Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

TL;DR

This paper characterizes sequences that preserve the order of sums in rearrangement inequalities and applies these results to Lorentz-type sequence spaces, resolving an open problem and showing Garling sequence spaces lack symmetric bases.

Contribution

It provides a complete characterization of sequences for which the rearrangement inequality holds in Lorentz-type spaces and settles an open problem about the structure of Garling sequence spaces.

Findings

Characterization of sequences preserving the rearrangement inequality.

Resolution of an open problem on Lorentz-type sequence spaces.

Proof that Garling sequence spaces have no symmetric basis.

Abstract

We characterize the sequences $(w_i)_{i=1}^\infty$ of non-negative numbers for which \[ \sum_{i=1}^\infty a_i w_i \quad \text{ is of the same order as } \quad \sup_n \sum_{i=1}^n a_i w_{1+n-i} \] when $(a_i)_{i=1}^\infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.

Equations

$$\sum _{i=1}^{\infty }{a}_i{w}_i\text{ is of the same order as }\underset{n}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}$$

$$\sum _{i=1}^n{a}_i{b}_{1+n-i}\le \sum _{i=1}^n{a}_i{b}_{\sigma (i)}\le \sum _{i=1}^n{a}_i{b}_i$$

$$\underset{n\in \mathbb{N}}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}\le \sum _{i=1}^{\infty }{a}_i{w}_i.$$

$$\sum _{i=1}^{\infty }{a}_i{w}_i\le C\underset{n\in \mathbb{N}}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}$$

$$\sum _{i=1}^{\infty }{a}_i{w}_i\le w_1\sum _{i=1}^{\infty }{a}_i=w_1\underset{n}{\sup }\sum _{i=1}^n{a}_i\le \frac{w_1}{w_{\infty }}\underset{n}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}.$$

$$\sum _{i=1}^{\infty }{a}_i{w}_i\le a_1\sum _{i=1}^{\infty }{w}_i=\frac{W}{w_1}{a}_1{w}_1\le \frac{W}{w_1}\underset{n}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}.$$

$$\sum _{i=1}^{\infty }{a}_i{w}_i\le C\underset{n\in \mathbb{N}}{\sup }\sum _{i=1}^n{a}_i{w}_{1+n-i}$$

$$A(f,\mathbf{w})$$

$$B(f,\mathbf{w})$$

$$B(f,\mathbf{w},n)=\sum _{i=1}^n{a}_i{w}_{1+n-i}.$$

$$S(\mathbf{w}):=\underset{f\in \mathcal{D}}{\sup }\frac{A(f,\mathbf{w})}{B(f,\mathbf{w})}=\infty .$$

$$\underset{n}{\lim }W(n)=\infty .$$

$$\underset{n\in \mathbb{N}}{\lim }(W(s+n)-W(n))=0$$

$$a_{i,r}=\{\begin{array}{ll}1/W(d_k)& \text{ if, for some 1≤k≤r, }{n}_{k-1}<i\le n_k\\ 0& \text{ if }i>n_r.\end{array}$$

$$A(f^{(r)},\mathbf{w})$$

$$=\sum _{k=1}^r\frac{W(n_k)-W(n_{k-1})}{W(d_k)}$$

$$\ge \sum _{k=1}^r\frac{W(d_k)-2^{-1}W(d_k)}{W(d_k)}$$

$$=\sum _{k=1}^r\frac{1}{2}$$

$$=\frac{r}{2}.$$

$$B(f^{(r)},\mathbf{w},n)=\sum _{i=1}^{n_r}{a}_{i,r}{w}_{1+n-i}\le \sum _{i=1}^{n_r}{a}_{i,r}{w}_{1+n_r-i}=B(f^{(r)},\mathbf{w},n_r).$$

$$B(f^{(r)},\mathbf{w},n)$$

$$\le \frac{W(n_k-n_{k-1})}{W(d_k)}+\sum _{j=1}^{k-1}\frac{W(n-n_{j-1})-W(n-n_j)}{W(d_j)}$$

$$=1+\sum _{j=1}^{k-1}\frac{W(n-n_{j-1})-W(n-n_j)}{W(d_j)}.$$

$$B(f^{(r)},\mathbf{w},n)$$

$$+\sum _{j=1}^{k-2}\frac{W(n-n_{j-1})-W(n-n_j)}{W(d_j)}$$

$$\le 1+\frac{W(n_{k-1}-n_{k-2})-W(n_{k-1}-n_{k-1})}{W(d_{k-1})}$$

$$+\sum _{j=1}^{k-2}\frac{W(n_{j+1}-n_{j-1})-W(n_{j+1}-n_j)}{W(d_j)}$$

$$=2+\sum _{j=1}^{k-2}\frac{W(d_{j+1}+d_j)-W(d_{j+1})}{W(d_j)}$$

$$\le 2+\sum _{j=1}^{k-2}{2}^{-j}$$

$$=3-2^{2-k}.$$

$$S(\mathbf{w})\ge \underset{r\in \mathbb{N}}{\sup }\frac{A(f^{(r)},\mathbf{w})}{B(f^{(r)},\mathbf{w})}\ge \underset{r\in \mathbb{N}}{\sup }\frac{r}{6}=\infty ,$$

$$\Vert f{\Vert }_{g(\mathbf{w},p)}=\underset{\varphi \in \mathcal{O}}{\sup }{\left(\sum _{i=1}^{\infty }|b_{\varphi (i)}{|}^p{w}_i\right)}^{1/p}$$

$$d(\mathbf{w},p):=\left\{(b_k{)}_{k=1}^{\infty }\in c_0:{\left(\sum _{i=1}^{\infty }(b_i^{\ast }{)}^p{w}_i\right)}^{1/p}<\infty \right\},$$

$$\mathcal{W}:=\left\{(w_i{)}_{i=1}^{\infty }\in c_0\setminus {\ell }_1:1=w_1\ge w_2\ge \cdots \ge w_i\ge w_{i+1}\ge \cdots >0\right\}$$

$$\Vert g{\Vert }_{g(\mathbf{w},p)}\le C\Vert f{\Vert }_{g(\mathbf{w},p)}$$

$$\sum _{i=1}^{\infty }{a}_i{w}_i$$

$$\le \underset{r\in \mathbb{N}}{\sup }{\Vert \sum _{i=1}^r{a}_i^{1/p}{\mathbf{e}}_i\Vert }_{g(\mathbf{w},p)}^p$$

Full text

[TABLE]

when $(a_{i})_{i=1}^{\infty}$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [AAW] and prove that Garling sequences spaces have no symmetric basis.

The research of the two firs-named authors was partially supported by the Spanish Research Grant Análisis Vectorial, Multilineal y Aplicaciones, reference number MTM2014-53009-P. F. Albiac also acknowledges the support of Spanish Research Grant Operators, lattices, and structure of Banach spaces, with reference MTM2016-76808-P. The research of the third author was partially supported by AcRF grant R-146-000-242-114.

1. Introduction

The rearrangement inequality states that, for $n\in\mathbb{N}$, if $(a_{i})_{i=1}^{n}$ and $(b_{i})_{i=1}^{n}$ are a pair of non-increasing $n$-tuples of non-negative scalars then we have

[TABLE]

for every permutation $\sigma$ of the set $\{1,\dots,n\}$ (see [HLP]*Theorem 368). Consequently, if $(a_{i})_{i=1}^{\infty}$ and $(w_{i})_{i=1}^{\infty}$ are non-increasing sequences of non-negative scalars,

[TABLE]

In this note we wonder about which are the non-increasing sequences $(w_{i})_{i=1}^{\infty}$ of non-negative scalars that verify a reverse inequality, i.e., in which cases there is a constant $C<\infty$ such that

[TABLE]

for every sequence $(a_{i})_{i=1}^{\infty}$ of non-negative scalars. For the time being some simple answers can be given. Indeed, on the one hand, if $w_{\infty}:=\inf_{i}w_{i}>0$ then

[TABLE]

On the other hand, if we consider $W:=\sum_{i=1}^{\infty}w_{i}<\infty$ and let $w_{1}>0$ (the case $w_{1}=0$ is trivial) then

[TABLE]

In fact, as we will show below, these two cases are the only ones for which (1.1) holds. This will be our main result as far as inequalities is concerned:

Theorem 1.1** (Main Theorem).**

Let $(w_{i})_{i=1}^{\infty}$ be a non-increasing sequence consisting of non-negative scalars. The following are equivalent:

• (i)

There is a constant $C<\infty$ such that

[TABLE]

for every sequence $(a_{i})_{i=1}^{\infty}$ of non-negative scalars.

• (ii)

Either $\sum_{i=1}^{\infty}w_{i}<\infty$ or $\inf_{i\in\mathbb{N}}w_{i}>0$.

Section 2 is devoted to proving Theorem 1.1. In Section 3 we use Theorem 1.1 to give some functional analytic properties of a recently introduced class of Lorentz-type spaces, called Garling sequence spaces. In particular, Theorem 1.1 is applied to show that Garling sequence spaces have no symmetric basis, answering thus a problem that was recently posed in [AAW].

Throughout this note we use standard terminology and notation in Banach space theory. As it is customary, we denote by $\ell_{q}$, $1\leq q\leq\infty$, the Banach space consisting of all $q$-summable sequences (bounded sequences in the case $q=\infty$) and by $c_{0}$ the subspace of $\ell_{\infty}$ consisting of all sequences converging to zero. For background on bases in Banach spaces we refer the reader to [AK2016].

2. Proof of the Main Theorem

Proof of Theorem 1.1.

As explained in the Introduction, we must only prove that (i) implies (ii).

Assume that (ii) does not hold, that is, $\mathbf{w}=(w_{i})_{i=1}^{\infty}\in c_{0}\setminus\ell_{1}$. Let us denote by $\mathcal{D}$ the set of (nonzero) non-increasing sequences of non-negative integers. For $f=(a_{i})_{i=1}^{\infty}\in\mathcal{D}$ and $n\in\mathbb{N}$ we put

[TABLE]

where, for $n\in\mathbb{N}$,

[TABLE]

With this notation we must prove that

[TABLE]

We will use the convention that $\sum_{i=1}^{0}c_{i}=0$ for all sequences of scalars $(c_{i})_{i=1}^{\infty}$.

For $n\in\mathbb{N}$ put $W(n)=\sum^{n}_{i=1}w_{i}$. Since $\mathbf{w}\notin\ell_{1}$ we have

[TABLE]

Moreover, since $\mathbf{w}\in c_{0}$,

[TABLE]

for any non-negative integer $s$. We use these properties to recursively construct an increasing sequence $(d_{k})_{k=0}^{\infty}$ of non-negative integers with $d_{0}=0$ verifying

1. (i)

$W(\sum^{k-1}_{j=1}d_{j})\leq 2^{-1}W(d_{k})$, and 2. (ii)

$W(d_{k-1}+d_{k})-W(d_{k})\leq 2^{1-k}W(d_{k-1})$

for $k=1,2,\dots$.

For every integer $k\geq 0$ put $n_{k}=\sum^{k}_{j=1}d_{j}$. For each $r\in\mathbb{N}$ we define a sequence $f^{(r)}=(a_{i,r})_{i=1}^{\infty}$ by

[TABLE]

It is clear that $f^{(r)}\in\mathcal{D}$ for all $r\in\mathbb{N}$. Taking into account the inequality in (i) we obtain

[TABLE]

Let $n\in\mathbb{N}$. In case that $n>n_{r}$ we have

[TABLE]

In case that $n\leq n_{r}$, pick $1\leq k\leq r$ with $n_{k-1}<n\leq n_{k}$. We have

[TABLE]

If $k=1$ we get $B(f^{(r)},\mathbf{w},n)\leq 1$. Assume that $k\geq 2$. Taking into account inequality (ii) and that, since $\mathbf{w}$ is non-increasing, the sequence $(W(n+t)-W(n+s))_{n=1}^{\infty}$ is non-increasing for any $s\leq t$, we obtain

[TABLE]

Therefore $B(f^{(r)},\mathbf{w})\leq 3$. Thus

[TABLE]

and the proof is over. ∎

3. Applications to Garling squence spaces

Let $1\leq p<\infty$ and let $\mathbf{w}=(w_{n})_{n=1}^{\infty}$ be a non-increasing sequence of positive scalars. Given a sequence of (real or comnplex) scalars $f=(b_{k})_{k=1}^{\infty}$ we put

[TABLE]

where $\mathcal{O}$ denotes the set of increasing functions from $\mathbb{N}$ to $\mathbb{N}$. The Garling sequence space $g(\mathbf{w},p)$ is the Banach space consisting of all sequences $f$ with $\|f\|_{g(\mathbf{w},p)}<\infty$.

Notice that if in (3) we replace “$\phi\in\mathcal{O}$” with “$\phi$ is a permutation of $\mathbb{N}$” we obtain the norm that defines the weighted Lorentz sequence space

[TABLE]

where $(b_{i}^{*})_{i=1}^{\infty}$ denotes the decreasing rearrangement of $(b_{k})_{k=1}^{\infty}$. So, the Garling sequence space $g(\mathbf{w},p)$ can be regarded as a variation of the weighted Lorentz sequence space $d(\mathbf{w},p)$.

Imposing the further conditions $\mathbf{w}\in c_{0}$ and $\mathbf{w}\notin\ell_{1}$ will prevent us, respectively, from having $g(\mathbf{w},p)=\ell_{p}$ or $g(\mathbf{w},p)=\ell_{\infty}$. We will assume as well that $\mathbf{w}$ is normalized, i.e., $w_{1}=1$. Thus, we put

[TABLE]

and we restrict our attention to weights $\mathbf{w}\in\mathcal{W}$.

For $n\in\mathbb{N}$, we will denote $\mathbf{e}_{n}=(\delta_{i,n})_{i=1}^{\infty}$, where $\delta_{i,n}=1$ if $n=i$ and $\delta_{i,n}=0$ otherwise. We have (see [AAW]*Theorem 3.1) that the canonical sequence $(\mathbf{e}_{n})_{n=1}^{\infty}$ is a Schauder basis for $g(\mathbf{w},p)$. A question posed and partially solved in [AAW] is to determine the weights $\mathbf{w}\in\mathcal{W}$ and the indices $p\in[1,\infty)$ for which $(\mathbf{e}_{n})_{n=1}^{\infty}$ is a symmetric basis of $g(\mathbf{w},p)$. Here we provide a complete intrinsic solution to this problem, in the sense that our approach is entirely based on Theorem 1.1.

Lemma 3.1**.**

*The canonical sequence $(\mathbf{e}_{n})_{n=1}^{\infty}$ is not a symmetric basis for $g(\mathbf{w},p)$ for any $\mathbf{w}\in\mathcal{W}$ and any $1\leq p<\infty$. *

Proof.

Assume that $(\mathbf{e}_{n})_{n=1}^{\infty}$ is a symmetric basis for $g(\mathbf{w},p)$. Then, there is a constant $C$ so that

[TABLE]

whenever the sequence $g$ is a permutation of the sequence $f$.

Given $r\in\mathbb{N}$ and $\phi\in\mathcal{O}$ let $n(r,\phi)$ be the largest integer $n$ such that $\phi(n)\leq r$. We have $\phi(i)\leq i+r-n(r,\phi)$ for $1\leq i\leq n(r,\phi)$. Given a non-increasing sequence $(a_{i})_{i=1}^{\infty}$ of non-negative numbers we have

[TABLE]

Theorem 1.1 yields the absurdity $\mathbf{w}\in\ell_{1}$ or $\mathbf{w}\notin c_{0}$. ∎

Now we are ready to establish the advertised structural properties of Garling sequence spaces.

Theorem 3.2**.**

Let $\mathbf{w}\in\mathcal{W}$ and $1\leq p<\infty$.

1. (i)

There is no symmetric basis for $g(\mathbf{w},p)$. 2. (ii)

$d(\mathbf{w},p)\subsetneq g(\mathbf{w},p)$. 3. (iii)

No subspace of $d(\mathbf{w},p)$ is isomorphic to $g(\mathbf{w},p)$. 4. (iv)

Let $I_{d,g}\colon d(\mathbf{w},p)\to g(\mathbf{w},p)$ be the natural inclusion map, and let $T\colon g(\mathbf{w},p)\to d(\mathbf{w},p)$ be a bounded linear operator. Then (despite the fact that $I_{d,g}$ is not a strictly singular operator) $T\circ I_{d,g}$ does not preserve a copy of $d(\mathbf{w},p)$, i.e., if $X$ is a subspace of $d(\mathbf{w},p)$ isomorphic to $d(\mathbf{w},p)$ then $T\circ I_{d,g}|_{X}$ is not an isomorphism.

Proof.

It follows using Lemma 3.1 in combination with [AAW]*Theorem 5.1. ∎

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