Some applications of the Regularity Principle in sequence spaces

TL;DR

This paper uses a recent Regularity Principle to provide a simple proof of Hardy--Littlewood inequalities for m-linear forms, establishing monotonicity of constants and their bounds across different parameters.

Contribution

It offers a straightforward proof of Hardy--Littlewood inequalities leveraging a new Regularity Principle, and derives monotonicity properties of the involved constants.

Findings

Proves the Hardy--Littlewood inequality using the Regularity Principle.

Shows that the constants are non-decreasing with respect to p.

Establishes bounds relating constants for different m and p values.

Abstract

The Hardy--Littlewood inequalities for $m$-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q.J. Math, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For $m<p\leq2m$ it asserts that there is a constant $D_{m,p}^{\mathbb{K}}\geq1$ such that \[ \left( \sum_{j_{1},\cdots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\cdots,e_{j_{m}})\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq D_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert , \] for all $m$--linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and all positive integers $n$. Using a Regularity Principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy--Littewood inequality and show that: (1) If $m<p_{1}<p_{2}\leq2m$ then $D_{m,p_{1}}^{\mathbb{K}}\leq D_{m,p_{2}}^{\mathbb{K}}$; (2) $D_{m,p}^{\mathbb{K}}\leq D_{m-1,p}^{\mathbb{K}}$ whenever $m<p\leq 2\left( m-1\right) $ for all $m\geq3$.

Equations

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

$${\left(\sum _{i,j=1}^n{\left|T(e_i,e_j)\right|}^{\frac{4}{3}}\right)}^{\frac{3}{4}}\le \sqrt{2}\underset{\Vert x\Vert ,\Vert y\Vert \le 1}{\sup }\left|T\left(x,y\right)\right|$$

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

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

$$D_{m,p}^{\mathbb{K}}\le {\left(\sqrt{2}\right)}^{m-1}.$$

$$D_{m,p}^{\mathbb{K}}\le 2^{\frac{\left(m-1\right)\left(p-m+1\right)}{p}}.$$

$$\frac{1}{p}+\frac{1}{p^{\ast }}=1.$$

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

$${\left(\sum _{j_1,\cdots ,j_m=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m})\right|}^{\frac{p_2}{p_2-m}}\right)}^{\frac{p_2-m}{p_2}}\le D_{m,p_2}^{\mathbb{K}}\Vert T\Vert$$

$$\frac{\left(\frac{p_2}{p_2-m}\right)\left(\frac{p_2}{p_2-1}\right)\left(\frac{p_1}{p_1-1}\right)}{\left(\frac{p_2}{p_2-1}\right)\left(\frac{p_1}{p_1-1}\right)+m\left(\frac{p_2}{p_2-m}\right)\left(\frac{p_2}{p_2-1}\right)-m\left(\frac{p_2}{p_2-m}\right)\left(\frac{p_1}{p_1-1}\right)}=\frac{p_1}{p_1-m},$$

$$p_1^{\ast }<\frac{m\left(\frac{p_2}{p_2-m}\right)\left(\frac{p_2}{p_2-1}\right)}{m\left(\frac{p_2}{p_2-m}\right)-\left(\frac{p_2}{p_2-1}\right)}=m^{\ast }\iff p_1>m$$

$${\left(\sum _{j_1,\cdots ,j_m=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m})\right|}^{\frac{p_1}{p_1-m}}\right)}^{\frac{p_1-m}{p_1}}\le D_{m,p_2}^{\mathbb{K}}\Vert T\Vert$$

$$D_{m,p_1}^{\mathbb{K}}\le D_{m,p_2}^{\mathbb{K}}.$$

$$D_{m+1,p}^{\mathbb{K}}\le D_{m,p}^{\mathbb{K}}.$$

$$\lfloor x\rfloor$$

$$\lceil x\rceil$$

$${\left(D_{m,p}^{\mathbb{K}}\right)}_{m=\lceil p/2\rceil }^{\lfloor p-1\rfloor }$$

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

$$\frac{p}{p-m}\ge 2$$

$${\left(\sum _{j_1,\cdots ,j_{m+1}=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m},x_{j_{m+1}})\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\le D_{m,p}^{\mathbb{K}}\Vert T\Vert \underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|$$

$$u_n:E\to {\ell }_{\frac{p-m}{p}}$$

$$u_n(x)={\left(T(e_{j_1},\cdots ,e_{j_m},x\right)}_{j_1,...,j_m=1}^n.$$

$$C_{\frac{p-m}{p}}\left({\ell }_{\frac{p-m}{p}}\right)=1.$$

$${\left(\sum _{j=1}^n{\Vert u_n(x_j)\Vert }^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}$$

$$\le C_{\frac{p-m}{p}}\left({\ell }_{\frac{p-m}{p}}\right){\left({\int }_{[0,1]}{\Vert \sum _{j=1}^n{r}_j(t)u_n(x_j)\Vert }^{2}dt\right)}^{1/2}$$

$$\le \underset{t\in [0,1]}{\max }\Vert \sum _{j=1}^n{r}_j(t)u_n(x_j)\Vert$$

$$\stackrel{\text{cite[cite][@@bibreflp, page 284]}}{\le }\underset{\omega \in B_{{\ell }_{\frac{p-m}{p}}^{\ast }}}{\sup }\sum _{j=1}^n\left|\omega (u_n(x_j))\right|$$

$$\le \Vert u_n\Vert \underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|$$

$$\le \underset{\Vert x\Vert \le 1}{\sup }{\left(\sum _{j_1,\cdots ,j_{m+1}=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m},x)\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\left(\underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|\right)$$

$$\le \underset{\Vert x\Vert \le 1}{\sup }{D}_{m,p}^{\mathbb{K}}\Vert T(\cdot ,...,\cdot ,x)\Vert \left(\underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|\right)$$

$$\le D_{m,p}^{\mathbb{K}}\Vert T\Vert \underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|.$$

$${\left(\sum _{j_1,\cdots ,j_{m+1}=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m},x_{j_{m+1}})\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}={\left(\sum _{j=1}^n{\Vert u_n(x_j)\Vert }^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}$$

$${\left(\sum _{j_1,\cdots ,j_{m+1}=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m},x_{j_{m+1}})\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\le D_{m,p}^{\mathbb{K}}\Vert T\Vert \underset{\phi \in B_{E^{\ast }}}{\sup }\sum _{j_{m+1}=1}^n\left|\phi (x_{j_{m+1}})\right|.$$

$${\left(\sum _{j_{m+1}=1}^n{\left(\sum _{j_1,\cdots ,j_{m+1}=1}^n{\left|T(e_{j_1},\cdots ,e_{j_m},x_{j_{m+1}})\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}\times \frac{p}{p-\left(m+1\right)}}\right)}^{\frac{p-\left(m+1\right)}{p}}$$

Full text

[TABLE]

for all $m$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and all positive integers $n$. Using a Regularity Principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy–Littewood inequality and show that:

(1) if $m<p_{1}<p_{2}\leq 2m$ then $D_{m,p_{1}}^{\mathbb{K}}\leq D_{m,p_{2}}^{\mathbb{K}}$;

(2) $D_{m,p}^{\mathbb{K}}\leq D_{m-1,p}^{\mathbb{K}}$ whenever $m<p\leq 2\left(m-1\right)$ for all $m\geq 3$.

Partially supported by Capes.

MSC2010: 46G25

1. Introduction

Littlewood’s $4/3$ inequality [14] is probably the forerunner of the by now so-called Hardy–Littlewood inequalities for multilinear forms. Published in 1930, it asserts that

[TABLE]

for all bilinear forms $T:\ell_{\infty}^{n}\times\ell_{\infty}^{n}\rightarrow\mathbb{C}$ for all positive integers $n$. One year later, Bohnenblust and Hille [8] generalized Littlewood’s inequality to $m$-linear forms and, in 1934, Hardy and Littlewood [12] extended Littlewood’s $4/3$ inequality to bilinear forms acting on $\ell_{p}^{n}$ spaces. In 1981, Praciano-Pereira [20] extended Hardy–Littlewood’s inequalities to $m$-linear forms and, in 2016 Dimant and Sevilla-Peris [11] completed the results of Praciano-Pereira.

These results can be summarized as follows: for any integer $m\geq 2$ there exist (optimal) 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 $m$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n$ (here, and henceforth, $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$). The exponents are optimal.

The investigation of the optimal constants of the Hardy–Littlewood inequalities (see [1, 3, 4, 5, 6]) is motivated by their connection with the important Bohnenblust-Hille inequality (see, for instance [9, 15, 18, 21] and the references therein).

The original estimates for $D_{m,p}^{\mathbb{K}}$ are

[TABLE]

Recently, Albuquerque *et al.[2] *(see also [16]) have proved that

[TABLE]

In this note we use a Regularity Principle recently proved in [17] to give a straightforward proof of (1.2) and we also show some new monotonicity properties of the constants $D_{m,p}^{\mathbb{K}}$.

2. A straightforward proof of the Hardy–Littlewood inequality and new

mototonicity properties

From now on, if $p\in(1,\infty),$ the number $p^{\ast}$ is the conjugate of $p$, i.e.,

[TABLE]

We start off by giving a straightforward proof of (1.2). It is folklore that the case $p=2m$ in (1.1) can be re-written as a coincidence theorem for multiple summing operators. It says that every continuous $m$-linear form is multiple $\left(2;\left(2m\right)^{\ast}\right)$-summing. By the Regularity Principle from [17] (more precisely, by the Inclusion Theorem [17, Proposition 3.4]) we know that every multiple $\left(2;\left(2m\right)^{\ast}\right)$-summing form is multiple $\left(\frac{p}{p-m};p^{\ast}\right)$-summing, when $m<p\leq 2m$ with standard domination of norms. This means that

[TABLE]

for all $m$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n.$ In other words, we have just proved (1.2). Moreover, we have shown that (1.2) is a consequence of (1.1).

Now we prove a monotonicity result:

Theorem 1**.**

If $m<p_{1}\leq p_{2}\leq 2m$ then $D_{m,p_{1}}^{\mathbb{K}}\leq D_{m,p_{2}}^{\mathbb{K}}$

.

Proof.

If

[TABLE]

for all $m$–linear forms $T:\ell_{p_{2}}^{n}\times\cdots\times\ell_{p_{2}}^{n}\rightarrow\mathbb{K}$ and all positive integers $n$, we know that every continuous $m$-linear form is multiple $\left(\frac{p_{2}}{p_{2}-m};p_{2}^{\ast}\right)$-summing with norm dominated by $D_{m,p_{2}}^{\mathbb{K}}.$ By [17, Proposition 3.4], since

[TABLE]

and

[TABLE]

we conclude that every continuous $m$-linear form is multiple $\left(\frac{p_{1}}{p_{1}-m};p_{1}^{\ast}\right)$-summing with norm dominated by $D_{m,p_{2}}^{\mathbb{K}}.$ Thus

[TABLE]

for all $m$–linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{1}}^{n}\rightarrow\mathbb{K}$ and all positive integers $n$. Thus

[TABLE]

∎

Remark 1**.**

The Inclusion Theorem [17, Proposition 3.4] was also proved recently and independently, with a different technique, by F. Bayart [7].

3. Further monotonicity properties

In this section we prove the following result:

Theorem 2**.**

Let $m\geq 2$ be a positive integer and $m+1<p\leq 2m.$ Then

[TABLE]

Denoting

[TABLE]

we have the following corollary:

Corollary 1**.**

Let $p\in(3,\infty).$ The finite sequence

[TABLE]

is decreasing.

3.1. Proof of Theorem 2

If $m<p\leq 2m$ we know that

[TABLE]

for all $m$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n$. If $E$ is any Banach space, let us denote its dual by $E^{\ast}$ and the closed unit ball of $E^{\ast}$ by $B_{E^{\ast}}.$ Since

[TABLE]

we can prove that

[TABLE]

for all $m+1$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\times E\rightarrow\mathbb{K}$ and all positive integers $n$. In fact, for all positive integers $n,$ define the linear operator

[TABLE]

by

[TABLE]

Since $\frac{p-m}{p}\geq 2$ we know that $\ell_{\frac{p-m}{p}}$ has cotype $\frac{p-m}{p}.$ Moreover, it is folklore that (see, for instance, [19, page 29]) the cotype constant is $1:$

[TABLE]

Therefore, by the definition of cotype, and letting, as usual, $r_{j}$ denote the Rademacher functions, we have

[TABLE]

Since

[TABLE]

we thus have

[TABLE]

By the Regularity Principle for sequence spaces [17], for $p>m+1,$ we have

[TABLE]

for all positive integers $n$. As a matter of fact, here we do not need the Regularity Principle in its whole generality, and we provide a direct proof of (3.2) for the sake of completeness. Note that (3.1) is equivalent to

[TABLE]

and by using a standard trick involving the Hölder inequality (see also [10, 10.4 (Inclusion Theorem)]) we have (3.2).

Choosing $E=\ell_{p}^{n}$, since

[TABLE]

and since

[TABLE]

we conclude that

[TABLE]

for all $m+1$–linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}$ and all positive integers $n$. Therefore

[TABLE]