A new estimate for the constants of an inequality due to Hardy and Littlewood

TL;DR

This paper introduces a new family of inequalities that extend a recent result by Albuquerque et al., providing improved estimates for the constants involved in an inequality originally due to Hardy and Littlewood.

Contribution

It offers a novel extension of Hardy-Littlewood type inequalities with new estimates for the involved constants.

Findings

Extended the family of inequalities related to Hardy-Littlewood

Provided improved bounds for the constants in these inequalities

Generalized previous results by Albuquerque et al.

Abstract

In this paper we provide a family of inequalities, extending a recent result due to Albuquerque et al.

Equations

$${\left(\sum _{j_1,\cdots ,j_m=1}^{\infty }{\left|T(e_{j_1},\cdots ,e_{j_m})\right|}^{\frac{2mp}{mp+p-2m}}\right)}^{\frac{mp+p-2m}{2mp}}\le C_{m,p}^{\mathbb{K}}\Vert T\Vert ,$$

$${\left(\sum _{j_1,\cdots ,j_m=1}^{\infty }{\left|T(e_{j_1},\cdots ,e_{j_m})\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\le D_{m,p}^{\mathbb{K}}\Vert T\Vert ,$$

$${\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\left|T\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^{\frac{p}{p-\left(m-1\right)}}\right)}^{\frac{p-\left(m-1\right)}{p}\cdot \frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\le 2^{\frac{\left(m-1\right)\left(p-m+1\right)}{p}}\Vert T\Vert .$$

$${\lambda }_{0}m<p\le \frac{2{\lambda }_0\left(m-1\right)}{2-{\lambda }_0},$$

$${\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\left|T\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^s\right)}^{\frac{1}{s}{\eta }_1}\right)}^{\frac{1}{{\eta }_1}}\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}\Vert T\Vert$$

$${\eta }_1$$

$$s$$

$$B_{p_1,\dots ,p_m}^{A,s,\alpha ,F,n}:=\inf \left\{C(n):{\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\Vert T\left(e_{j_1},\dots ,e_{j_m}\right)\Vert }^s\right)}^{\frac{1}{s}\alpha }\right)}^{\frac{1}{\alpha }}\le C(n),\text{ for all }i\in A\right\},$$

$$\sum _{j=1}^m\left(\frac{1}{p_j}-\frac{1}{q_j}\right)<\frac{1}{{\lambda }_0}\text{ and }s\ge {\left[\frac{1}{{\lambda }_0}-\sum _{j=1}^{m-1}\left(\frac{1}{p_j}-\frac{1}{q_j}\right)\right]}^{-1}=:{\eta }_2$$

$$B_{p_1,\dots ,p_m}^{\{m\},s,{\eta }_1,F,n}\le B_{q_1,\dots ,q_m}^{I_m,s,{\lambda }_0,F,n},$$

$${\eta }_1:={\left[\frac{1}{{\lambda }_0}-\sum _{j=1}^m\left(\frac{1}{p_j}-\frac{1}{q_j}\right)\right]}^{-1}$$

$$A_q{\left(\sum _{j=1}^n|a_j{|}^2\right)}^{\frac{1}{2}}\le {\left({\int }_{[0,1]}{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^{q}dt\right)}^{\frac{1}{q}}\le B_q{\left(\sum _{j=1}^n|a_j{|}^2\right)}^{\frac{1}{2}},$$

$$\underset{\begin{array}{c}{i}_k=1,\dots ,N\\ k=1,\dots ,m\end{array}}{\max }\left|a_{i_1,\dots ,i_m}\right|\le {\left({\int }_{[0,1{]}^m}{\left|\sum _{i_1,\dots ,i_m=1}^N{r}_{i_1}(t_1)\cdots r_{i_m}(t_m)a_{i_1,\dots ,i_m}\right|}^{t}d{t}_1\cdots d{t}_m\right)}^{1/t}$$

$${\left({\int }_{[0,1{]}^m}{\left|\sum _{i_1,\dots ,i_m=1}^N{r}_{i_1}(t_1)\cdots r_{i_m}(t_m)a_{i_1,\dots ,i_m}\right|}^{t}d{t}_1\cdots d{t}_m\right)}^{1/t}$$

$$={\left({\int }_{[0,1{]}^{m-1}}{\left[{\left({\int }_0^1{\left|\sum _{i_1=1}^N{r}_{i_1}(t_1)\left(\sum _{i_2,\dots ,i_m=1}^N{r}_{i_2}(t_2)\cdots r_{i_m}(t_m)a_{i_1,\dots ,i_m}\right)\right|}^{t}d{t}_1\right)}^{\frac{1}{t}}\right]}^{t}d{t}_2\cdots d{t}_m\right)}^{1/t}$$

$$\ge {\left({\int }_{[0,1{]}^{m-1}}{\left|\sum _{i_2,\dots ,i_m=1}^N{r}_{i_2}(t_2)\cdots r_{i_m}(t_m)a_{i_1,\dots ,i_m}\right|}^{t}d{t}_2\cdots d{t}_m\right)}^{1/t}$$

$$\ge \left|a_{i_1,\dots ,i_m}\right|,$$

$$s={\left[\frac{1}{{\lambda }_0}-\sum _{j=1}^{m-1}\left(\frac{1}{p_j}-\frac{1}{q_j}\right)\right]}^{-1}\ge 2$$

$${\lambda }_0\in [1,2].$$

$${\left(\sum _{j_1=1}^n{\left(\sum _{\widehat{j_1}=1}^n{\left|S\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^s\right)}^{\frac{1}{s}{\lambda }_0}\right)}^{\frac{1}{{\lambda }_0}}$$

$$\le {\left(\sum _{j_1=1}^n{\left({\left({\left(\sum _{\widehat{j_1}=1}^n{\left|S\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^2\right)}^{\frac{1}{2}}\right)}^{\theta }{\left(\underset{\widehat{j_1}}{\max }\left|S\left(e_{j_1},\dots ,e_{j_m}\right)\right|\right)}^{1-\theta }\right)}^{{\lambda }_0}\right)}^{\frac{1}{{\lambda }_0}}$$

$$\le {\left(\sum _{j_1=1}^n{\left({\left(A_{{\lambda }_0}^{-\left(m-1\right)}{R}_n\right)}^{\frac{2}{s}}{R}_n^{1-\frac{2}{s}}\right)}^{{\lambda }_0}\right)}^{\frac{1}{{\lambda }_0}}$$

$$=A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}{\left(\sum _{j_1=1}^n{\int }_{[0,1{]}^{m-1}}{\left|\sum _{\widehat{j_1}=1}^n{r}_{j_2}(t_2)\cdots r_{j_m}(t_m)S\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^{{\lambda }_0}d{t}_2\cdots d{t}_m\right)}^{1/{\lambda }_0}$$

$$=A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}{\left({\int }_{[0,1{]}^{m-1}}\sum _{j_1=1}^n{\left|S\left(e_{j_1},\sum _{j_2=1}^n{r}_{j_2}(t_2)e_{j_2},\dots ,\sum _{j_m=1}^n{r}_{j_m}(t_m)e_{j_m}\right)\right|}^{{\lambda }_0}d{t}_2\cdots d{t}_m\right)}^{1/{\lambda }_0}$$

$$\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}{\left(\underset{t_2,\dots ,t_m\in [0,1]}{\sup }\sum _{j_1=1}^n{\left|S\left(e_{j_1},\sum _{j_2=1}^n{r}_{j_2}(t_2)e_{j_2},\dots ,\sum _{j_m=1}^n{r}_{j_m}(t_m)e_{j_m}\right)\right|}^{{\lambda }_0}\right)}^{1/{\lambda }_0}$$

$$\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}\Vert S\Vert ,$$

$$R_n:={\left({\int }_{[0,1{]}^{m-1}}{\left|\sum _{\widehat{j_1}=1}^n{r}_{j_2}(t_2)\cdots r_{j_m}(t_m)S\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^{{\lambda }_0}d{t}_2\cdots d{t}_m\right)}^{\frac{1}{{\lambda }_0}}.$$

$${\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\left|S\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^s\right)}^{\frac{1}{s}{\lambda }_0}\right)}^{\frac{1}{{\lambda }_0}}\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}\Vert S\Vert$$

$${\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\left|T\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^s\right)}^{\frac{1}{s}{\eta }_1}\right)}^{\frac{1}{{\eta }_1}}\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}\Vert T\Vert$$

$${\eta }_1:={\left[\frac{1}{{\lambda }_0}-\sum _{j=1}^m\left(\frac{1}{p_j}-\frac{1}{q_j}\right)\right]}^{-1}.$$

$$s={\left[\frac{1}{{\lambda }_0}-\frac{m-1}{p}\right]}^{-1}\ge 2$$

$${\lambda }_0\in [1,2]$$

$$p>{\lambda }_{0}m$$

$${\eta }_1=\frac{{\lambda }_{0}p}{p-{\lambda }_{0}m},$$

$${\left(\sum _{j_i=1}^n{\left(\sum _{\widehat{j_i}=1}^n{\left|T\left(e_{j_1},\dots ,e_{j_m}\right)\right|}^s\right)}^{\frac{1}{s}{\eta }_1}\right)}^{\frac{1}{{\eta }_1}}\le A_{{\lambda }_0}^{\frac{-2\left(m-1\right)}{s}}\Vert T\Vert$$

Full text

Partially supported by Capes.

MSC2010: 46G25

1. Introduction

The Hardy–Littlewood inequalities [12] for $m$–linear forms and polynomials (see [2, 3, 4, 5, 9, 15, 17]) are perfect extensions of the Bohnenblust–Hille inequality [6] when the sequence space $c_{0}$ is replaced by the sequence space $\ell_{p}$. These inequalities assert that for any integer $m\geq 2$ there exist constants $C_{m,p}^{\mathbb{K}},D_{m,p}^{\mathbb{K}}\geq 1$ such that

[TABLE]

when $2m\leq p\leq\infty$, and

[TABLE]

when $m<p\leq 2m$, for all continuous $m$–linear forms $T:\ell_{p}\times\cdots\times\ell_{p}\rightarrow\mathbb{K}$ (here, and henceforth, $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$). Both exponents are optimal, $i.e.,$ cannot be smaller without paying the price of a dependence on $n$ arising on the respective constants. Following usual convention in the field, $c_{0}$ is understood as the substitute of $\ell_{\infty}$ when the exponent $p$ goes to infinity.

The investigation of the optimal constants of the Hardy–Littlewood inequalities is closely related to the fashionable, mysterious and puzzling investigation of the optimal Bohnenblust–Hille inequality constants (see, for instance [15] and the references therein).

In this note we extend the following result of [1, Theorem 3]:

Theorem 1** (Albuquerque et al.).**

Let $m\geq 2$ be a positive integer and $m<p\leq 2m-2.$ Then, for all continuous $m$-linear forms $T:\ell_{p}\times\cdots\times\ell_{p}\rightarrow\mathbb{K}$, we have

[TABLE]

More precisely, using a different technique we find a family of inequalities extending the above result. Our result reads as follows, where $A_{\lambda_{0}}$ is the optimal constant of the Khinchin inequality (defined in Section 2):

Theorem 2**.**

If $\lambda_{0}\in[1,2]$ and

[TABLE]

then

[TABLE]

for

[TABLE]

all $m$-linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$, and all positive integers $n.$

Our main technique is based on an original argument developed in [1], with some slight technical changes.

2. The proof of Theorem 2

Let $m\geq 2$ be a positive integer, $F$ be a Banach space, $A\subset I_{m}:=\{1,\dots,m\},p_{1},\dots,p_{m},s,\alpha\geq 1$ and

[TABLE]

in which $\widehat{j_{i}}$ means that the sum runs over all indexes but $j_{i}$, and the infimum is taken over all norm-one $m$-linear operators $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow F$. We begin by recalling the following lemma proved in [1]:

Lemma 1**.**

Let $1\leq p_{k}<q_{k}\leq\infty,\,k=1,\dots,m$ and $\lambda_{0},s\geq 1$. If

[TABLE]

then

[TABLE]

where

[TABLE]

We also need to recall the Khinchin inequality: for any $0<q<\infty$, there are positive constants $A_{q}$, $B_{q}$ such that regardless of the positive integer $n$ and of the scalar sequence $(a_{j})_{j=1}^{n}$ we have

[TABLE]

where $r_{j}$ are the classical Rademacher functions (random variables).

The best constants $A_{q}$ are the following ones (see [11]):

• •

$\displaystyle A_{q}=\sqrt{2}\left(\frac{\Gamma\left(\frac{1+q}{2}\right)}{\sqrt{\pi}}\right)^{\frac{1}{q}}$ if $q>q_{0}\cong 1.85$;

• •

$\displaystyle A_{q}=2^{\frac{1}{2}-\frac{1}{q}}$ if $q<q_{0},$

where $q_{0}\in(1,2)$ is the only real number such that $\Gamma\left(\frac{p_{0}+1}{2}\right)=\frac{\sqrt{\pi}}{2}$. For complex scalars, using Steinhaus variables instead of Rademacher functions it is well known that a similar inequality holds, but with better constants. In this case the optimal constant is

• •

$\displaystyle A_{q}=\Gamma\left(\frac{q+2}{2}\right)^{\frac{1}{q}}$ if $q\in[1,2]$.

The notation of the constants $A_{q}$ above will be used in all this paper. The following result is a variant of [1], and is based on the Contraction Principle (see [8, Theorem 12.2]). From now on $r_{i}(t)$ are the Rademacher functions.

Lemma 2**.**

Regardless of the choice of the positive integers $m,N$ and the scalars $a_{i_{1},\dots,i_{m}}$, $i_{1},\dots,i_{m}=1,\dots,N$,

[TABLE]

for all $t\geq 1$.

Proof.

The proof is an adaptation of an argument used in [1]. Essentially, we have to use the Contraction Principle inductively. The case $m=1$ is nothing else than the standard version of Contraction Principle (see [8, Theorem 12.2]). For all positive integers $i_{1},\dots,i_{m}$,

[TABLE]

where we used the Contraction Principle and the induction hypothesis on the first and second inequalities, respectively. This concludes the proof of the lemma. ∎

Now we are able to complete the proof. Let $S:\ell_{\infty}^{n}\times\cdots\times\ell_{\infty}^{n}\rightarrow\mathbb{K}$ be an $m$-linear form and consider

[TABLE]

and

[TABLE]

Since $s\geq 2$, from Lemma 2, Hölder’s inequality and Khinchin’s inequality for multiple sums ([16]), choosing $\theta=2/s$ we obtain

[TABLE]

where

[TABLE]

Repeating the same procedure for the other indexes we have

[TABLE]

for all $i=1,...,m.$ Hence, from Lemma 1, we conclude that

[TABLE]

for all $m$-linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n,$ where

[TABLE]

We thus conclude that if

[TABLE]

and

[TABLE]

with

[TABLE]

and

[TABLE]

then

[TABLE]

for all $m$-linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n.$ In other words, if

[TABLE]

with

[TABLE]

and

[TABLE]

then

[TABLE]

for all $m$-linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n.$

Remark 1**.**

If $\lambda_{0}=1$ then we get

[TABLE]

with $m<p\leq 2m-2$ and

[TABLE]

and we recover [1, Theorem 3].