text{mbox}{\mathrm{Id}}_E\Vert }_{L_E^p}=\infty ,$$

$${\Vert T_{{|}_{X_p}}\otimes \text{mbox}{\mathrm{Id}}_E\Vert }_{L_E^p}\cdot {\Vert T_{{|}_{T(X_p)}}^{-1}\otimes \text{mbox}{\mathrm{Id}}_E\Vert }_{L_E^p}<\infty ,$$

$$\mathcal{D}=\{[(k-1)2^{-n},k{2}^{-n}[,1\le k\le 2^n,n\in \mathbb{N}\}.$$

$${\mathcal{E}}^{\ast }=\bigcup _{I\in \mathcal{E}}I.$$

$$h_I=\{\begin{array}{ll}1& \text{ on the left half of }I;\\ -1& \text{ on the right half of }I.\end{array}$$

$$S(f)={\left(\sum _{I\in \mathcal{D}}⟨f,\frac{h_I}{|I|}{⟩}^2{1}_I\right)}^{1/2}.$$

$$c_p\Vert f{\Vert }_{L^p}\le \Vert S(f){\Vert }_{L^p}\le C_p\Vert f{\Vert }_{L^p},$$

$$\Vert f{\Vert }_{H^1}=\Vert S(f){\Vert }_{L^1}.$$

$${\Vert \sum \pm x_I{h}_I\Vert }_{L_E^p}\le C_p{\Vert \sum x_I{h}_I\Vert }_{L_E^p}.$$

$${\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)a_n{x}_n{\Vert }_{E}dt\le {\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)x_n{\Vert }_{E}dt.$$

$${\left({\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)x_n{\Vert }_E^{p}dt\right)}^{1/p}\le C_p{\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)x_n{\Vert }_{E}dt,1<p<\infty .$$

$${\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)\mathbb{E}(f_n|{\mathcal{F}}_n){\Vert }_{L_E^p}^{p}dt\le C_p{\int }_0^1\Vert \sum _{n=1}^N{r}_n(t)f_n{\Vert }_{L_E^p}^{p}dt.$$

$$\sum _{n=1}^{N_j}{a}_{n,j}=1\text{ and }{y}_j=\sum _{n=1}^{N_j}{a}_{n,j}{x}_n,j\in \mathbb{N},$$

$$|\tau (I)|=|I|,I\in \mathcal{D}.$$

$$T:h_I\to h_{\tau (I)}.$$

$$|\tau (Q(I){)}^{\ast }|\le C|I|,I\in \mathcal{D}.$$

$$\tau (I)=I+|I|m(|I|),I\in \mathcal{D}.$$

$${\tau }_m(I)=I+m|I|,I\in \mathcal{D}.$$

$$\Vert T:L_E^p\to L_E^p\Vert \le C_p(E)(1+\log m).$$

$$|I|/C\le |E_I|\le C|I|.$$

$$E_{I_1}\cup E_{I_2}\subseteq E_I\text{and}{E}_{I_1}\cap E_{I_2}=\varnothing .$$

$$H_I\subseteq E_I\text{and}|H_I|\ge \delta |E_I|.$$

$$A_I\cap A_J\ne \varnothing \text{implies that}{A}_I\subseteq A_J\text{or}{A}_J\subseteq A_I.$$

$$|A_I|\le C|I|,$$

Full text

We prove the existence of a large subspace $X_{p}\subset L^{p}$ on which a bounded rearrangement operator acts as an isomorphism.

Moreover, we study winning strategies for a combinatorial two person game played with coloured collections of dyadic intervals.

MSC 2000: 46B25, 46E40, 46B09, 91A05

Key words: Haar system, Rearrangement Operators, UMD spaces, 2-person games

1. 1 Introduction

2. 2 Preliminaries

3. 3 Subspace Theorems for Rearrangement Operators

4. 4 Shift operators

5. 4.1 Semenov condition for shifts.

6. 4.2 Shifts and nested collections.

7. 5 Combinatorics of coloured intervals

8. 5.1 Winning Strategies for Player B.

9. 5.2 Winning Strategies for Player A.

1 Introduction

We study the connections between rearrangement operators of the Haar system and martingale transforms. We restrict to rearrangements $\tau$ acting on dyadic intervals such that $|\tau(I)|=|I|$ and operators given by

[TABLE]

Our chief interest lies in obtaining workable and directly verifyable descriptions of those rearrangements $\tau$ for which $T\otimes\mbox{ \rm Id}_{E}$ is bounded on $L^{p}_{E}$ for all Banach spaces $E$ in the UMD class.

We isolate first a purely combinatorial condition on $\tau$ that implies the boundedness of $T\otimes\mbox{ \rm Id}_{E}$ on $L^{p}_{E}$ for $1<p<\infty,$ and $E$ in the UMD class.Thereby we convert the analytic question into a combinatorial problem. We let ${\cal F}$ be a collection of dyadic intervals and $\{A_{I}:\,I\in{\cal F}\}$ be a tree of measurable sets such that

[TABLE]

We say that $\tau:{\cal F}\to\tau({\cal F})$ admitts the supporting tree $\{A_{I}:\,I\in{\cal F}\}$ if there exists $c>0,\delta>0$ so that $|A_{I}|\leq c|I|$ and

[TABLE]

If $\tau:{\cal F}\to\tau({\cal F})$ admitts a supporting tree, then, by an application of Stein’s martingale inequality, $T\otimes\mbox{ \rm Id}_{E}$ is bounded on

[TABLE]

Hence if we ask for the $L^{p}_{E}$ boundedness of $T\otimes\mbox{ \rm Id}_{E}$ the ensueing combinatorial problem consists in decomposing the collection of all dyadic intervals into ${\cal F}_{1},\dots,{\cal F}_{N}$ so that the restrictions $\tau:{\cal F}_{i}\to\tau({\cal F}_{i})$ admit a supporting tree. For such a decomposition to exist it is necessary that $T$ (scalar valued) is an isomorphism on some $L^{p}$ with $1<p\neq 2<\infty.$ The results of this paper are all related to the open problem whether this is also a sufficient condition. That is, we are concerned with the following extension problem for rearrangement operators:

Conjecture 1.1

Assume that the scalar valued rearrangement operator $T$ is an isomorphism on $L^{p},\,p\neq 2.$ Is it true that $T\otimes\mbox{ \rm Id}_{E}$ is an isomorphism on $L^{p}_{E},$ for $1<p<\infty$ and any $E$ in the UMD class.

The vector valued extension problem for rearrangement operators as formulated above is meaningful only within the class of isomorphisms on $L^{p}.$ Indeed [9] and [20] contain examples of bounded rearrangement operators on $L^{p},1<p<2$ and asssociated examples of UMD spaces $E$ so that the vector valued extension $T\otimes\mbox{ \rm Id}_{E}$ is not a bounded operator on $L^{p}_{E}.$

Semenov’s theorem [22] provides an intrinsic criterion for $\tau$ so that $T$ is an $L^{p}$ isomorphism. Thus by considering $T$ *together with its * vector valued extension $T\otimes\mbox{ \rm Id}_{E}$ we are led to the following combinatorial problem concerning tree structures and rearrangements.

Conjecture 1.2

Does the hypothesis

[TABLE]

for any dyadic $I$ imply, that the entire collection of dyadic intervals can be decomposed into ${\cal F}_{1},\dots,{\cal F}_{N},$ with $N=N(C),$ so that the restrictions

[TABLE]

In this paper we give partial solutions to the conjectures LABEL:con1 and LABEL:con2. Our results are connected to a wider set of problems as follows:

Seeking understanding of scalar valued operators by studying simultaneously their vector valued extensions is a central line of investigation in Banach space theory. Classical and authoritative accounts thereof are [5] by T. Figiel and [25] by A. Pełczyński. In the context of rearrangement operators, the search for supporting trees is just motivated by our attempts to prove the boundedness of vector valued rearrangements. 2. 2.

Section 4.7. in [26] by G. Pisier contains the question of describing the class of $L^{p}$ bounded operators $R$ so that $R\otimes\mbox{ \rm Id}_{E}$ is bounded on $L^{p}_{E},$ for any UMD space $E.$ Calderon-Zygmund integrals and martingale transforms share this property. The theorems in the present paper aim at a description of the rearrangement operators in this class. 3. 3.

The rearrangement problems arising with the unconditionality of the Franklin system and its generalisations [6] , [14] are special cases respectively model cases for the problems treated in Section LABEL:sem.s1. 4. 4.

The recent extrapolation theorems [9], [20], for bounded and invertible vector valued rearrangement operators on the $L_{E}^{p}\,(1<p<\infty)$ scale provide the structural support for the conjectures LABEL:con1 and LABEL:con2. Indeed, if the latter were true the former extrapolation results would simply follow from the well known scalar case. 5. 5.

The examples in [9] and [20] of $L^{p}$ bounded rearrangement operators $T$ for which

[TABLE]

provide the motivation for proving the subspace theorem in Section LABEL:sub. Depending on $\tau$ we determine a large subspace $X_{p}\subseteq L^{p}$ on which nevertheless the restricted operator $T_{|_{X_{p}}}\otimes\mbox{ \rm Id}_{E}$ acts as an isomorphism so that,

[TABLE]

for $1<p<\infty$ and any UMD space $E.$

The organisation of the paper is as follows. In Section LABEL:prel we review the concepts and theorems used in the paper. In Section LABEL:sub we prove a subspace theorem for rearrangement operators. For an $L^{p}$ bounded rearrangment operator $T$ we determine a block basis $\{\widetilde{h_{I}}\},$ equivalent to the entire Haar system, such that $T$ acts as an isomorphism on the subspaces spanned by $\{\widetilde{h_{I}}\}.$ This done by constructing a tree that simultaneously supports $\{\widetilde{h_{I}}\}$ and its image $\{T(\widetilde{h_{I}})\}.$ Section LABEL:sem.s1 treats special shift operators $S.$ We show that $S\otimes\mbox{ \rm Id}_{E}$ is $L^{p}_{E}$ bounded for $E$ in the UMD class, provided that the associated shift parameters form a decomposable sequence in the sense of Defintion LABEL:sem.d1. In Section LABEL:c.s1 we study thoroughly the problem of finding $(d,\eta)$ homogeneous decompositions of a collection ${\cal C}$ of dyadic intervals, that preserves a pre-existing $(d,\eta)$ homogenous decomposition of a fixed subcollection ${\cal C}^{\prime}$ in ${\cal C}.$ We obtain conditions for its existence and determine examples for which this problem is without solution. While -we think- this is a combinatorial problem interesting in itself we present it here since it should support the construction of trees for rearrangements.

2 Preliminaries

Dyadic intervals and the Haar system.

See e.g. [19]. We let ${\cal D}$ denote the collection of (half-open) dyadic intervals contained in [0,1]. Thus

[TABLE]

For $n\in{\mathbb{N}}$ write ${\cal D}_{n}=\{I\in{\cal D}:|I|=2^{-n}\}.$ For a collection of dyadic intervals ${\cal E}$ we use the $*-$ notation to denote the pointset covered by ${\cal E}$ thus

[TABLE]

For $I\in{\cal D}$ denote by $Q(I)$ the collection of all dyadic intervals that are contained in $I.$

Denote by $\{h_{I}:I\in{\cal D}\}$ the $L^{\infty}-$ normalized Haar system, where $h_{I}$ is supported on $I$ and

[TABLE]

The Haar system is an unconditional basis in $L^{p},$ $(1<p<\infty).$ For $f\in L^{p}$ we define its dyadic square function as

[TABLE]

The Marcinkiewicz [15] interpretation of R.E.A.C. Paley’s theorem [23] asserts that

[TABLE]

where $C_{p}\sim p^{2}/(p-1)$ and $c_{p}=C_{p}^{-1}.$ Recall also the definition of dyadic $H^{1},$ defined by the relation $f\in H^{1}$ if $S(f)\in L^{1}$ and

[TABLE]

The Haar system in Bochner-Lebesgue spaces.

[8], [17]. Let $1<p<\infty.$ For a Banach space $E$ we denote by $L_{E}^{p},$ the Bochner Lebesgue space of E valued and $p-$ integrable functions on the unit interval. We say that a Banach space $E$ satisfies the UMD property if there exists $C_{p}>0$ so that for any finite collection $x_{I}\in E$

[TABLE]

Kahane’s principle of contraction and Kahane’s inequality.

[13], [28]. Let $\{r_{n}\}$ denote the sequence of independent $\{+1,-1\}$ valued Rademacher functions. Let $x_{n}\in E$ be a sequence in a Banach space $E$ and let $a_{n}\in{\mathbb{R}}$ so that $|a_{n}|\leq 1.$ Then, ,

[TABLE]

where $C>0$ is independent of $N.$ We apply the above principle of contraction in combination with the Kahane’s inequality asserting that

[TABLE]

A Martingale Inequality.

[19] [8]. The following vector valued martingale inequality is due to Bourgain and known as Bourgain’s version of E.M. Stein’s martingale inequality. It asserts the following. Let $E$ be a Banach space with the UMD property. For any increasing sequence of $\sigma-$algebras ${\cal F}_{n}$ and any sequence $f_{n}\in L^{p}_{E}$ with $1<p<\infty,$

[TABLE]

where ${\mathbb{E}}(f|{\cal F})$ denotes conditional expectation with respect to ${\cal F}$ and where as above $\{r_{n}\}$ denotes the sequence of independent Rademacher functions.

The Theorem of Mazur.

[28]. Let $E$ be Banach space and $(x_{n})$ be a sequence in $E$ with weak limit $x\in E.$ Then there exist a sequence $a_{n,j}\in[0,1]$ $N_{j}\in{\mathbb{N}}$ so that

[TABLE]

converges to $x$ in norm of $E,$ that is, $\lim_{j\to\infty}\|y_{j}-x\|_{E}=0.$

Semenov’s Theorem.

[22]. Let $\tau:{\cal D}\to{\cal D}$ be bijective satisfying

[TABLE]

The induced rearrangement operator is the linear extension of the map

[TABLE]

Let $1<p<2.$ Semenov’s theorem asserts that $T$ is bounded on $L^{p}$ if there exists $C>0$ so that

[TABLE]

Moreover, condition (LABEL:10okt2) characterizes the boundedness of $T$ on $H^{1}.$ Specializing further we consider shifts defined by

[TABLE]

For shifts, Semenov’s condition specializes as follows: Shift operators are bounded on $L^{p}$ if there exists $K$ so that the set $\tau(Q(I))^{*}$ can be covered by $K$ dyadic intervals of the same length as $I.$ The best known of these shifts are those used by T. Figiel [6, 7] to study of Calderon-Zygmund operators,

[TABLE]

Their norm estimates are given by T.Figiel’s theorem [6, 7]. Below we apply it for fixed and small values of $m.$

Theorem 2.1

The linear extension of $Th_{I}=h_{\tau_{m}(I)}$ defines a bounded operator on $L^{p}_{E}\,(1<p<\infty)$ for each Banach space $E$ with UMD property, and

[TABLE]

Dyadic trees.

Let ${\cal D}$ be a collection of dyadic intervals We say that $\{E_{I}:I\in{\cal D}\}$ is a dyadic tree of sets if, the following conditions hold:

Each of the sets $E_{I}$ is a finite union of dyadic intervals. 2. 2.

There exists $C>0$ so that

[TABLE] 3. 3.

If $I_{1}$ is the left half of $I$ and $I_{2}$ is its right half then

[TABLE]

Let $\{E_{I}:I\in{\cal D}\}$ be a dyadic tree and $\alpha\in{\mathbb{R}}.$ Then we also use the term dyadic tree for the translates $\{\alpha+E_{I}:I\in{\cal D}\}.$ We say that a collection of measurable sets $\{H_{I}:I\in{\cal D}\}$ supports a dyadic tree $\{E_{I}:I\in{\cal D}\}$ if there exists $\delta>0$ so that for $I\in{\cal D}$

[TABLE]

Trees and nested collections.

Let ${\cal F}$ be a subset of all dyadic intervals. We say that $\{A_{I}:I\in{\cal F}\}$ is a tree (or equivalently a nested collection) of measurable sets if for $I,J\in{\cal F},$

[TABLE]

Rearrangements with supporting trees.

Recall that the rearrangements we consider satisfy $|\tau(I)|=|I|.$ We say that $\tau:{\cal F}\to{\cal D}$ admitts the supporting tree $\{A_{I}:I\in{\cal F}\}$ if there exists $C>0,\delta>0$ so that

[TABLE]

[TABLE]

The interest in the notion of rearrangements admitting a supporting tree comes from the following observation, obtained by merging [12, Proposition 6.9] with [6, 7].

Theorem 2.2

Let $\tau:{\cal F}\to{\cal D}$ be a rearrangement admitting a supporting tree (with constants $C>0,\delta>0.$) Let

[TABLE]

Then

[TABLE]

extends to an isomorphism

[TABLE]

so that

[TABLE]

Proof. The proof is based on the contraction principle and Stein’s martingale inequality. By hypothesis $\tau:{\cal F}\to{\cal D}$ has a supporting tree, say $\{A_{I}:I\in{\cal F}\}.$ Let $N\in{\mathbb{N}}$ and define two families of increasing $\sigma-$algebras,

[TABLE]

We translate the hypothesis into pointwise estimates for conditional expectations. Let $I\in{\cal F}$ with $|I|=2^{-N},$ then

[TABLE]

Now fix $x_{I}\in E$ for $I\in{\cal F}.$ For $I\in{\cal D}\setminus{\cal F}$ put $x_{I}=0.$ With the UMD property on $E,$ the contraction principle, and Bourgain’s version of Stein’s Martingale inequality we get,

[TABLE]

Next exploit the second part of the hypothesis

[TABLE]

and continue again with the contraction principle and the martingale inequality.

[TABLE]

The reverse estimate follows in a similar fashion.

3 Subspace Theorems for Rearrangement Operators

Subspace theorems are concerned with the following phenomenon. For a well behaved linear transformation $T$ on a Banach space $E$ there exists a –large– subspace $F\subseteq E$ on which $T$ is much better behaved. The best known examples of subspace theorems include bounded and non weakly compact operators on the spaces $C(K),$ $A,$ $L^{\infty},$ $H^{\infty},$ and embedding operators on $L^{p}$ spaces.

If $T$ is a bounded operator on $C(K)$ and not weakly compact, then there exists a subspace $F\subseteq C(K)$ isomorphic to $c_{0}$ so that $T_{|F}$ is an isomorphism. [24] The same holds for the disk algebra $A.$ [3] If T is a bounded operator on $L^{\infty}$ and not weakly compact, then there exists a subspace $F\subseteq L^{\infty}$ isomorphic to $\ell^{\infty}$ so that $T_{|F}$ is an isomorphism. [27] The same assertion holds for operators on the space $H^{\infty}.$ [2] 2. 2.

Another class of subspace theorems concern embedding operators $T:L^{p}\to L^{p}$ with $1\leq p<\infty.$ To any such embedding there exist a subspace $F\subseteq L^{p}$ so that $F$ and $T(F)$ are complemented in $L^{p},$ and $F$ is isomorphic to the ambient space $L^{p}.$ [4], [16], [12]. Extensions of this theorem hold for rearrangement invariant Banach spaces in which the Haar system is an unconditional basis. [12]

In this section we prove a subspace theorem for a rearrangement operator acting on the Haar system

[TABLE]

where $\tau:{\cal D}\to{\cal D}$ is bijective satisfying

[TABLE]

Theorem 3.1

Assume that $T$ is bounded on $L^{p_{0}}$ for some $1<p_{0}\neq 2<\infty.$Then for any $1<p<\infty\,(sic!)$ there exists a closed subspace $X_{p}\subseteq L^{p}$ isomorphic to $L^{p},$ so that

[TABLE]

is an isomorphism,

[TABLE]

The subspaces $X_{p}$ and $T(X_{p})$ are complemented in $L^{p}.$

Theorem LABEL:summ is a direct consequence of Theorem LABEL:select and Theorem LABEL:equivalence. In the course of its proof we use martingale techniques, most notably the inequalities of Paley respectively Stein (in Burkholder’s respectively Bourgain’s version, [8], [19]). Thus the method yields extensions to vector valued rearrangement operators so that

[TABLE]

whenever $E$ satisfies the UMD property. The significance of this remark is connected with the examples in [9] and [20] of scalar valued $L^{p}$ bounded rearrangement operator $T$ satisfying

[TABLE]

The combinatorial core.

The following result, was developed for the proof that the spaces $L^{p},1\leq p<\infty$ are primary. We refer to the work of Enflo and Starbird [4], Johnson, Maurey, Schechtman, and Tzafriri [12], and Enflo via Maurey [16]. It is the main combinatorial tool by which we find two tree structures (one in the domain, another in the range of the operator $T$) that are compatible with the action of rearrangement operators.

Proposition 3.2

Let $\nu$ be a measure on $[0,1]$ taking values in $\{f\in L^{1}[0,1]:f\geq 0\}.$ Assume that

[TABLE]

Then there exist dyadic trees $\{G_{I}:I\in{\cal D}\}$ and $\{F_{I}:I\in{\cal D}\}$ so that

[TABLE]

where $\delta=\delta(C).$

Our definition of a dyadic tree (as given in the preliminaries section) includes the requirement that $G_{I}$ and $F_{I}$ can be written as finite unions of dyadic intervals. Lemma 9.8 in [12] states just that $F_{I}$ is a finite union of dyadic intervals. However, the proof of [12, Lemma 9.8] can easily be modified to yield that also $G_{I}$ is a finite union of dyadic intervals. See [18]. In the applications of Proposition LABEL:30okt1 the vector valued measure $\nu$ carries well structured information on weak limits of non linear functionals [12], [1], [18]. The non linearities arise by composing the linear operator under investigation with the dyadic square function.

The hypothesis of Proposition LABEL:30okt1 are easily verified with the following criterion [12, Chapter 9].

Proposition 3.3

Let $\nu$ be a measure on $[0,1]$ taking values in $\{f\in L^{1}[0,1]:f\geq 0\}.$ Assume that

[TABLE]

and

[TABLE]

then

[TABLE]

Proof. First observe that by (LABEL:15okt1) and the additivity of the vector measure,

[TABLE]

Next by arithmetic and the Cauchy-Schwarz inequality,

[TABLE]

By (LABEL:15okt2) we get for the second term in the above expression

[TABLE]

Combining (LABEL:15okt4) with (LABEL:15okt5) and using Hoelder’s inequality gives

[TABLE]

Theorem 3.4

To each rearrangement $\tau:{\cal D}\to{\cal D}$ satisfying Semenov’s condition

[TABLE]

there exist pairwise disjoint collections of dyadic intervals $\{{\cal H}_{I}:I\in{\cal D}\}$ so that

The family ${\cal H}_{I}$ consists of pairwise disjoint dyadic intervals of equal length, that is, if $J_{1},J_{2}\in{\cal H}_{I}$ then $|J_{1}|=|J_{2}|$ and $J_{1}\cap J_{2}=\emptyset.$ 2. 2.

*The families $\{\tau({\cal H}_{I})^{*}:I\in{\cal D}\}$ respectively *$\{{\cal H}_{I}^{*}:I\in{\cal D}\}$support dyadic trees.

Proof. In the course of selecting the families $\{{\cal H}_{I}:I\in{\cal D}\}$ we exploit Proposition LABEL:30okt1 and take advantage of the fact that we are working with rearrangment operators for which (LABEL:1feb091) holds.

Selecting two trees.

We review the construction of the non negative $L^{2}$ valued vector measure describing the limiting behavior of the operator $T.$ Here we refer to Chapter 9 of [12].

Fix $I\in{\cal D},n\in{\mathbb{N}}.$ Let

[TABLE]

Since $0\leq d_{n}(I)\leq 1,$ the sequence $\{d_{n}(I):n\in{\mathbb{N}}\}$ has a weak cluster point in $L^{2}.$ By a diagonal argument there exists a subsequence $(n_{k})$ so that for $I\in{\cal D}$ the sequence $\{d_{n_{k}}(I)\}$ converges weakly in $L^{2}.$ For $I\in{\cal D}$ define $\nu(I)$ as $L^{2}-$weak limit,

[TABLE]

Using that $\tau$ is bijective and $|\tau(I)|=|I|$ it is straightforward to observe that

[TABLE]

Since $\tau$ satisfies Semenov’s condition, the linear extension of $T(h_{I})=h_{\tau(I)}$ defines a bounded operator on $H^{1}.$ We use the boundedness of $T$ on $H^{1}$ to prove that

[TABLE]

Mazur’s theorem asserts that there exist $N\in{\mathbb{N}},$ $a_{n}\in[0,1]$ so that $\sum_{n\in A_{I}}\alpha_{n}=1$ and

[TABLE]

Define next

[TABLE]

Since $d_{n}(I)^{2}=d_{n}(I),$ we have the identity

[TABLE]

By (LABEL:app) and (LABEL:id) we get immediately

[TABLE]

Invoking that $\|k_{I}\|_{H^{1}}=|I|$ we obtain that and $\|Tk_{I}\|_{H^{1}}\leq\|T\|_{H^{1}}|I|$ hence

[TABLE]

Since $\epsilon>0$ is arbitrary we obtain (LABEL:sqrt) as claimed. Combining (LABEL:obs) and (LABEL:sqrt) yields these estimates

[TABLE]

Since $\nu$ is a finitely additive set function satisfying $\int_{0}^{1}\nu(I)dt\leq|I|,$ we may extend the mapping

[TABLE]

to an absolutely continuous vector measure on the $\sigma-$ algebra generated by ${\cal D}$ so that (LABEL:input) holds and

[TABLE]

By Proposition LABEL:30okt1 there exist two dyadic trees $\{G_{I}:I\in{\cal D}\}$ and $\{F_{I}:I\in{\cal D}\}$ so that

[TABLE]

This completes our summary of the selection process in [12].

Defining ${\cal H}_{I}$.

We turn to drawing consequences of (LABEL:comb). Here we exploit that the operator generating the vector measure $\{\nu(I):I\in{\cal D}\}$ is defined as a rearrangement

[TABLE]

There exists $N(I)\in{\mathbb{N}}$ so that $G_{I}$ is a finite union of intervals in ${\cal D}_{N(I)}.$ We test the weak limit $\nu(F_{I})$ by integrating against the function $1_{G_{I}}.$ There exists $M(I)\geq N(I)$ so that

[TABLE]

Define now the collection

[TABLE]

Supporting trees.

We first verify that $\{\tau({\cal H}_{I})^{*}:I\in{\cal D}\}$ supports the tree $\{G_{I}:I\in{\cal D}\}.$ Recall that by definition of ${\cal H}_{I}$ we have the inclusion

[TABLE]

It remains for verify the measure estimate: Here we use the identity

[TABLE]

together with (LABEL:comb) and (LABEL:16okt1). This gives

[TABLE]

Next we verify that $\{{\cal H}_{I}^{*}:I\in{\cal D}\}$ supports the tree $\{F_{I}:I\in{\cal D}\}.$ Note first the inclusion

[TABLE]

Next using the identity $|\tau(I)|=|I|$ we get a reduction to (LABEL:me2) treated above:

[TABLE]

The collections $\{{\cal H}_{I}:I\in{\cal D}\}$ are chosen so that Stein’s Martingale Inequality yields the estimates of Theorem LABEL:equivalence below. As a result the scalar valued estimates of Theorem LABEL:equivalence remain true when the coefficients are taken from a Banach space with the UMD property.

Theorem 3.5

Let $\tau:{\cal D}\to{\cal D}$ be a rearrangement satisfying $|\tau(I)|=|I|,\quad I\in{\cal D}$ and define

[TABLE]

Assume that $\{{\cal H}_{I}:I\in{\cal D}\}$ are pairwise disjoint collections of dyadic intervals so that

If $J_{1},J_{2}\in{\cal H}_{I}$ then $|J_{1}|=|J_{2}|$ and $J_{1}\cap J_{2}=\emptyset.$ 2. 2.

The families $\{{\cal H}_{I}^{*}:I\in{\cal D}\}$ and $\{\tau({\cal H}_{I})^{*}:I\in{\cal D}\}$ support dyadic trees of sets.

Then the blocked system

[TABLE]

satisfies the following estimates: For $1<p<\infty,$ and any choice of $x_{I}\in{\mathbb{R}},$

[TABLE]

[TABLE]

Proof. Let $\{F_{I}\}$ be the dyadic tree supported by $\{{\cal H}_{I}^{*}:I\in{\cal D}\}$ and $\{G_{I}\}$ be the dyadic tree supported by $\{\tau({\cal H}_{I})^{*}:I\in{\cal D}\}$. Without loss of generality we may assume that the tree of sets $\{F_{I}\}$ is contained in the interval $[0,1]$ and that $\{G_{I}\}$ is contained in $[1,2].$ Let ${\cal E}_{N}$ be the $\sigma-$algebra generated by the sets

[TABLE]

Since $\{F_{I}\}$ and $\{G_{I}\}$ are trees and moreover $G_{J}\cap F_{I}=\emptyset,I,J\in{\cal D}$ it follows that $\{{\cal E}_{N},N\in{\mathbb{N}}\}$ is an increasing sequence of $\sigma-$algebras. Let ${\mathbb{E}}_{N}$ denote the conditional expectation operator induced by ${\cal E}_{N}.$ Note that the following pointwise estimates hold true.

[TABLE]

Exploiting the estimate (LABEL:crucial) together with the unconditionality of the Haar system and Bourgain’s version of E.M. Stein’s martingale inequality we get the following estimates. Let $x_{I}\in{\mathbb{R}}$ with $I\in{\cal D}$ be a sequence with finitely many entries different from zero.

[TABLE]

Observe also the pointwise estimates.

[TABLE]

By (LABEL:crucial11) together with the unconditionality of the Haar system and Bourgain’s version of E.M. Stein’s martingale inequality we get similarly,

[TABLE]

This proves (LABEL:jan10.e1).

To prove (LABEL:jan10.e2), let ${\cal F}_{N}$ be the $\sigma-$algebra generated by the sets

[TABLE]

Since $\{F_{I}\}$ is a tree $\{{\cal F}_{N},N\in{\mathbb{N}}\}$ is an increasing sequence of $\sigma-$algebras. Let ${\mathbb{F}}_{N}$ denote the conditional expectation operator induced by ${\cal F}_{N}.$ The collection $\{{\cal H}_{I}^{*}:I\in{\cal D}\}$ supports the tree $\{{\cal F}_{I},I\in{\cal D}\}$. Therefore the following pointwise estimates hold true.

[TABLE]

[TABLE]

Finally since both since both $\{I\}$ and $\{F_{I}\}$ are trees of sets we obtain by a measure preserving transformation that

[TABLE]

The estimates of Theorem LABEL:equivalence assert that in $L^{p},1<p<\infty$ the blocked system is equivalent to the Haar system, hence the subspace $X_{p}\subseteq L^{p}$ defined to be the $L^{p}-$closure of $\mbox{\rm span }\{\widetilde{h_{I}}:I\in{\cal D}\}$ is isomorphic to the ambient space $L^{p}.$ And furthermore on the subspace $X_{p}$ the rearrangement operator $T$ acts as an isomorphism.

Proof of Theorem LABEL:summ :

Consider first $1<p_{0}<2.$ By Semenov’s theorem $\|T\|_{p_{0}}<\infty$ implies that

[TABLE]

Apply Theorem LABEL:select to the rearrangement $\tau:{\cal D}\to{\cal D}.$ Let $\{{\cal H}_{I}:I\in{\cal D}\}$ denote the collections of dyadic intervals satisfying the conclusion of Theorem LABEL:select. Put

[TABLE]

Next fix $1<p<\infty.$ Let $X_{p}$ denote the closure of $X$ in $L^{p},$ and similarly let $(T(X))_{p}$ be the closure of $T(X)$ in $L^{p}.$ Theorem LABEL:equivalence asserts that $X_{p}$ is isomorphic to $L^{p},$ and that the map

[TABLE]

extends uniquely to an isomorphism with domain $X_{p}$ and range $(T(X))_{p}.$ Denoting the extension still by $T$ we have $(T(X))_{p}=T(X_{p}),$ and

[TABLE]

Next we turn to the case $2<p_{0}<\infty.$ Note that for rearrangement operators the transposed operator coincides with the inverse $T^{-1}$ defined by $\tau^{-1}:{\cal D}\to{\cal D}.$ Thus $S=T^{*}=T^{-1}$ is a bounded operator on $L^{p_{0}^{*}}$ where $1/p_{0}^{*}+1/p_{0}=1.$ Now $p_{0}^{*}<2.$ By the first part of the theorem applied to the rearrangement operator $S,$ for $1<p<\infty$ there exists $X_{p}$ isomorphic to $L^{p},$ so that

[TABLE]

Since $S=T^{-1},S^{-1}=T$ with $Y_{p}=S(X_{p})$ we obtain

[TABLE]

4 Shift operators

We consider rearrangements of ${\cal D}$ of particular type, namely shift operators. Let

[TABLE]

and let $M=\{m_{j},j\geq 1\}$ be a sequence of integers satisfying $|m_{j}|\leq 2^{j}$. Consider $\tau_{M}:{\cal D}\to{\cal D}$ given by

[TABLE]

where $k+m_{j}$ is understood ${\rm mod}\;2^{j}$. This rearrangement is called shift generated by $M$.

In this section, we give a version of Semenov’s theorem suitably adapted to the special nature of shift operators, see Proposition LABEL:sem.p1. We isolate a class of shift operators for which Conjecture LABEL:con1 and Conjecture LABEL:con2 hold true, Theorems LABEL:sem.t2 and LABEL:sem.t1. We prove directly, without using the machinery of Section LABEL:sub a subspace theorem for shift operators, Theorem LABEL:sem.t3.

4.1 Semenov condition for shifts.

We next give a version of Semenov’s criterion that holds specifically for shift operators. Let $j\in{\mathbb{N}}$ and put

[TABLE]

so that

[TABLE]

Then define

[TABLE]

The next Proposition LABEL:sem.p1 relates Semenov’s condition to the boundedness of the sequence $N_{j}(M)$.

Proposition 4.1

Let $\tau_{M}$ be the shift generated by the sequence $M=\{m_{j},j\geq 1\}$. Then we have for $I\in{\cal D}_{j}$

[TABLE]

Consequently, $\tau_{M}$ satisfies Semenov’s condition if and only if there is a constant $K>0$ such that $N_{j}(M)\leq K$ for all $j\geq 1$.

Proof. Let $N_{j}=N_{j}(M)$. For fixed $j$, let

[TABLE]

be the enumeration of the set of indices $\{k:I_{j,k}\cap\{x_{l},l\geq j\}\neq\emptyset\}$. Take $I\in{\cal D}_{j}$ and $l\geq j$. Let $k_{i}$ be such that $x_{l}\in I_{j,k_{i}}$. Then we have

[TABLE]

and consequently

[TABLE]

To prove the other estimate, let $J,L\in\{I_{j,k_{i}},1\leq i\leq N_{j}\}$ be such that ${\rm dist}(J,L)>0$. Then for $l_{1},l_{1}$ such that $x_{l_{1}}\in J$ and $x_{l_{2}}\in L$ we have

[TABLE]

Clearly, for each $l\geq j$ we have $|\tau(Q(I)\cap{\cal D}_{l})^{*}|=|I|$. Combining these observations we get

[TABLE]

4.2 Shifts and nested collections.

Proposition LABEL:sem.p1 implies that if $\tau_{M}$ satisfies Semenov’s condition, then the collection of accumulation points of sequence $X=\{x_{j},j\geq 1\}$ is finite. Therefore, without loss of generality we assume $\lim_{j\to\infty}x_{j}=0$.

We turn our attention to sequences $M$ with $\limsup_{j\to\infty}N_{j}(M)\leq 2$. We are able to describe the structure of such sequences:

Proposition 4.2

Let $M=\{m_{j},j\geq 1\}$ and $X=\{x_{j}={m_{j}\over 2^{j}},j\geq 1\}$ be such that $\lim_{j\to\infty}x_{j}=0$ and $\limsup_{j\to\infty}N_{j}=2$. Then exist sequences

[TABLE]

with the following properties: $\{j_{k},k\geq 0\}$ is increasing and if

[TABLE]

then

[TABLE]

and if

[TABLE]

then

[TABLE]

Proof. First, note that if $N_{j}=1$ then $0\leq x_{l}<{1\over 2^{j}}$ for all $l\geq j$. Therefore, in case $\limsup_{j\to\infty}N_{j}=1$ the above condition is clearly satisfied.

Consider the case $\limsup_{j\to\infty}N_{j}=2$. We construct sequences $\{a_{k},k\geq 1\}$ and $\{j_{k},k\geq 0\}$ inductively. We put an additional requirement that $N_{j_{k}}=2$. Let

[TABLE]

For $n_{1}<j<j_{1}$ we have $N_{j}=1$, and consequently $\{x_{l},l\geq j\}\subset[0,{1\over 2^{j}})$. Taking $j=j_{1}-1$ we get $x_{l}\leq{1\over 2^{j_{1}-1}}$ for $l\geq j_{1}$.

Then, take $l$ such that $j_{0}\leq l\leq n_{1}$. Since $N_{l}=2$, there are only two intervals from ${\cal D}_{l}$ which have nonempty intersection with $\{x_{i},i\geq l\}$: one of them is $I_{l,1}$, and the other contains $x_{n_{1}}$. This implies that either $x_{l}\in I_{l,1}$ (in such case, $x_{l}=0$), or $x_{l}$ and $x_{n_{1}}$ must be in the same interval from ${\cal D}_{l}$, which implies $|x_{n_{1}}-x_{l}|<{1\over 2^{l}}$. Therefore, it is enough to put $a_{1}=x_{n_{1}}$. Note that $N_{j_{1}}=2$.

Having defined $j_{k}$ and $a_{k}$, we define $j_{k+1}$ and $a_{k+1}$. Since $N_{j_{k}}=2$, there is $j\geq j_{k}$ such that $x_{j}>{1\over 2^{j_{k}}}$