Two Remarks on Primary Spaces

TL;DR

This paper investigates the factorization properties of operators on certain Banach spaces, proving that the identity operator factors through any operator or its complement, and offers combinatorial proofs replacing previous reliance on Szemeredi's theorem.

Contribution

It extends factorization results to spaces (H^p(T)) for 1<p<, using combinatorics of dyadic intervals instead of Szemeredi's theorem.

Findings

Identity factors through any operator or its complement on (H^1(T)).

Analogous results are established for (H^p(T))) spaces for 1<p<.

Combinatorial methods replace the dependence on Szemeredi's theorem.

Abstract

We prove that for any operator $T$ on $ \ell^\infty(H^1 (\bT))$, the identity factores through $T$ or $\Id - T$. We re-prove analogous results of H.M. Wark for the spaces $\ell^infty(H^p(\bT))$, $1<p <\infty$. In the present paper direct combinatorics of colored dyadic intervals replaces the dependence on Szemeredi's theorem in the work of H. M. Wark.

Equations

$${\left(\sum _{n=1}^{\infty }{H}_n^p(\mathbb{T})\right)}_{\infty }.$$

$${}\begin{array}[]{ccc}H^{p}_{n}&\stackrel{{\scriptstyle\mbox{ \rm Id}}}{{\longrightarrow}}&H^{p}_{n}\\ E\downarrow&&\uparrow P\\ H^{p}_{N}&\stackrel{{\scriptstyle H}}{{\longrightarrow}}&H^{p}_{N}\end{array}$$

$$T(e_j)=\lambda e_j+\text{ small L∞ error},j\in \Lambda .$$

$$T(x_j)={\lambda }_j{x}_j+\text{ small ℓmp error},j\in \sigma .$$

$$T(b_J)={\lambda }_J{b}_J+\text{ small mboxBMO error},J\in {\mathcal{D}}_n.$$

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

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

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

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

$$J:H_n^1(\mathbb{T})\to H_n^1(\delta ),s_I\to h_I$$

$$\Vert J:H_n^1(\mathbb{T})\to H_n^1(\delta )\Vert \cdot \Vert J^{-1}:H_n^1(\delta )\to H_n^1(\mathbb{T})\Vert \le C.$$

$$D(x_1,\dots ,x_n,\dots )=(T_1(x_1),\dots ,T_n(x_n),\dots )$$

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

$${\left(\sum a_i^2\Vert y_i{\Vert }_{H^1}^2\right)}^{1/2}\le 4\Vert \sum a_i^2{y}_i{\Vert }_{H^1}.$$

$$[[\mathcal{E}]]=\underset{E_I\in \mathcal{E}}{\sup }\frac{1}{|E_I|}\sum _{E_J\in \mathcal{E},E_J\subseteq E_I}|E_J|.$$

$$E_I=\bigcup _{J\in {\mathcal{E}}_I}J\text{and}{b}_I=\sum _{J\in {\mathcal{E}}_I}{h}_J$$

$$\frac{|I|}{2}\le |E_I|/|E_{[0,1[}|\le |I|.$$

$$E_{I_1}\subseteq \{t:b_I(t)=+1\}\text{and}{E}_{I_2}\subseteq \{t:b_I(t)=-1\},$$

$$Q(f)=\sum _{J\in {\mathcal{D}}_m}⟨f,\frac{b_J}{\Vert b_J{\Vert }_2}⟩\frac{b_J}{\Vert b_J{\Vert }_2},b_J=\sum _{I\in {\mathcal{B}}_J}{h}_I,$$

$$Y_p=\left(\text{mbox}\mathrm{span}\{b_I:I\in \mathcal{L}\},\Vert {\Vert }_p\right).$$

$$\text{mbox}{\mathrm{Id}}_{L_n^p}=R\text{mbox}{\mathrm{Id}}_{Y_p}I\text{with}\Vert R{\Vert }_p\cdot \Vert E{\Vert }_p\le C_p.$$

$${}\begin{array}[]{ccc}\ell^{\infty}(H^{1}({\mathbb{T}}))&\stackrel{{\scriptstyle\mbox{ \rm Id}}}{{\longrightarrow}}&\ell^{\infty}(H^{1}({\mathbb{T}}))\\ E\downarrow&&\uparrow P\\ \ell^{\infty}(H^{1}({\mathbb{T}}))&\stackrel{{\scriptstyle H}}{{\longrightarrow}}&\ell^{\infty}(H^{1}({\mathbb{T}}))\end{array},$$

$$X=(\sum H_n^1(\delta ){)}_{\infty }.$$

$$D=PTE\text{and (!)}\text{mbox}{\mathrm{Id}}_X-D=P\left(\text{mbox}{\mathrm{Id}}_{{\ell }^{\infty }(H^1(\mathbb{T}))}-T\right)E,$$

$$Q(f)=\sum _{J\in {\mathcal{D}}_m}⟨f,\frac{b_J}{\Vert b_J{\Vert }_2}⟩\frac{b_J}{\Vert b_J{\Vert }_2},b_J=\sum _{I\in {\mathcal{B}}_J}{h}_I,$$

$$\Vert Q(x){\Vert }_{H^1}\le ϵ\Vert x{\Vert }_{H^1}.$$

$${\tau }^{-1}\Vert x_i\Vert$$

$$\ge c|{\mathcal{L}}_i{|}^{1/2},$$

$$[[{\mathcal{G}}_0]]\ge N-\ln (cn{\tau }^{-2}).$$

$$\int (\sum _{J\in {\mathcal{G}}_M}⟨x_i,\frac{h_J}{|J|}{⟩}^2{1}_J{)}^{1/2}\le C\tau \Vert x_i{\Vert }_{H^1}.$$

$$n_1=\min \{n\le N_0:\int (\sum _{k=n_0}^n⟨x_i,\frac{h_{J_k}}{|J_k|}{⟩}^2{1}_{J_k}{)}^{1/2}\le \tau \Vert x_i{\Vert }_{H^1}\}.$$

Full text

We re-prove analogous results of H.M. Wark for the spaces $\ell^{\infty}(H^{p}({\mathbb{T}})),$ $1<p<\infty.$ In the present paper direct combinatorics of colored dyadic intervals replaces the dependence on Szemeredi’s theorem in [8].

MSC 2000: 46B25, 46E40

Key words: Primary Spaces, Factorization of Operators, Haar System.

1. 1 Introduction
2. 2 Notation and Preliminaries
3. 3 Factorization through Operators on $\ell^{\infty}(H^{1}({\mathbb{T}}))$.
4. 4 Factorization through Operators on $\ell^{\infty}(L^{p}),$ $1<p<\infty.$

1 Introduction

Consider $\ell^{\infty}(H^{p}({\mathbb{T}})),$ consisting of bounded $H^{p}({\mathbb{T}})-$valued sequences, and let $T$ be a continuous linear operator acting on that space. The identity on $\ell^{\infty}(H^{p}({\mathbb{T}})),$ factores then through $T$ or $\mbox{ \rm Id}-T.$ This assertion holds true for the range $1<p\leq\infty$ and also when $H^{p}({\mathbb{T}})$ is replaced by the dual space ${H^{1}({\mathbb{T}})}^{*}.$ See J. Bourgain [2], H.M. Wark [8] and [6, 5]. As a consequence for each $1<p\leq\infty$ the space $\ell^{\infty}(H^{p}({\mathbb{T}})),$ is primary. When $p=\infty$ this class of spaces arises when the Pelczynski decomposition method is applied to $H^{\infty}$ or its complemented subspaces.

The conceptual framework for proving these results was introduced by J . Bourgain [2]. It consists of two basic steps and aims at replacing the factorization problem on $\ell^{\infty}(H^{p}({\mathbb{T}}))$ by its localized, finite dimensional counterpart. The first step is to show that general operators on $\ell^{\infty}(H^{p}),$ can be replaced by diagonal operators on

[TABLE]

The second (and often harder) part of the argument is to verify the following property for operators acting on finite dimensional spaces. To each $n\in{\mathbb{N}}$ and $1<p\leq\infty,$ there exists $N=N(n,p)$ so that for any linar operator on $H^{p}_{N}({\mathbb{T}})$, the identity on $H^{p}_{n}({\mathbb{T}})$ can be well-factored through $T$ or through $\mbox{ \rm Id}-T.$ That is,

[TABLE]

where $H=T$ or $H=\mbox{ \rm Id}-T$ and $\|E\|\cdot\|P\|\leq 16.$

The methods yielding (LABEL:factorization) fall roughly into two classes. The first one works is by restriction: One selects a large subset of a basis in $H^{p}_{N}({\mathbb{T}})$ on which the operator acts as a multiplier. The second method uses averaging or blocking : One attempts to find *block bases * forming a set of approximate eigenvectors. The three cases $p=\infty,$ $1<p<\infty,$ and ${H^{1}_{n}({\mathbb{T}})}^{*},$ were treated by separate techniques as summarized below:

The case ${p=\infty.}$ By harmonic analysis J. Bourgain [2] proves that there exists $\lambda\in{\mathbb{C}}$ and an arithmetic progression $\Lambda$ of length $n$ in $\{1,\dots,N\}$ so that the restriction of $T$ to the exponentials $e_{j}(t)=e^{ijt}$ satisfies

[TABLE] 2. 2.

The case ${1<p<\infty.}$ As an application of Szemeredi’s theorem [7], H.M. Wark [8] obtains $\lambda\in{\mathbb{C}}$ and an arithmetic progression $\Lambda$ of length $n$ in $\{1,\dots,N\}$ so that the restriction of $T$ to the exponentials $e_{j}(t)=e^{ijt}$ satisfies similarly $T(e_{j})\sim\lambda e_{j}$ with $j\in\Lambda.$ Here the error is small in $L^{p}.$

The $L^{p}$ boundedness of the Hilbert transform [12] allows us to replace in (LABEL:factorization) the spaces of analytic polynomials $H_{n}^{p}({\mathbb{T}})$ respectively $H_{n}^{p}({\mathbb{T}})$ finite dimensional Minkowski spaces $\ell_{m}^{p}$ of the corresponding dimensions. Now apply J. Bourgain’s and L. Tzafriri’s results on restricted invertibility. To any linear operator $T:\ell_{m}^{p}\to\ell_{m}^{p}$ there exist [3] random subsets $\sigma$ of $\{1,\dots,m\}$ of cardinality $m/C$ so that the restriction of $T$ to the corresponding $\ell_{m}^{p}-$unit vectors $\{x_{j},j\in\sigma\}$ satisfies

[TABLE] 3. 3.

The case of ${H^{1}_{n}({\mathbb{T}})}^{*}.$ By Bochkarev’s theorem [1, Section 4], the spaces ${H^{1}_{n}({\mathbb{T}})}^{*}$ admit an unconditional basis that is equivalent to the Haar basis in dyadic $BMO_{n}.$ Exploiting this connection [5], yields a block basis $\{b_{I}:I\in{\cal D}_{n}\}$ in ${H^{1}_{N}({\mathbb{T}})}^{*},$ well-equivalent to the Bochkarev basis in ${H^{1}_{n}({\mathbb{T}})}^{*}$ on which the given operator $T:{H^{1}_{N}({\mathbb{T}})}^{*}\to{H^{1}_{N}({\mathbb{T}})}^{*}$ acts as

[TABLE]

In this note we adapt the blocking of the Haar basis to the spaces $\ell^{\infty}(H^{p}({\mathbb{T}})),$ $1\leq p<\infty.$ First we settle the case of $\ell^{\infty}(H^{1}({\mathbb{T}})),$ left open in [8]. After the reduction to diagonal operators, this is obtained with Bochkarev’s theorem [1, Section 4], and reduction to [5, Theorem 2]. Second, we re-prove results of H. M. Wark [8]. We find our block bases directly, working with the Haar system; thus we provide an alternative to the method based on Szemeredi’s theorem, and also to the restricted invertibility methods.

2 Notation and Preliminaries

The Haar System.

We let ${\cal D}$ denote the collection of (half-open) dyadic intervals contained in the unit interval

[TABLE]

For $n\in{\mathbb{N}}$ write ${\cal D}_{n}=\{I\in{\cal D}:|I|\geq 2^{-n}\}.$ 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 a Schauder basis in $L^{p},$ $(1\leq p<\infty)$ and an unconditional Schauder basis when $1<p<\infty.$ For $f\in L^{p}$ define its dyadic square function as

[TABLE]

Then, $c_{p}\|f\|_{L^{p}}\leq\|S(f)\|_{L^{p}}\leq C_{p}\|f\|_{L^{p}},$ where $C_{p}\sim p^{2}/(p-1)$ and $c_{p}=C_{p}^{-1};$ see e.g., [6, Chapter 1].

To $1\leq p\leq\infty$ and $n\in{\mathbb{N}}$ we denote by $L^{p}_{n}$ the linear span of $\{h_{I}:I\in{\cal D}_{n}\},$ equipped with the norm in $L^{p}.$

$H^{p}$ spaces.

Dyadic $H^{1}$ is defined by the relation $f\in H^{1}$ if $S(f)\in L^{1}$ and its norm is

[TABLE]

Let $n\in{\mathbb{N}}.$ We denote by $H^{1}_{n}(\delta)$ the subspace of $H^{1}$ that is spanned by $\{h_{I}:|I|\geq 2^{-n}\}$ endowed with the norm in $H^{1}.$

Let ${\mathbb{T}}=\{e^{i\theta}:\theta\in[-\pi,\pi]\}.$ For $1\leq p<\infty$ let $H^{p}({\mathbb{T}})$ denote the norm closure in $L^{p}({\mathbb{T}})$ of $\mbox{\rm span }\{e^{in\theta}:n\in{\mathbb{N}}\}.$ The space $H^{\infty}({\mathbb{T}})$ is the weak$-*$ closure in $L^{\infty}({\mathbb{T}})$ of $\mbox{\rm span }\{e^{in\theta}:n\in{\mathbb{N}}\}.$ As a consequence of the $L^{p}$ boundedness of the Hilbert transform [12] the spaces $H^{p}({\mathbb{T}})$ and $L^{p}({\mathbb{T}})$ are isomorphic. By B. Maurey’s theorem [4] the spaces $H^{1}({\mathbb{T}})$ and dyadic $H^{1}$ are isomorphic.

For $1\leq p\leq\infty$ we denote by $H^{p}_{n}({\mathbb{T}})$ the subspace of $H^{p}({\mathbb{T}})$ that is spanned by the polynomials $\{e^{i\theta},\dots,e^{i(2^{n+1}-1)\theta}\}.$ The norm on $H^{p}_{n}({\mathbb{T}})$ is the one induced by $H^{p}({\mathbb{T}}).$

The Theorem of Bochkarev.

The trigonometric system does not form a Schauder basis in $H^{1}_{n}({\mathbb{T}})$ with basis constant independent of $n,$ see [12]. Nevertheless the theorem of S.V. Bochkarev [1] asserts that the spaces $H^{1}_{n}({\mathbb{T}})$ admit an unconditional basis well equivalent to the Haar basis in $H^{1}_{n}(\delta).$ That is, there exists a system $\{s_{I}\in H^{1}_{n}({\mathbb{T}}):|I|\geq 2^{-n}\}$ of functions so that the linear extension of the operator

[TABLE]

is bijective and defines an isomorphism with norm (and norm of the inverse) bounded independent of $n.$ Thus

[TABLE]

Diagonal operators.

Let $Z_{p}=\left(\sum_{n=1}^{\infty}H^{p}_{n}({\mathbb{T}})\right)_{\infty}.$ We say that $D:Z_{p}\to Z_{p}$ is a diagonal operator if there exists a sequence of opertors $T_{n}:H^{p}_{n}({\mathbb{T}})\to H^{p}_{n}({\mathbb{T}})$ so that

[TABLE]

Dyadic Trees.

Let $m\in{\mathbb{N}}.$ We say that $\{E_{I}:I\in{\cal D}_{m}\}$ is a dyadic tree of sets if, the following conditions hold for every $I\in{\cal D}_{m-1}.$ If $I_{1}$ is the left half of $I$ and $I_{2}=I\setminus I_{1}$ then

[TABLE]

Lower $\ell^{2}$ estimates.

[6, Chapter 5] Let $y_{i}\in H^{1}$ , $a_{i}\in{\mathbb{R}}$ and suppose that the sequence $\{y_{i}\}$ is disjointly supported over the Haar system. Then

[TABLE]

The Carleson Constant.

Let $m\in{\mathbb{N}}$ and ${\cal E}=\{E_{I}:I\in{\cal D}_{m}\}$ a dyadic tree. The Carleson constant of ${\cal E}$ is defined as

[TABLE]

Projections and Large Carleson Constants.

See [6, Chapter 1, 3]. Fix $N>>m,$ and let ${\cal E}_{I}\subseteq{\cal D}_{N},$ for $I\in{\cal D}_{m}$ be pairwise disjoint collections consisting of pairwise disjoint dyadic intervals, so that

[TABLE]

satisfy the following conditions:

The collection $\{E_{I}:I\in{\cal D}_{m}\}$ is a dyadic tree satisfying

[TABLE] 2. 2.

For each $I\in{\cal D}_{m-1},$

[TABLE]

where $I_{1}$ is the left half of $I$ and $I_{2}=I\setminus I_{1}.$

Then [6, Chapter 1] the orthogonal projection

[TABLE]

is a bounded operator on $L^{p}$ and on $H^{1}$ with norm $\|Q\|_{p}\leq C_{p},$ respectively $\|Q\|_{H^{1}}\leq C.$

Let next ${\cal L}\subseteq\{E_{I}:I\in{\cal D}_{m}\}.$ and

[TABLE]

By the condensation lemma in combination with the Gamlen-Gaudet theorem the following implication holds true, see [6, Chapter 3]. If $n\in{\mathbb{N}}$ is such that $[\![{\cal L}]\!]\geq n2^{n},$ then there exist linear operators $E:L^{p}_{n}\to Y_{p}$ and $R:Y_{p}\to L_{n}^{p}$ so that

[TABLE]

3 Factorization through Operators on

$\ell^{\infty}(H^{1}({\mathbb{T}}))$.

In this section we prove that for any operator $T$ on $\ell^{\infty}(H^{1}({\mathbb{T}}))$ the identity factores through $T$ or $\mbox{ \rm Id}-T.$

Theorem 3.1

Let $T$ be a bounded operator on $\ell^{\infty}(H^{1}({\mathbb{T}}))$ with $\|T\|\leq 1.$ Then there exist an embedding $E$ and a projection $P$ so that

[TABLE]

where $H=T$ or $H=\mbox{ \rm Id}-T$ and $\|E\|\cdot\|P\|\leq C.$

Infer from Maurey’s theorem [4] that $H^{1}({\mathbb{T}})$ can be replaced by dyadic $H^{1},$ and from Wojtaszczyk’s theorem [10, 11] that $\ell^{\infty}(H^{1}({\mathbb{T}}))$ is isomorphic to $(\sum H^{1}_{n}(\delta))_{\infty}.$

First we give the reduction of Theorem LABEL:27okt1 to diagonal operators on

[TABLE]

Theorem 3.2

Let $T$ be a bounded operator on $\ell^{\infty}(H^{1}({\mathbb{T}}))$ with $\|T\|\leq 1.$ Then there exist an embedding $E:\ell^{\infty}(H^{1}({\mathbb{T}}))\to X,$ a projection $P:X\to\ell^{\infty}(H^{1}({\mathbb{T}}))$ and a diagonal operator $D:X\to X$ satisfying $\|D\|\leq C$ and

[TABLE]

where $\|E\|\cdot\|P\|\leq C.$

It is well known that the diagonal operator $D$ together with its embedding $E$ and projection $P$ can be constructed in straightforward manner from the assertions of the following proposition.

Proposition 3.3

To $\epsilon>0$ and $n\in{\mathbb{N}}$ there exists $N=N(\epsilon,n)$ with the following property: To each $n-$dimensional subspace $E\subseteq H^{1}_{N}$ there exist pairwise disjount collections of dyadic intervals $\{{\cal B}_{J}:J\in{\cal D}_{n}\}$ so that

The sets $B_{J}=\bigcup_{I\in{\cal B}_{J}}I$ form a dyadic tree satisfying $|J|/2\leq|B_{J}|/|B_{[0,1[}\leq|J|.$ 2. 2.

The orthogonal projection

[TABLE]

satisfies $\|Q\|\leq 4.$ 3. 3.

For each $x\in E,$

[TABLE]

Proof. Let $\eta>0$ and $\{x_{1},\dots,x_{M}\}$ be an $\eta-$net in $\{x\in E:\|x\|_{H^{1}}=1\}.$ This can be done with $M\leq\exp(2n/\eta).$

Fix $\tau=\tau(\epsilon,n)>0$ and put ${\cal L}_{i}=\{J\in{\cal D}_{N}:|\langle x,h_{J}\rangle|\geq\tau\}.$ Apply (LABEL:lowerl2) to obtain an upper estimate for the cardinality of ${\cal L}_{i}.$

[TABLE]

for some $c>0.$ Hence the union ${\cal L}=\bigcup_{i=1}^{M}{\cal L}_{i}$ is of cardinality $\leq Cn/\tau^{2}.$ Put ${\cal G}_{0}={\cal D}_{N}\setminus{\cal L}.$ Its Carleson constant is bounded from below as

[TABLE]

In the next steps we select a subcollection ${\cal G}_{M}\subseteq{\cal G}_{0}$ with the following two properties

$[\![{\cal G}_{M}]\!]\geq[\![{\cal G}_{0}]\!](\tau^{2M}c)$ 2. 2.

For $i\leq M,$

[TABLE]

The construction of $G_{M}$ is done inductively over $M$ steps. In the first step we fix $x_{1}$ and let $\{J_{k}:1\leq k\leq N_{0}\}$ with $N_{0}\leq 2^{N-1}$ be an enumeration of the intervals in ${\cal G}_{0}.$ Define inductively a stopping time sequence $n_{0},\dots,n_{L}\leq N_{0}$ as follows. Put $n_{o}=0$ and

[TABLE]

Assume that $n_{0},n_{1},\dots,n_{\ell}$ have been determined and that $n_{\ell}<\infty.$ Then define

[TABLE]

Finally define

[TABLE]

We use again (LABEL:lowerl2). Split $x_{1}$ along the Haar basis as $x_{1,\ell}=\sum_{k=n_{\ell}+1}^{n_{\ell+1}}\langle x_{1},\frac{h_{J_{k}}}{|J_{k}|}\rangle h_{J_{k}}.$ Note that $x_{1,\ell}$ are disjointly supported over the Haar system so that the lower $\ell^{2}$ estimate (LABEL:lowerl2) gives

[TABLE]

Hence $L\leq C\tau^{-2}.$ Define next the partition of ${\cal G}_{0}$ as ${\cal G}_{0}=\bigcup_{\ell=1}^{L}{\cal G}_{0,\ell},$ where ${\cal G}_{0,\ell}=\bigcup_{k=n_{\ell}+1}^{n_{\ell+1}}\{J_{k}\}.$ Consequently there exists $\ell_{0}\leq L$ so that $[\![{\cal G}_{0,\ell_{0}}]\!]\geq[\![{\cal G}_{0}]\!]/L.$ Put ${\cal G}_{1}={\cal G}_{0,\ell_{0}}.$ Summing up the first step, we obtained ${\cal G}_{1}\subseteq{\cal G}_{0}$ so that

[TABLE]

and

[TABLE]

This completes the first step. In the second step we repeat the argument of the first step with $x_{1}$ replaced by $x_{2}$ and ${\cal G}_{0}$ replaced by ${\cal G}_{1}.$ This implies the existence of ${\cal G}_{2}\subseteq{\cal G}_{1}$ so that

[TABLE]

and

[TABLE]

Iterating $M-$times we obtain a decreasing chain of collections ${\cal G}_{M}\subseteq\dots\subseteq{\cal G}_{0},$ so that

[TABLE]

and

[TABLE]

We specify now $\tau,\eta$ so that $(\tau+\eta)<\epsilon 2^{-n}(\log n)^{-1}.$ Consequently there exists $N=N(\epsilon,n)$ so that

[TABLE]

Next we apply the condensation lemma and the Gamlen-Gaudet selection process [6, Chapter 3] to ${\cal G}_{M}.$

This implies that there exist $\{{\cal B}_{J}\subseteq{\cal G}_{M}:J\in{\cal D}_{n}\}$ so that $B_{J}=\bigcup_{I\in{\cal B}_{J}}I$ and $b_{J}=\sum_{I\in{\cal B}_{J}}h_{I}$ satisfy the following conditions:

$\{B_{J}:J\in{\cal D}_{n}\}$ is a dyadic tree and

[TABLE] 2. 2.

$B_{J_{1}}\subseteq\{t:h_{I}(t)=+1\}$ and $B_{J_{2}}\subseteq\{t:h_{I}(t)=-1\},$ where $J_{1}$ is the left half of $J$ and $J_{2}=J\setminus J_{1}.$

Next we claim that moreover

[TABLE]

To see this fix $x\in E$ with $\|x\|_{H^{1}}=1$ Choose $x_{i}\in\{x_{1},\dots,x_{m}\}$ so that $\|x-x_{i}\|_{H^{1}}<\eta,$ and estimate using the defining properties of ${\cal G}_{M}.$

[TABLE]

It remains to verify that $\|Q(x)\|_{H^{1}}\leq\epsilon$ for $x\in E.$ Using the above coefficient estimate and triangle inequality we get,

[TABLE]

We have now reduced the problem of factoring the identity on $\ell^{\infty}(H^{1}({\mathbb{T}}))$ to its finite dimensional counterpart. The next step in the proof is factorization of the identity on $H^{1}_{N}({\mathbb{T}}).$ The following theorem settles a point left open in [8].

Theorem 3.4

For any $n\in{\mathbb{N}}$ there exists $N=N(n)$ such that for any linear operator $T:H^{1}_{N}({\mathbb{T}})\to H^{1}_{N}({\mathbb{T}})$ the identity on $H^{1}_{n}({\mathbb{T}})$ factores through $H=T$ or $H=\mbox{ \rm Id}_{H^{1}_{N}({\mathbb{T}})}-T,$ as

[TABLE]

where $E:H^{1}_{n}({\mathbb{T}})\to H^{1}_{n}({\mathbb{T}})$ and $P:H^{1}_{n}({\mathbb{T}})\to H^{1}_{n}({\mathbb{T}})$ are bounded linear operators saitsfying

[TABLE]

Proof. Apply the theorem of S.V. Bochkarev [1, Section 4], and dualize [5, Theorem 2].

4 Factorization through Operators on

$\ell^{\infty}(L^{p}),$ $1<p<\infty.$

In this section we re-prove the theorem of H.M. Wark [8] that for any bounded operator $T$ on $\ell^{\infty}(L^{p})$ the identity factores through $T$ or $\mbox{ \rm Id}-T.$

Theorem 4.1

Let $T$ be a bounded operator on $\ell^{\infty}(L^{p}({\mathbb{T}}))$ with $\|T\|\leq 1.$ Then there exist an embedding $E$ and a projection $P$ so that

[TABLE]

where $H=T$ or $H=\mbox{ \rm Id}-T$ and $\|E\|\cdot\|P\|\leq C_{p}.$

It is well known that Theorem LABEL:37okt1 follows from the next theorem.

Theorem 4.2

For any $1<p<\infty$ and any $n\in{\mathbb{N}}$ there exists $N=N(n,p)$ so that for any linear operator $T:L^{p}_{N}\to L^{p}_{N}$ the identity on $L^{p}_{n}$ factores through $H=T$ or $H=\mbox{ \rm Id}_{L^{p}_{N}}-T,$ as

[TABLE]

where $E:L^{p}_{n}\to L^{p}_{N}$ and $P:L^{p}_{N}\to L^{p}_{n}$ are bounded linear operators satisfying $\|E\|\cdot\|P\|\leq C_{p}.$

We obtain the finite dimensional factorization of Theorem LABEL:Lp by exhibiting a block-basis of the Haar system on which $T$ acts as a multiplier. We use combinatorics of colored dyadic intervals as in [6]. (For an alternative derivation see the appendix.) The desired block bases are constructed below.

Theorem 4.3

For every $1<p<\infty,$ and $m\in{\mathbb{N}}$ there exists $N=N(m,p)$ so that the following holds true: To every linear operator $H:L^{p}_{N}\to L^{p}_{N}$ bounded by $\|H\|\leq 1$ there exist pairewise disjoint collections

[TABLE]

consisting of pairwise disjoint intervals so that

[TABLE]

satisfy the following conditions:

The collection $\{E_{I}:I\in{\cal D}_{m}\}$ is a dyadic tree satisfying

[TABLE] 2. 2.

For each $I\in{\cal D}_{m-1},$

[TABLE]

where $I_{1}$ is the left half of $I$ and $I_{2}=I\setminus I_{1}.$ 3. 3.

For each $I\in{\cal D}_{m},$

[TABLE]

The following Lemma LABEL:thin is the main component in the proof of Theorem LABEL:gg. Fix $1<p<\infty,$ $I\in{\cal D}$ and $k\in{\mathbb{N}}.$ Let $x\in L^{p},$ $y\in L^{q}$ and $1/p+1/q=1$ so that

[TABLE]

Define ${\cal B}$ to be the collection of dyadic intervals for which the Haar coefficients of $x$ resectively $y$ are above the critical threshold $|J|/k,$ thus

[TABLE]

Lemma 4.4

Let $\ell\in{\mathbb{N}}$ and put

[TABLE]

There exists $1\leq j\leq A_{p}$ so that the collection ${\cal B}_{j}=\{J\in{\cal B}:|J|=2^{-j}|I|\}$ satisfies

[TABLE]

Proof. Assume that the lemma is false. That is, for each $j\in A_{p}$ we have

[TABLE]

For $J\in{\cal B}_{j}$ there holds $|\langle x,h_{J}\rangle|+|\langle y,h_{J}\rangle|>|J|/k.$ Hence by summing the lower estimate (LABEL:16okt2) we obtain

[TABLE]

On the other hand it follows from the unconditionality of the Haar system in $L^{p},1<p<\infty$ that for any choice of $\alpha_{J}\in[-1,+1]$

[TABLE]

Invoking the assumptions that $\|x\|_{p}\leq|I|^{1/p}$ and $\|y\|_{q}\leq|I|^{1/q}$, the unconditionality of the Haar system (LABEL:16okt4) yields the following upper estimate for the Haar coefficients competing with (LABEL:16okt3),

[TABLE]

Comparison of (LABEL:16okt3) with (LABEL:16okt5) gives an upper estimate for $A_{p}$ as follows,

[TABLE]

By cancellation and arithmetic we get $A_{p}\leq\ell^{2}k^{2}(C_{q}+C_{p})^{2},$ contradicting the initial choice of $A_{p}.$

Proof of Theorem LABEL:gg.

Put $b_{1}=h_{[0,1]},$ and ${\cal E}_{1}=\{[0,1[\}.$ At stage $i$ we are given pairwise disjoint collections of dyadic intervals in ${\cal D}_{N},$ ${\cal E}_{1},\dots,{\cal E}_{i}$ and functions $b_{1},\dots,b_{N}$ so that

[TABLE]

Fix ${\cal E}_{i}$ and $J\in Q({\cal E}_{i}).$ Put

[TABLE]

and define the collections of good intervals as follows

[TABLE]

By Lemma LABEL:thin there exists

[TABLE]

so that the collection of dyadic intervals

[TABLE]

satisfies the lower estimate

[TABLE]

Next we define the collection of dyadic intervals ${\cal E}_{i+1}$ and the associated function $b_{i+1}$ using to the well known Gamlen Gaudet construction. In case $i$ is even we put

[TABLE]

Similarly if $i$ is odd, we put

[TABLE]

Since ${\cal E}_{i+1}\subseteq{\cal C}\subseteq{\cal G}$ our construction so far, implies that $b_{i+1}$ satisfies the almost orthogonality relation

[TABLE]

In view of (LABEL:16okt6) and (LABEL:16okt7) we may now repeat the induction step. Taking into account the (cummulative effect of) the lower estimate (LABEL:16okt8) and (LABEL:27okt11) it is easy to see that we may choose $N=N(\epsilon,n)$ large enough so that we may continue iterating until we reach step $2^{m+1}-1.$ The system $\{b_{1},\dots,b_{2^{m+1}-1}\}$ satisfies then the full set of almost orthogonalty relations

[TABLE]

It remains to relabel the collections $\{{\cal E}_{i}\}$ and functions $\{b_{i}\}$ appropriately. Since $\mu\leq m$ and $1\leq k\leq 2^{\mu}$ define uniquely

[TABLE]

we obtain a canonical bijection between ${\cal D}_{m}$ and $\{1,\dots,2^{m+1}-1\}.$ We relabel accordingly using this correspondence,

[TABLE]

As a result of the Gamlen Gaudet construction the sets

[TABLE]

form a dyadic tree, and the conditions (LABEL:16okt8) translate into a measure estimate

[TABLE]

By (LABEL:22okt1)—(LABEL:22okt3), the orthogonal projection

[TABLE]

is bounded operator on $L^{p}$ with norm $\|Q\|_{p}\leq C_{p}.$

Proof of Theorem LABEL:Lp.

Fix $n\in{\mathbb{N}}$ and put $m=C_{p}n2^{n}.$ Determine $N=N(m,p)$ so that the conclusion of Theorem LABEL:gg holds true. Accordingly we may select pairewise disjoint collections $\{{\cal E}_{I}\subseteq{\cal D}_{N}:I\in{\cal D}_{m}\}$ consisting of pairwise disjoint intervals so that conditions (LABEL:22okt11)–(LABEL:22okt14) hold true.

Next define the collections

[TABLE]

and ${\cal R}={\cal L}\setminus\{E_{J}:J\in{\cal D}_{m}\}.$ Clearly, then ${\cal L}\cup{\cal R}=\{E_{J}:J\in{\cal D}_{m}\},$ and hence, one of these collections satisfies a lower estimate for its Carleson Constant,

[TABLE]

We continue under the assumption that the first alternative holds and remark that otherwise we would replace $T$ by $\mbox{ \rm Id}-T$ and continue with the argument given below. Define, then, the space

[TABLE]

Since $[\![{\cal L}]\!]\geq n2^{n}/2$, (LABEL:22okt1)–(LABEL:22okt4) imply that there exist linear operators

[TABLE]

so that $\mbox{ \rm Id}_{L^{p}_{n}}=R\mbox{ \rm Id}_{Y_{p}}I$ and $\|R\|_{p}\cdot\|I\|_{p}\leq C_{p}.$ On $L^{p}_{n}$ define the operator

[TABLE]

By choice of ${\cal L}$ we get with (LABEL:22okt1)–(LABEL:22okt3) that

[TABLE]

We will verify next that for $g\in Y_{p},$

[TABLE]

To this end fix $I\in{\cal L}$ and expand $PT(b_{I})$ as

[TABLE]

Next let $g\in Y_{p},$ hence $g=\sum_{I\in{\cal L}}c_{I}b_{I}.$ Put

[TABLE]

and define the error term

[TABLE]

Then, by arithmetic we get

[TABLE]

Since $\beta(J)\leq C_{p}^{-1}|J|^{4}\|g\|_{p},$ and $|\langle Tb_{J},b_{J}\rangle|\geq\|b_{J}\|_{2}^{2}/2,$ whenever $J\in{\cal L},$ we obtain with triangle inequality, the following estimate for the off-diagonal term,

[TABLE]

which gives (LABEL:22okt21).

Appendix: Reviewing restricted invertibility.

A proof of Theorem LABEL:Lp follows directly from the random methods of J. Bourgain and L. Tzafriri [3] on restricted invertibility of matrices.

Theorem 4.5

For any $1<p<\infty$ and any $n\in{\mathbb{N}}$ there exists $N=N(n,p)$ so that for any linear operator $T:\ell^{p}_{N}\to\ell^{p}_{N}$ the identity on $\ell^{p}_{n}$ factores through $H=T$ or $H=\mbox{ \rm Id}_{\ell^{p}_{N}}-T,$ as

[TABLE]

where $E:\ell^{p}_{n}\to\ell^{p}_{N}$ and $P:\ell^{p}_{N}\to\ell^{p}_{n}$ are bounded linear operators satisfying

[TABLE]

Proof. The operator $T$ has matrix representation with respect to $\{e_{j}:j\leq n\}$, the unit vector basis of $\ell^{p}_{N}.$ Define ${\cal L}=\{i\leq n:|(Te_{i},e_{i})|\geq 1/2\}$ and ${\cal R}=\{1,\dots,n\}\setminus{\cal L}.$ Thus the cardinality of ${\cal L}$ or ${\cal R}$ is at least $N/2.$ Assume without loss of generality that $|{\cal L}|\geq N/2.$ (Otherwise replace $T$ by $\mbox{ \rm Id}-T$ and proceed as below.) Put $N_{1}=|\{1,\dots,N\}\cap{\cal L}|.$ Clearly, there exist now linear operators $E_{1}:\ell^{p}_{N}\to\ell^{p}_{N_{1}}$ and $P_{1}:\ell^{p}_{N_{1}}\to\ell^{p}_{N},$ and an operator $S:\ell^{p}_{N_{1}}\to\ell^{p}_{N_{1}}$ satisfying the following conditions: First, $S$ factores through $T$ as

[TABLE]

and, second,

[TABLE]

Now we apply the theorem of J. Bourgain and L. Tzafriri [3] on restricted invertibility to the operator $S:\ell^{p}_{N_{1}}\to\ell^{p}_{N_{1}}.$ To each given $\epsilon>0$ it yields the existence of a constant $C=C(\epsilon,p)$ and subset $\sigma\subseteq\{1,\dots,N_{1}\}$ so that $|\sigma|\geq N_{1}/C$ and $R_{\sigma}SR_{s}$ is invertible (on its range) and satisfies

[TABLE]

(Here $R_{\sigma}$ denotes the operator of restriction to basis vectors $\{e_{j}:j\in\sigma\}.$ ) Now put $N_{2}=|\{1,\dots,N_{1}\}\cap\sigma|.$ By restricted invertibility there are now operators $E_{2}:\ell^{p}_{N_{2}}\to\ell^{p}_{N_{1}}$ and $P_{1}:\ell^{p}_{N_{1}}\to\ell^{p}_{N_{2}},$ so that the identity on $\ell^{p}_{N_{2}}$ factores through $S$ as

[TABLE]

In summary, we obtained the factorization through $T,$

[TABLE]

Address:

Institut für Analysis

J. Kepler Universität

A-4040 Linz

Austria.

pfxm@bayou.uni-linz.ac.at