lacunary Walsh series in rearrangement invariant spaces

TL;DR

This paper extends classical results about Rademacher systems in rearrangement invariant spaces to lacunary Walsh series, showing similar subspace properties hold for these series.

Contribution

It demonstrates that properties known for Rademacher systems also apply to lacunary Walsh series within rearrangement invariant spaces, broadening the scope of these classical results.

Findings

Classical results extend to lacunary Walsh series

Subspace properties hold for lacunary Walsh series

Similar behavior as Rademacher systems in rearrangement invariant spaces

Abstract

We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series.

Equations

$$r_k(t):=\mathrm{sign}\sin (2^k\pi t),t\in [0,1],k\ge 1.$$

$$A_{p}\Big{(}\sum_{k\geq 1}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}r_{k}\Big{\|}_{L^{p}([0,1])}\leq B_{p}\Big{(}\sum_{k\geq 1}a_{k}^{2}\Big{)}^{1/2},$$

$$A_{X}\Big{(}\sum_{k\geq 1}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}r_{k}\Big{\|}_{X}\leq B_{X}\Big{(}\sum_{k\geq 1}a_{k}^{2}\Big{)}^{1/2}$$

$$A(p,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k=1}^{\infty}a_{k}w_{n_{k}}\Big{\|}_{L^{p}([0,1])}\leq B(p,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}$$

$$A^{\prime}_{2}\Big{(}\sum_{k\geq N}a_{k}^{2}\Big{)}^{1/2}\leq\Big{(}\int_{E}\Big{|}\sum_{k\geq N}a_{k}r_{k}\Big{|}^{2}\frac{dm}{m(E)}\Big{)}^{1/2}\leq B^{\prime}_{2}\Big{(}\sum_{k\geq N}a_{k}^{2}\Big{)}^{1/2},$$

$$A^{\prime}(p,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{(}\int_{E}\Big{|}\sum_{k\geq N}a_{k}w_{n_{k}}\Big{|}^{p}\frac{dm}{m(E)}\Big{)}^{1/p}\leq B^{\prime}(p,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}$$

$$\|g\|_{X^{\prime}}:=\sup\Big{\{}\Big{|}\int_{0}^{1}fg\,dm\,\Big{|}:\|f\|_{X}\leq 1\Big{\}}$$

$$\Phi (s)={\int }_0^s\varphi (t)dt,$$

$$\|f\|_{L^{\Phi}}:=\inf\Big{\{}\lambda>0:\int_{0}^{1}\Phi(|f|/\lambda)\,dm\leq 1\Big{\}}$$

$$w_k:=r_1^{a_1}\cdot \dots \cdot r_n^{a_n}.$$

$$\Vert f{\Vert }_{X|E}:=\Vert f\circ {\rho }_E^{-1}{\Vert }_X$$

$$m_E(A):=\frac{m(E\cap A)}{m(E)},A\subset E,A\text{ measurable}.$$

$$\|f\|_{L^{p}|E}=\Big{(}\int_{E}|f|^{p}\frac{dm}{m(E)}\Big{)}^{1/p},$$

$$\|f\|_{L^{M_{p}}|E}:=\inf\Big{\{}\lambda>0:\int_{E}M_{p}(|f|/\lambda)\,\frac{dm}{m(E)}\leq 1\Big{\}}.$$

$$A^{\prime}(M_{2},q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{L^{M_{2}}|E}\leq B^{\prime}(M_{2},q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$B^{\prime }(2n,q)\le (1+\sqrt{2})(2+2\alpha {)}^{1/2n}{n}^{1/2}$$

$$C_{1}{A^{\prime}(1,q)}\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq C_{1}\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{L^{1}|E}\leq\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{L^{M_{2}}|E},$$

$$f:=\sum _{k\ge 1}{a}_k{w}_{n_k}.$$

$$\begin{split}\int_{E}\big{(}\exp|f(t)/\lambda|^{2}-1\big{)}\frac{dt}{m(E)}&=\sum_{n\geq 1}\frac{1}{n!\lambda^{2n}}\int_{E}\Big{|}\sum_{k\geq 1}a_{k}w_{n_{k}}(t)\Big{|}^{2n}\frac{dt}{m(E)}\\ &\leq\sum_{n\geq 1}\frac{B^{\prime}(2n,q)^{2n}}{n!\lambda^{2n}}\|(a_{k})_{1}^{\infty}\|_{2}^{2n}.\end{split}$$

$$\begin{split}&\int_{E}\big{(}\exp|f(t)/\lambda|^{2}-1\big{)}\frac{dt}{m(E)}\\ &\qquad\leq\sum_{n\geq 1}\frac{\big{(}(1+\sqrt{2})(2+2\alpha)^{1/2n}n^{1/2}\big{)}^{2n}}{n!\lambda^{2n}}\|(a_{k})_{1}^{\infty}\|_{2}^{2n}\\ &\qquad=(2+2\alpha)\sum_{n\geq 1}\frac{(1+\sqrt{2})^{2n}n^{n}}{n!\lambda^{2n}}\|(a_{k})_{1}^{\infty}\|_{2}^{2n}\\ &\qquad\leq C_{2}(2+2\alpha)\sum_{n\geq 1}\Big{(}\frac{(1+\sqrt{2})^{2}e}{\lambda^{2}}\|(a_{k})_{1}^{\infty}\|_{2}^{2}\Big{)}^{n}.\end{split}$$

$$\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{L^{M_{2}}|E}\leq B^{\prime}(M_{2},q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$A^{\prime}(X,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{X|E}\leq B^{\prime}(X,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$R_M:=\sum _{k=1}^M{a}_k{r}_k\text{and}{W}_M:=\sum _{k=1}^M{a}_k{w}_{n_k}$$

$$W_M=\sum _{{\epsilon }_1,\dots ,{\epsilon }_M=\pm 1}({\epsilon }_1{a}_1+\dots {\epsilon }_M{a}_M){\chi }_{A({\epsilon }_1,\dots ,{\epsilon }_M)},$$

$$\Vert f{\Vert }_X\le C\Vert f{\Vert }_{L^{M_2}}$$

$$A(X,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{X}\leq B(X,q)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$A(X)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2}\leq\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{X}\leq B(X)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$\Big{\|}\sum_{k\geq 1}a_{k}w_{n_{k}}\Big{\|}_{X}\leq B(X)\Big{(}\sum_{k=1}^{\infty}a_{k}^{2}\Big{)}^{1/2},$$

$$s_n:=\frac{1}{\sqrt{n}}\sum _{k=1}^n{w}_{m_k},v_n:=\frac{1}{\sqrt{n}}\sum _{k=1}^n{r}_k.$$

$$\Vert v_n{\Vert }_X=\Vert s_n{\Vert }_X\le B(X),$$

Full text

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1. Introduction

The Rademacher functions are

[TABLE]

The Rademacher system $(r_{k})$ is orthonormal, not complete, independent and identically distributed on $[0,1]$. Its behavior in rearrangement invariant function spaces has been pretty well studied. Khintchine inequality, [8], and the theorems by Rodin and Semenov and Lindenstrauss and Tzafriri are among the core results on this topic.

Theorem 1** (Khintchine inequality).**

Given $0<p<\infty$, there exist constants $A_{p},B_{p}>0$ such that

[TABLE]

for all $(a_{k})\in\ell^{2}$.

Rodin and Semenov characterized those rearrangement invariant spaces $X$ for which Khintchine inequality holds with $X$ in place of $L^{p}$, [11, Theorem 6]. Denote by $G$ the closure in $L^{\infty}$ of the Orlicz space $L^{M_{2}}$ generated by $M_{2}(t):=\exp(t^{2})-1$.

Theorem 2** (Rodin and Semenov).**

Let $X$ be a rearrangement invariant space on $[0,1]$. The following conditions are equivalent:

1. i)

There exist constants $A_{X},B_{X}>0$ such that the inequality

[TABLE]

holds for all $(a_{k})\in\ell^{2}$. 2. ii)

The continuous embedding $G\subset X$ holds.

The complementability of the closed linear subspace $[r_{k}]_{X}$ generated by the Rademacher system in a rearrangement invariant space $X$ was also characterized by means of the space $G$, [10, Theorem 2.b.4] and [12]. Denote by $X^{\prime}$ the associate space of $X$.

Theorem 3** (Rodin and Semenov, Lindenstrauss and Tzafriri).**

Let $X$ be a rearrangement invariant space on $[0,1]$. The following conditions are equivalent:

1. i)

The subspace $[r_{k}]_{X}$ is complemented in $X$. 2. ii)

The continuous embeddings $G\subset X$ and $G\subset X^{\prime}$ hold.

The Walsh system $(w_{k})$ on $[0,1]$ consists of all finite products of Rademacher functions. Unlike the Rademacher system, the Walsh system is complete and not independent. However, Sagher and Zhou showed that Khintchine inequality also holds for lacunary sequences of Walsh functions, [14, Theorem 1].

Theorem 4** (Sagher and Zhou).**

Given $0<p<\infty$ and $q>1$, there exist constants $A(p,q),B(p,q)>0$ such that, for any sequence $(w_{n_{k}})$ of Walsh functions with $n_{k+1}/n_{k}\geq q>1$ for all $k\geq 1$, the inequalities

[TABLE]

hold for all $(a_{k})_{1}^{\infty}\in\ell^{2}$.

In this paper we show that Theorem 2 and Theorem 3 also hold for lacunary sequences of Walsh functions (Theorem 11 and Theorem 13).

We also consider local versions of Khintchine inequality. The first local result was given by Zygmund for $L^{2}$, [16, Lemma V.8.3].

Lemma 5** (Zygmund).**

There exist constants $A^{\prime}_{2},B^{\prime}_{2}>0$ such that, for any set $E\subset[0,1]$ with $m(E)>0$, there exists $N=N(E)$ such that

[TABLE]

for all $(a_{k})$ in $\ell^{2}$.

Zygmund’s local result has been generalized in two different directions: first, by considering rearrangement invariant function spaces $X$ with $G\subset X\subset L^{2}$ (see, for example, [2], [3],[5], [6], [13] and [15]); and second, by considering lacunary sequences of Walsh functions in place of the Rademacher system. In this regard, Sagher and Zhou proved the following result, [14, Theorem 2].

Theorem 6** (Sagher and Zhou).**

Given $0<p<\infty$ and $q>1$, there exist constants $A^{\prime}(p,q),B^{\prime}(p,q)>0$ such that for any set $E\subset[0,1]$ of positive measure, there exists $N=N(E,q)$ so that for any sequence $(w_{n_{k}})$ of Walsh functions with $n_{k+1}/n_{k}\geq q>1$ for all $k\geq 1$, the inequalities

[TABLE]

hold for all $(a_{k})_{1}^{\infty}\in\ell^{2}$.

We extend this local result for lacunary sequences of Walsh functions on $L^{p}$ to the space $L^{M_{2}}$ of functions of square exponential integrability (Theorem 7).

2. Preliminaries

A Banach function space over $[0,1]$ is a linear subspace $X$ of measurable functions on $[0,1]$, endowed with a complete norm $\|\cdot\|_{X}$, such that $g\in X$ and $|f|\leq|g|$ a.e. implies $f\in X$ and $\|f\|_{X}\leq\|g\|_{X}$. The associate space $X^{\prime}$ of a Banach function space $X$ consists of all measurable functions $g$ on $[0,1]$ for which the associate functional

[TABLE]

is finite. The inclusion $X^{\prime}\subset X^{*}$ always holds between the associate space $X^{\prime}$ and the dual Banach space $X^{*}$. The spaces $X^{\prime}$ and $X^{*}$ are isomorphic if and only if $X$ has absolutely continuous norm (that is, order bounded increasing sequences are norm convergent).

We denote the distribution function of a measurable function $f$ on $[0,1]$ by $m_{f}(\lambda):=m(\{x\in[0,1]:|f(x)|>\lambda\})$, for all $\lambda>0$. A Banach function space $X$ over $[0,1]$ is rearrangement invariant (r.i.) if $m_{f}=m_{g}$ and $f\in X$ imply $g\in X$ and $\|g\|_{X}=\|f\|_{X}$. The associate space $X^{\prime}$ of an r.i. space $X$ is an r.i. space.

The second associate space of $X$ is $X^{\prime\prime}:=(X^{\prime})^{\prime}$. The embedding $X\subset X^{\prime\prime}$ holds for any Banach function space $X$. A Banach function space $X$ satisfies the Fatou property if $f_{n}\in X$ with $\|f_{n}\|_{X}\leq M$ for all $n\geq 1$ and $0\leq f_{n}\leq f_{n+1}\nearrow f$ a.e. implies that $f\in X$ and $\|f\|_{X}=\sup_{n}\|f_{n}\|_{X}$. A Banach function space $X$ satisfies the Fatou property if and only if $X^{\prime\prime}$ coincides with $X$.

For $X$ an r.i. space on $[0,1]$, the embeddings $L^{\infty}\subset X\subset L^{1}$ hold. Denote by $X_{0}$ the closure of $L^{\infty}$ in $X$.

A Banach space $X$ on $[0,1]$ for which the continuous embeddings $L^{\infty}\subset X\subset L^{1}$ hold is called an intermediate space between $L^{1}$ and $L^{\infty}$. The space $X$ is called an interpolation space if, for every linear operator $T$ such that $T\colon L^{1}\to L^{1}$ and $T\colon L^{\infty}\to L^{\infty}$ are continuous, then $T\colon X\to X$ is continuous.

Interpolation spaces between $L^{1}([0,1])$ and $L^{\infty}([0,1])$ are (after renorming if necessary) rearrangement invariant, and rearrangement invariant spaces which satisfy the Fatou property or are separable are interpolation spaces between $L^{1}([0,1])$ and $L^{\infty}([0,1])$ (for precise details, see [9, Chp. II, §4]).

A Young function is as function of the form

[TABLE]

where $\phi\colon[0,+\infty)\to[0,+\infty)$ is increasing, left–continuous and $\phi(0)=0$. The Orlicz space $L^{\Phi}$ generated by a Young function $\Phi$ consists of all measurable functions $f$ on $[0,1]$ for which the norm

[TABLE]

is finite. The space $G:=(L^{M_{2}})_{0}$, where $L^{M_{2}}$ is the Orlicz space generated by $M_{2}(t):=\exp(t^{2})-1$, is of particular interest in the study of the Rademacher system.

We consider the Walsh system according to Paley’s numbering, that is, $w_{0}:=1$, and for $k=a_{1}2^{0}+\ldots+a_{n}2^{n-1}$ with $a_{0},\ldots,a_{n}\in\{0,1\}$,

[TABLE]

A sequence $(w_{n_{k}})$ of the Walsh system is $q$–lacunary if $n_{k+1}/n_{k}\geq q>1$ for all $k\geq 1$. Since $w_{2^{n}}=r_{n+1}$, the Rademacher system $(r_{k})$ is a $2$–lacunary sequence of Walsh functions.

For details on the theory of rearrangement invariant spaces, see [4], [9] and [10].

3. When is the subspace $[w_{n_{k}}]_{X}$ isomorphic to $\ell^{2}$?

We start proving an extension of the local version of Khintchine inequality for lacunary sequences of Walsh functions on $L^{p}$ by Sagher and Zhou (Theorem 6). To this aim we introduce the following concept.

Given a set $E\subset[0,1]$ with $m(E)>0$, consider the mapping $\rho_{E}\colon E\to[0,1]$ defined as $\rho_{E}(x):=m(E\cap[0,x])/m(E)$, $x\in[0,1]$. There exist sets $A_{1}\subset E$ and $A_{2}\subset[0,1]$ of measure zero such that $\rho_{E}\colon E\setminus A_{1}\to[0,1]\setminus A_{2}$ is bijective. For $X$ an r.i. space on $[0,1]$, the local space $X|E$ consists of all measurable functions $f$ on $E$ for which the norm

[TABLE]

is finite. Here, $\rho_{E}^{-1}$ is the left continuous inverse of the increasing function $\rho_{E}$. The space $X|E$ is an r.i. space on $E$ endowed with the measure

[TABLE]

The spaces $L^{p}|E$ and $L^{M_{p}}|E$ coincide with the spaces in the local results by Zygmund for $L^{2}$ (Lemma 5), Sagher and Zhou for $L^{p}$ and $L^{M_{1}}$ (see [13], [14] and [15]) and Carrillo–Alanís for $L^{M_{2}}$ [5, Theorem 4], that is, for $0<p<\infty$,

[TABLE]

and, for $M_{p}(t):=\exp(t^{p})-1$, $p>0$,

[TABLE]

The definition of $X|E$ allows us to consider a local space on $E$ in the cases when an explicit expression of the norm of $X$ is not available (see [6] for details).

Then, we have the following result for $L^{M_{2}}|E$.

Theorem 7**.**

Let $q>1$ and $E\subset[0,1]$ be a set of positive measure. Consider the space $L^{M_{2}}$ (of functions of square exponential integrability). There exist constants $A^{\prime}(M_{2},q)$, $B^{\prime}(M_{2},q)>0$ and $N=N(E)$ such that, for any $q$–lacunary sequence $(w_{n_{k}})$ of Walsh functions with $n_{k}\geq N$ for all $k\geq 1$, we have

[TABLE]

for all $(a_{k})_{1}^{\infty}\in\ell^{2}$.

Proof.

We will use the inequality

[TABLE]

for the constant $B^{\prime}(p,q)$ in Theorem 6, where $\alpha$ is the least integer such that $q^{\alpha}\geq 2$ when $1<q<2$, and $\alpha=0$ when $q\geq 2$. This inequality is not explicitly stated in [14], but it follows from the proof of Theorem 6.

Let $N=N(E)$ be as in Theorem 6. The left–hand side inequality follows from the embedding $L^{M_{2}}\subset L^{1}$, which implies $L^{M_{2}}|E\subset L^{1}|E$, and from Theorem 6 for $p=1$. Thus, for some constant $C_{1}>0$,

[TABLE]

for any $q$–lacunary sequence $(w_{n_{k}})$ with $n_{k}\geq N$ and $(a_{k})_{1}^{\infty}\in\ell^{2}$.

In order to prove the right–hand side inequality, let

[TABLE]

We proceed as in the proof of Theorem 4 of [5], replacing [5, Lemma 5] by (1). From the power series expansion of $\exp(t^{2})-1$ and Theorem 6 for $p=2n$, we have

[TABLE]

It follows, applying Stirling’s formula, that for some absolute constant $C_{2}>0$,

[TABLE]

From this inequality it follows, as in the proof of Theorem 4 of [5], that there exists a constant $B^{\prime}(M_{2},q)>0$ such that

[TABLE]

and so the proof is complete. ∎

Remark 8**.**

(i) Note that $\alpha$ establishes the dependence between $B^{\prime}(M_{2},q)$ and $q$ in Theorem 7. In particular, since $q\geq 2$ implies $\alpha=0$, the constant $B^{\prime}(M_{2},q)$ is the same for all $q\geq 2$.

(ii) The non–local case of Theorem 7, i.e. for $E=[0,1]$, is referred to in [7, p. 247].

The next result is a consequence of Theorem 7.

Corollary 9**.**

Let $q>1$ and $E\subset[0,1]$ be a set of positive measure. Given an r.i. space $X$ with $G\subset X$, there exist constants $A^{\prime}(X,q)$, $B^{\prime}(X,q)>0$ and $N=N(E)$ such that, for any $q$–lacunary sequence $(w_{n_{k}})$ of Walsh functions with $n_{k}\geq N$ for all $k\geq 1$, we have

[TABLE]

for all $(a_{k})_{1}^{\infty}\in\ell^{2}$.

Proof.

It follows from Theorem 7, Theorem 6 for $p=1$, and from the fact that $G\subset X\subset L^{1}$ implies $G|E\subset X|E\subset L^{1}|E$. ∎

Note that, in the particular case when $X=L^{p}$, Corollary 9 coincides with the local result for lacunary Walsh series in Theorem 6. On the other hand, since $r_{n+1}=w_{2^{n}}$, Corollary 9 also allows us to recover the local results for the Rademacher system in [5], [6], [13] and [15].

Our next aim is to extend Theorem 2 to lacunary series of Walsh functions. The next technical result is needed. The dyadic intervals of order $n$ are $I^{n}_{k}:=(k/2^{n},(k+1)/2^{n})$, for $n\geq 0$ and $0\leq k\leq 2^{n}-1$.

Lemma 10**.**

Let $q\geq 2$ and $(w_{n_{k}})$ be a $q$–lacunary sequence of Walsh functions. Then, for any $M\geq 1$ and $(a_{k})\in\ell^{2}$, the functions

[TABLE]

have the same distribution function.

Proof.

We proceed by induction on $M$. For $M=1$, we have $|W_{1}|=|R_{1}|$.

Since $W_{M}$ is a finite sum of Walsh functions, it is constant on the dyadic intervals of order $N$, where $N$ is such that $2^{N-1}\leq n_{M}<2^{N}$. For $\varepsilon_{i}=\pm 1$, consider the set $A(\varepsilon_{1},\ldots,\varepsilon_{M})$, where $w_{M}$ takes the value $a_{1}\varepsilon_{1}+\ldots+a_{M}\varepsilon_{M}$. Since $A(\varepsilon_{1},\ldots,\varepsilon_{M})$ consists of a finite union of dyadic intervals of order less or equal than $2^{N}$, we have

[TABLE]

with $m(A(\varepsilon_{1},\ldots,\varepsilon_{M}))=1/2^{M}$.

Assume that $W_{M}$ and $R_{M}$ have the same distribution function. From the fact that $n_{M+1}/n_{M}\geq 2$ it follows that $n_{M+1}\geq 2^{N}$, and so the order of $w_{M+1}$ is greater or equal than $N+1$. It follows that each set $A(\varepsilon_{1},\ldots,\varepsilon_{M})$ can be divided into two sets, $A(\varepsilon_{1},\ldots,\varepsilon_{M},1)$ and $A(\varepsilon_{1},\ldots,\varepsilon_{M},-1)$, consisting on finite unions of dyadic intervals of order less or equal than $N+1$, both of the same measure, where $w_{M+1}$ takes values $1$ and $-1$. Thus, $W_{M+1}$ has the same distribution function that $R_{M+1}$. ∎

We prove a version of Theorem 2 for lacunary sequences of Walsh functions.

Theorem 11**.**

Let $X$ be an r.i. space on $[0,1]$. The following conditions are equivalent.

1. (i)

The continuous embedding $G\subset X$ holds, that is, there exists a constant $C>0$ such that

[TABLE]

for all $f\in L^{\infty}$. 2. (ii)

For any $q>1$, there exist constants $A(X,q),B(X,q)>0$ such that

[TABLE]

for all $(a_{k})_{1}^{\infty}\in\ell^{2}$, and any $q$–lacunary system $(w_{n_{k}})$ of Walsh functions. 3. (iii)

There exist a sequence $(n_{k})$ and constants $A(X),B(X)>0$ such that

[TABLE]

for all $(a_{k})_{1}^{\infty}\in\ell^{2}$.

Proof.

$\emph{(i)}\Rightarrow\emph{(ii)}$ Assume that $G\subset X$. Since for any r.i. space $X$ the continuous embedding $X\subset L^{1}$ holds, then (ii) follows from Theorem 4 for $p=1$ and from Theorem 7 with $E=[0,1]$.

$\emph{(ii)}\Rightarrow\emph{(iii)}$ Is clear.

$\emph{(iii)}\Rightarrow\emph{(i)}$ To show (i) it suffices to assume that the right–hand side inequality in (iii) holds, that is, for some constant $B(X)>0$,

[TABLE]

for any $(a_{k})_{1}^{\infty}\in\ell^{2}$. Consider a subsequence $(m_{k})\subset(n_{k})$ such that $m_{k+1}/m_{k}\geq 2$ for all $k\geq 1$, and let

[TABLE]

Then, from Lemma 10, $s_{n}$ and $v_{n}$ have the same distribution function, for all $n\geq 1$. From (iii), we have $\|s_{n}\|_{X}\leq B(X)$. Thus,

[TABLE]

and so $v_{n}\in X$, and $v_{n}$ are uniformly bounded in norm. Following the steps of the proof of Theorem 2 by Rodin and Semenov (see [11, Theorem 6]), $v_{n}\in X$ with $\|v_{n}\|_{X}\leq B(X)$ for all $n\geq 1$ implies, via the Central Limit Theorem, that $G\subset X$. ∎

Remark 12**.**

Let $(w_{n_{k}})$ be a $q$–lacunary sequence of Walsh functions, $X$ an r.i. space on $[0,1]$ and $(a_{k})\in\ell^{2}$ with $\sum_{k\geq 1}a_{k}r_{k}\in X$.

If $q\geq 2$, then it follows from Lemma 10 that

[TABLE]

Combining this fact together with the results mentioned in Introduction, we get at once Theorems 7 and 11 in the case when $q\geq 2$.

Consider now the case $1<q<2$. From [1, Theorem 8.1(d)] and Theorem 6 we have that $(w_{n_{k}})$ is majorized in distribution by $(r_{k})$, that is, there exists a constant $C>1$ such that

[TABLE]

for all $\lambda>0$, $M\in\mathbb{N}$ and $a_{1},\ldots,a_{M}\in\mathbb{R}$. From the boundedness on any r.i. space $X$ of the dilation operator $\sigma_{1/C}$,

[TABLE]

with norm $\|\sigma_{1/C}\|_{X}\leq C$, it follows that

[TABLE]

Note that this inequality suffices in order to prove Theorem 7 for $E=[0,1]$, and it also holds even in the case when $G\nsubseteq X$. The opposite majoration, that is, $(r_{k})$ being majorized in distribution by $(w_{n_{k}})$, is an open problem for which we have not found any references. It is a relevant question in the context of this paper, since it would imply that lacunary Walsh series and Rademacher series have equivalent norms in any r.i. space.

4. Complementability

The main result of this section is the following.

Theorem 13**.**

Let $X$ be an r.i. space on $[0,1]$ which is an interpolation space between $L^{1}([0,1])$ and $L^{\infty}([0,1])$. The following conditions are equivalent.

1. (i)

The continuous embeddings $G\subset X$ and $G\subset X^{\prime}$ hold, that is, there exist constants $C,C^{\prime}>0$ such that

[TABLE]

for all $f\in L^{\infty}$. 2. (ii)

For any $q>1$ and any $q$–lacunary sequence $(w_{n_{k}})$ of Walsh functions, the space $[w_{n_{k}}]_{X}$ is complemented in $X$. 3. (iii)

There exists $q>1$ and a $q$–lacunary sequence $(w_{n_{k}})$ of Walsh functions such that $[w_{n_{k}}]_{X}$ is complemented in $X$.

The proof follows the ideas of [10, Theorem 2.b.4] and [12]. We need some auxiliary results.

Proposition 14**.**

Let $X$ be an r.i. space on $[0,1]$ which is an interpolation space between $L^{1}([0,1])$ and $L^{\infty}([0,1])$, and $(w_{n_{k}})$ a $q$–lacunary subsequence of the Walsh system with $q>1$. Then, $(w_{n_{k}})$ is a basic sequence in $X$.

Proof.

The case when $q\geq 2$ follows from Lemma 10 and from the fact that $(r_{k})$ is a basic sequence in $X$.

Let $1<q<2$. For $s\in\mathbb{N}$, denote by $I^{s}_{j}=(j/2^{s},(j+1)/2^{s})$ the dyadic intervals of order $s$, $0\leq j\leq 2^{s}-1$, and consider the averaging operator

[TABLE]

The operators $A_{s}\colon X\to X$ are uniformly bounded (see [9, II,§3.2]).

Let $0\leq k<2^{s}$. Since $w_{k}$ is constant on $\chi_{I^{s}_{j}}$ for $0\leq j\leq 2^{s}-1$, we have

[TABLE]

On the other hand, noting that for $k\geq 2^{s}$ and $0\leq j\leq 2^{s}-1$,

[TABLE]

it follows that $A_{s}(w_{k})=0$, $k\geq 2^{s}$.

Let $M,N\in\mathbb{N}$ with $M<N$. Note, for $1\leq j\leq N$, that

[TABLE]

where $C_{1}$ denotes the constant in the continuous embedding $X\subset L^{1}$.

There are two cases. Suppose first that there exists $s\in\mathbb{N}$ such that $2^{s-1}\leq n_{M}<2^{s}\leq n_{N}$. Let $L$ such that $n_{L}<2^{s}\leq n_{L+1}$, with $M\leq L<L+1\leq N$. Then,

[TABLE]

Note that the last sum vanishes in the case when $M=L$. It follows that

[TABLE]

where $\varphi_{X}(t):=\|\chi_{[0,t]}\|_{X}$, $0\leq t\leq 1$, denotes the fundamental function of the space $X$.

Suppose next that there exists $s\in\mathbb{N}$ such that $2^{s}\leq n_{M}<n_{N}<2^{s+1}$ (in this case necessarily $1<q<2$). Let $L$ such that $n_{L}<2^{s}\leq n_{L+1}$, with $L+1\leq M$. Then,

[TABLE]

and so we have

[TABLE]

Let $\alpha\in\mathbb{N}$ be such that $q^{\alpha}\geq 2$. From the lacunary condition, we have in both cases that $|L-M|\leq\alpha$. Taking (2) into account, it follows that

[TABLE]

which shows that $(w_{n_{k}})$ is a basic sequence in $X$. ∎

Lemma 15**.**

Let $X$ be an r.i. space on $[0,1]$ and $(w_{n_{k}})$ a $q$–lacunary sequence of Walsh functions. The following conditions are equivalent.

1. (i)

The operator $T\colon X\to\ell^{2}$ given by

[TABLE]

is continuous. 2. (ii)

The continuous embedding $G\subset X^{\prime}$ holds.

Proof.

$\emph{(i)}\Rightarrow\emph{(ii)}$ Assume that $T:X\to\ell^{2}$ is continuous. Then, the adjoint operator $T^{\prime}:\ell^{2}\to X^{\prime}$ is continuous. Denote by $(e_{k})$ the canonical basis of $\ell^{2}$. Let $f\in X$. Then,

[TABLE]

and so $T^{\prime}e_{k}=w_{n_{k}}$. Hence, for $b=(b_{k})_{1}^{\infty}\in\ell^{2}$, we have

[TABLE]

Together with the continuity of $T^{\prime}$, it follows that

[TABLE]

for all $(b_{k})\in\ell^{2}$. This condition, as in the proof of Theorem 11, implies $G\subset X^{\prime}$.

$\emph{(ii)}\Rightarrow\emph{(i)}$ If $(w_{n_{k}})$ is a $q$–lacunary subsequence of Walsh functions and $G\subset X^{\prime}$, we have from Theorem 11 applied to $X^{\prime}$ that there exists a constant $B(X^{\prime},q)>0$ such that, for $N\geq 1$, we have

[TABLE]

Let $f\in X$. Fix $N\geq 1$. Then, from Hölder’s inequality for $X$ and $X^{\prime}$,

[TABLE]

It follows that

[TABLE]

that is, $\|Tf\|_{\ell^{2}}\leq B(X^{\prime},q)\|f\|_{X}$, and so (i) is established. ∎

For $n\geq 1$ and any $0\leq j,k\leq 2^{n}-1$, since $w_{k}$ is constant on the dyadic intervals of order $n$, we have that $w_{k}(I^{n}_{j})$ is well–defined and takes values $1$ or $-1$. The following property appears in [10, Theorem 2.b.4]:

[TABLE]

but no detail of its proof is given. For the sake of completeness, we give a proof of this fact.

Lemma 16**.**

For $n\geq 1$ and $0\leq j,k\leq 2^{n}-1$, we have

[TABLE]

Proof.

The case when either $k=0$ or $j=0$ follows from the fact that $w_{k}(I^{n}_{0})=1$ and $w_{0}(I^{n}_{k})=1$ for $0\leq k\leq 2^{n}-1$.

Fix $n\geq 1$ and $1\leq j,k\leq 2^{n}-1$. Let

[TABLE]

Fix $x\in I^{n}_{j}$, and denote by $\{t\}$ the fractional part of $t\in\mathbb{R}$. From

[TABLE]

we have

[TABLE]

Now we use the fact that $x\in I^{n}_{j}$ if and only if

[TABLE]

with $0<\varepsilon<1/2^{n}$. Then, for any $0\leq i\leq n-1$,

[TABLE]

It follows that

[TABLE]

Hence,

[TABLE]

which together with (4) implies that

[TABLE]

The symmetry in $a_{i}$ and $b_{i}$ of the expression above implies that $w_{k}(I^{n}_{j})=w_{j}(I^{n}_{k})$. ∎

We can now proceed to the proof of the main result.

Proof of Theorem 13.

It is clear that $\emph{(ii)}\Rightarrow\emph{(iii)}$.

$\emph{(i)}\Rightarrow\emph{(ii)}$ Let $(w_{n_{k}})$ be a $q$–lacunary subsequence of Walsh functions. From the assumption that $G\subset X^{\prime}$, Lemma 15 implies that the operator $T:X\to\ell^{2}$ in (3) is continuous, that is,

[TABLE]

Denote by $P:X\to[w_{n_{k}}]_{X}$ the projection

[TABLE]

From $G\subset X$ and Theorem 11 we have

[TABLE]

It follows that

[TABLE]

that is, the subspace $[w_{n_{k}}]_{X}$ is complemented in $X$.

$\emph{(iii)}\Rightarrow\emph{(i)}$ We follow the ideas in the proof of the complementability result for Rademacher functions (Theorem 3) by Lindenstrauss and Tzafriri [10, Theorem 2.b.4]. Assume that $(w_{n_{k}})$ is a lacunary sequence with $q>1$ such that $[w_{n_{k}}]_{X}$ is complemented in $X$. Then, there exists a bounded linear operator $Q:X\to[w_{n_{k}}]_{X}$ with $Q^{2}=Q$. Since $(w_{n_{k}})$ is a basic sequence, there exists $(q_{n_{k}})\subset X^{*}$ such that

[TABLE]

Let $n\geq 1$ and $X_{n}$ be the linear subspace generated in $X$ by the characteristic functions of the dyadic intervals of order $n$. From the linear independence of the Walsh functions and from the fact that $w_{k}$ is constant on the dyadic intervals of order $n$ for $0\leq k\leq 2^{n}-1$, we have that $X_{n}$ coincides with the linear space generated by $\{w_{k}:k=0,\ldots,2^{n}-1\}$, that is,

[TABLE]

Let $w_{m_{1}},\ldots,w_{m_{N}}$ be the Walsh functions of the sequence $(w_{n_{k}})$ with order less or equal than $n$. Note that $N$ and $m_{1},\ldots m_{N}$ depend on $n$. Denote by $Q_{n}$ the restriction of $Q$ to $X_{n}$, $Q_{n}:X_{n}\to X_{n}$, that is,

[TABLE]

For $0\leq j,k\leq 2^{n}-1$, denote

[TABLE]

Let $T_{j}:X_{n}\to X_{n}$ be the operator defined by

[TABLE]

Since $X$ is r.i. and the distribution function of $f$ and $T_{j}(f)$ coincide for $f\in X_{n}$, we have $\|T_{j}\|=1$ for $1\leq j\leq 2^{n}$.

Denote by $P_{n}:X_{n}\to X_{n}$ the restriction of the projection in (5) to $X_{n}$, that is,

[TABLE]

We will prove that

[TABLE]

From (5),

[TABLE]

Since $\langle w_{m_{k}},w_{i}\rangle=\delta_{m_{k},i}$, it follows that $P_{n}(w_{i})=w_{i}$ for $i\in\{m_{1},\ldots,m_{N}\}$. Otherwise, $P_{n}(w_{i})=0$.

On the other hand,

[TABLE]

Since $Q_{n}$ is a projection, we have from (6) that $q_{m_{k}}(w_{i})=\delta_{m_{k},i}$. Thus, (7) follows from

[TABLE]

Given integers $k,n$ and $m$, we write $k=n\oplus m$ whenever the following relation holds:

[TABLE]

Then, from Lemma 16,

[TABLE]

and so it follows that

[TABLE]

which proves (7).

Let us see that the operators $P_{n}:X_{n}\to X_{n}$ have uniformly bounded norm (in $n$). From (7), since $\|T_{j}\|=1$, we have for any $f\in X_{n}$ that

[TABLE]

that is, $\|P_{n}f\|\leq\|Q\|$ for all $n\geq 1$.

Next we show that $G\subset X^{\prime}$ follows from the fact that $P_{n}:X_{n}\to X_{n}$ have uniformly bounded norm. Denote by $T_{n}:X_{n}\to\ell^{2}$ the operator given by

[TABLE]

From $X\subset L^{1}$, there exists a constant $C_{1}>0$ such that $C_{1}\|f\|_{L^{1}}\leq\|f\|_{X}$ for all $f\in X$. Together with Theorem 4 for $p=1$,

[TABLE]

It follows, for some constant $C_{2}>0$, that

[TABLE]

for all $n\geq 1$. Thus, the adjoint operator

[TABLE]

is bounded with $\|T^{\prime}_{n}\|\leq C_{2}$, $n\geq 1$. Since, for any $(b_{k})\in\ell^{2}$,

[TABLE]

we have, as in the proof of Lemma 15, that

[TABLE]

Since this inequality holds for all $n\geq 1$, we have

[TABLE]

This inequality, together with the embedding $X^{\prime}\subset L^{1}$ and Theorem 4, shows that the subspace $[w_{n_{k}}]_{X^{\prime}}$ is isomorphic to $\ell^{2}$. From Theorem 11, this is equivalent to $G\subset X^{\prime}$ .

Now we show that $G\subset X$. Let $n\geq 1$. For $f\in X_{n}$ and $g\in X_{n}^{\prime}$,

[TABLE]

Thus,

[TABLE]

It follows that

[TABLE]

The argument above, together with $X^{\prime}_{n}$ in place of $X_{n}$, shows that $G\subset X^{\prime\prime}$ follows from the fact that $P_{n}:X^{\prime}_{n}\to X^{\prime}_{n}$ is uniformly bounded. Thus, for the separable parts we have that $G_{0}\subset(X^{\prime\prime})_{0}$. Since $(X^{\prime\prime})_{0}=X_{0}$ it follows that

[TABLE]

which concludes the proof.

∎

Acknowledgements.

This work is part of the Ph.D. thesis of the author which has been written at University of Sevilla under the supervision of Prof. G. P. Curbera.