Anisotropic Regularity Principle in sequence spaces and applications

TL;DR

This paper introduces a generalized anisotropic regularity principle in sequence spaces, extending existing results and applying it to improve Hardy--Littlewood inequalities for multilinear forms.

Contribution

It refines a recent technique to establish a broad anisotropic regularity principle and generalizes previous Hardy--Littlewood inequality results for multilinear forms.

Findings

Established a general anisotropic regularity principle in sequence spaces.

Extended Hardy--Littlewood inequalities for multilinear forms.

Unified and improved previous results in the literature.

Abstract

We refine a recent technique introduced by Pellegrino, Santos, Serrano and Teixeira and prove a quite general anisotropic regularity principle in sequence spaces. As applications we generalize previous results of several authors regarding Hardy--Littlewood inequalities for multilinear forms.

Equations

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

$${\left(\sum _{j_1=1}^{\infty }{\left(\dots {\left(\sum _{j_m=1}^{\infty }{\Vert T\left(e_{j_1},\dots ,e_{j_m}\right)\Vert }^{s_m}\right)}^{\frac{s_{m-1}}{s_m}}\dots \right)}^{\frac{s_1}{s_2}}\right)}^{\frac{1}{s_1}}\le {\left(\sqrt{2}\right)}^{m-1}\Vert T\Vert$$

$$s_1=\frac{p}{p-m},\dots ,s_m=\frac{2mp}{mp+p-2m},$$

$${\left(\sum _{j=1}^{\infty }{\Vert T(x_j)\Vert }^r\right)}^{\frac{1}{r}}\le C{\Vert (x_j{)}_{j\in \mathbb{N}}\Vert }_{w,p}$$

$${\ell }_p^w(X):=\left\{(x_j{)}_{j\in \mathbb{N}}\in X^{\mathbb{N}}:{\Vert (x_j{)}_{j\in \mathbb{N}}\Vert }_{w,p}:=\underset{\phi \in B_{X^{\prime }}}{\sup }{\left(\sum _{j=1}^{\infty }{\left|\phi (x_j)\right|}^p\right)}^{\frac{1}{p}}<\infty \right\}.$$

$${\ell }_{\mathbf{p}}(X):={\ell }_{p_1}\left({\ell }_{p_2}\left(\cdots \left({\ell }_{p_m}\left(X\right)\right)\cdots \right)\right)$$

$${\Vert \mathbf{x}\Vert }_{\mathbf{p}}:={\left(\sum _{j_1=1}^{\infty }{\left(\sum _{j_2=1}^{\infty }{\left(\cdots {\left(\sum _{j_{m-1}=1}^{\infty }{\left(\sum _{j_m=1}^{\infty }{\Vert x_{\mathbf{j}}\Vert }^{p_m}\right)}^{\frac{p_{m-1}}{p_m}}\right)}^{\frac{p_{m-2}}{p_{m-1}}}\dots \right)}^{\frac{p_2}{p_3}}\right)}^{\frac{p_1}{p_2}}\right)}^{\frac{1}{p_1}}<\infty .$$

$${\Vert {\left(T(x_{\mathbf{j}})\right)}_{\mathbf{j}\in {\mathbb{N}}^m}\Vert }_{\mathbf{r}}:={\left(\sum _{j_1=1}^{\infty }{\left(\dots {\left(\sum _{j_m=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^{r_m}\right)}^{\frac{r_{m-1}}{r_m}}\dots \right)}^{\frac{r_1}{r_2}}\right)}^{\frac{1}{r_1}}\le C\prod _{k=1}^m\Vert x^k{\Vert }_{w,p_k},$$

$$\frac{1}{r}-\frac{m}{p}+\frac{m}{q}>0.$$

$${\Pi }_{(r;p)}^m\left(X_1,\dots ,X_m;Y\right)\subset {\Pi }_{(s;q)}^m\left(X_1,\dots ,X_m;Y\right),$$

$$\frac{1}{s}-\frac{m}{q}=\frac{1}{r}-\frac{m}{p},$$

$$\frac{1}{r}-\left|\frac{1}{\mathbf{p}}\right|+\left|\frac{1}{\mathbf{q}}\right|>0.$$

$${\Pi }_{(r;\mathbf{p})}^m\left(X_1,\dots ,X_m;Y\right)\subset {\Pi }_{(s;\mathbf{q})}^m\left(X_1,\dots ,X_m;Y\right),$$

$$\frac{1}{s}-\left|\frac{1}{\mathbf{q}}\right|=\frac{1}{r}-\left|\frac{1}{\mathbf{p}}\right|.$$

$$\frac{1}{r}-\left|\frac{1}{\mathbf{p}}\right|+\left|\frac{1}{\mathbf{q}}\right|>0.$$

$${\Pi }_{(r;\mathbf{p})}^m\left(X_1,\dots ,X_m;Y\right)\subset {\Pi }_{(\mathbf{s};\mathbf{q})}^m\left(X_1,\dots ,X_m;Y\right),$$

$$\frac{1}{s_k}-{\left|\frac{1}{\mathbf{q}}\right|}_{\ge k}=\frac{1}{r}-{\left|\frac{1}{\mathbf{p}}\right|}_{\ge k},$$

$${\left(\sum _{i=1}^{\infty }{\left(\sum _{j=1}^{\infty }|a_{ij}{|}^p\right)}^{q/p}\right)}^{1/q}\le {\left(\sum _{j=1}^{\infty }{\left(\sum _{i=1}^{\infty }|a_{ij}{|}^q\right)}^{p/q}\right)}^{1/p}.$$

$${\left(\sum _{j_3=1}^{\infty }{\left(\sum _{j_1,j_2=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^r\right)}^{\frac{1}{r}\cdot r}\right)}^{\frac{1}{r}}={\Vert {\left(T(x_{\mathbf{j}})\right)}_{\mathbf{j}\in {\mathbb{N}}^3}\Vert }_r\le C\prod _{k=1}^3\Vert x^k{\Vert }_{w,p_k},$$

$$v_3(x_3)={\left(T\left(x_{j_1}^1,x_{j_2}^2,x_3\right)\right)}_{j_1,j_2\in \mathbb{N}},$$

$${\left(\sum _{j_3=1}^{\infty }{\Vert v_3\left(x_{j_3}^3\right)\Vert }^r\right)}^{\frac{1}{r}}\le C_3{\Vert x^3\Vert }_{w,p_3},$$

$$\frac{1}{p_3}-\frac{1}{r}\le \frac{1}{q_3}-\frac{1}{s_3}.$$

$${\left(\sum _{j_2=1}^{\infty }{\left(\sum _{j_1=1}^{\infty }{\left(\sum _{j_3=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^{s_3}\right)}^{\frac{s_3}{s_3}}\right)}^{\frac{s_3}{s_3}}\right)}^{\frac{1}{s_3}}$$

$$\le {\left(\sum _{j_3=1}^{\infty }{\left(\sum _{j_1,j_2=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^r\right)}^{\frac{s_3}{r}}\right)}^{\frac{1}{s_3}}$$

$$\le C{\Vert x^3\Vert }_{w,q_3}\prod _{k=1}^2{\Vert x^k\Vert }_{w,p_k}.$$

$$v_2(x_2)={\left(T\left(x_{j_1}^1,x_2,x_{j_3}^3\right)\right)}_{j_1,j_3\in \mathbb{N}},$$

$${\left(\sum _{j_2=1}^{\infty }{\Vert v_2\left(x_{j_2}^2\right)\Vert }^{s_3}\right)}^{\frac{1}{s_3}}\le C_2{\Vert x^2\Vert }_{w,p_2},$$

$${\left(\sum _{j_2=1}^{\infty }{\left(\sum _{j_1=1}^{\infty }{\left(\sum _{j_3=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^{s_3}\right)}^{\frac{s_3}{s_3}}\right)}^{\frac{s_2}{s_3}}\right)}^{\frac{1}{s_2}}\le C_2{\Vert x^2\Vert }_{w,q_2}=C{\Vert x^1\Vert }_{w,p_1}\prod _{k=2}^3{\Vert x^k\Vert }_{w,q_k}.$$

$$\frac{1}{s_2}=\frac{1}{s_3}-\frac{1}{p_2}+\frac{1}{q_2}=\frac{1}{r}-\frac{1}{p_2}-\frac{1}{p_3}+\frac{1}{q_2}+\frac{1}{q_3},$$

$${\left(\sum _{j_1=1}^{\infty }{\left(\sum _{j_2=1}^{\infty }{\left(\sum _{j_3=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^{s_3}\right)}^{\frac{s_2}{s_3}}\right)}^{\frac{s_2}{s_2}}\right)}^{\frac{1}{s_2}}$$

$$\le {\left(\sum _{j_2=1}^{\infty }{\left(\sum _{j_1=1}^{\infty }{\left(\sum _{j_3=1}^{\infty }{\Vert T\left(x_{\mathbf{j}}\right)\Vert }^{s_3}\right)}^{\frac{s_3}{s_3}}\right)}^{\frac{s_2}{s_3}}\right)}^{\frac{1}{s_2}}$$

$$\le C{\Vert x^1\Vert }_{w,p_1}\prod _{k=2}^3{\Vert x^k\Vert }_{w,q_k}.$$

Full text

1. 1 Introduction

2. 1.1 Summability of multilinear operators

3. 1.2 Inclusion Theorems

4. 2 The new Inclusion Theorem

5. 2.1 The proof of Theorem 3

6. 3 A new Regularity Principle for sequence spaces

7. 4 Applications: Hardy–Littlewood’s inequalities

1. Introduction

Regularity techniques are crucial in many fields of pure and applied sciences. Recently, Pellegrino, Teixeira, Santos and Serrano addressed a regularity problem in sequence spaces with deep connections with the Hardy–Littlewood inequalities. In this paper we follow this vein, exploring the ideas from the Regularity Principle proven by Pellegrino et al. [21] and providing a couple of applications.

The paper is organized as follows. In Section 2, borrowing ideas from [21] we prove an anisotropic inclusion theorem for summing operators that will be useful along the paper. The techniques and arguments explored in Section 2 paves the way to the statement of a kind of anisotropic regularity principle for sequence spaces/series, in Section 3, completing results from [21]. In the final section the bulk of results are combined to prove new Hardy–Littlewood inequalities for multilinear operators.

1.1. Summability of multilinear operators

The theory of multiple summing multilinear mappings was introduced in [18, 23]; this class is certainly one of the most useful and fruitful multilinear generalizations of the concept of absolutely summing linear operators, with important connections with the Bohnenblust–Hille and Hardy-Littlewood inequalities and its applications in applied sciences. For recent results on absolutely summing operators and these classical inequalities we refer to [7, 11, 17] and the references therein.

The Hardy–Littlewood inequalities have its starting point in 1930 with Littlewood’s $4/3$ inequality [16]. In 1934 Hardy and Littlewood extended Littlewood’s $4/3$ inequality [15] to more general sequence spaces. Both results are for bilinear forms. In 1981 Praciano-Pereira [24] extended these results to the $m$-linear setting and recently various authors have revisited this subject. In 2016 Dimant and Sevilla-Peris [13] proved the following inequality: for all positive integers $m$ and all $m<p\leq 2m$ we have

[TABLE]

for all continuous $m$-linear forms $T:\ell_{p}\times\cdots\times\ell_{p}\rightarrow\mathbb{K}$. It is also proved that the exponent $\frac{p}{p-m}$ cannot be improved. However, this optimality seems to be just apparent, as remarked in some previous works (see [4, 11]). Following these lines, the exponent can be potentially improved in the anisotropic viewpoint. In order to do that, the theory of summing operators shall play a fundamental role.

Along the years, somewhat puzzling inclusion results for multilinear summing operators were obtained [8, 9, 23]. In this note we prove an inclusion result for multiple summing operators generalizing recent approaches of Pellegrino, Santos, Serrano and Teixeira [21] and Bayart [5]. It is interesting to note that, en passant, a sharper version of the Hardy–Littlewood inequalities for $m$-linear forms, not encompassed by several recent attempts (for instance, [4]), is provided as application. In fact, we show that under the same hypothesis of (1.1) we have

[TABLE]

with

[TABLE]

which is quite better than (1.1). Despite the huge advance recently obtained on this direction in [4], our results and techniques were not encompassed by its techniques.

Throughout this paper $X,Y$ shall stand for Banach spaces over the scalar field $\mathbb{K}$ of real or complex numbers. The topological dual of $X$ and its unit closed ball are denoted by $X^{\prime}$ and $B_{X^{\prime}}$, respectively. For $r,p\geq 1$, a linear operator $T:X\rightarrow Y$ is said $(r;p)$-summing if there exists a constant $C>0$ such that

[TABLE]

for any weakly $p$-summable vector sequence $(x_{j})_{j\in\mathbb{N}}\in\ell_{p}^{w}(X)$, where

[TABLE]

For $\mathbf{p}:=(p_{1},\dots,p_{m})\in[1,\infty]^{m}$, using the definitions of mixed norm $L_{\mathbf{p}}$ spaces from [6], the mixed norm sequence space

[TABLE]

gathers all multi-index vector valued matrices $\mathbf{x}:=\left(x_{\mathbf{j}}\right)_{\mathbf{j}\in\mathbb{N}^{m}}$ with finite $\mathbf{p}$-norm; here $\mathbf{j}:=(j_{1},\dots,j_{m})$ stands for a multi-index as usual. Notice that each norm $\|\cdot\|_{p_{k}}$ is taken over the index $j_{k}$ and that each index $j_{k}$ is related to the $\|\cdot\|_{p_{k}}$ norm. For instance, when ${\mathbf{p}}\in[1,\infty)^{m}$ a vector matrix $\mathbf{x}$ belongs to $\ell_{\mathbf{p}}(X)$ if, and only if,

[TABLE]

Over the last years, many different generalizations of the theory of absolutely and multiple summing operators were obtained. A natural anisotropic approach to multiple summing operators is the following: For $\mathbf{r},\mathbf{p}\in[1,+\infty)^{m}$, a multilinear operator $T:X_{1}\times\cdots\times X_{m}\longrightarrow Y$ is said to be multiple $(\mathbf{r},\mathbf{p})$-summing if there exists a constant $C>0$ such that for all sequences $x^{k}:=(x_{j}^{k})_{j\in\mathbb{N}}\in\ell_{p_{k}}^{w}(X_{k}),\,k=1,\dots,m$,

[TABLE]

where $T(x_{\mathbf{j}}):=T\left(x_{j_{1}}^{1},\dots,x_{j_{m}}^{m}\right)$. The class of all multiple $(\mathbf{r},\mathbf{p})$-summing operators is a Banach space with the norm defined by the infimum of all previous constants $C>0$. This norm is denoted by $\pi_{(\mathbf{r},\mathbf{p})}(\cdot)$ and the space that gathers all such operators by $\Pi_{(\mathbf{r};\mathbf{p})}^{m}(X_{1},\ldots,X_{m};Y)$. When $r_{1}=\cdots=r_{m}=r,\,s_{1}=\cdots=s_{m}=s$ we simply write $(r;\mathbf{p}),\,(\mathbf{r};s)$, respectively.

1.2. Inclusion Theorems

Basic results from the theory of summing operators are inclusion theorems. For linear operators, it is folklore that $p$-summability implies $q$-summability whenever $1\leq p\leq q$. More generally, although basic, the following is quite useful (see [12]).

Linear Inclusion Theorem**.**

If $s\geq r,\,q\geq p$ and $\frac{1}{p}-\frac{1}{r}\leq\frac{1}{q}-\frac{1}{s}$, then every absolutely $(r;p)$-summing linear operator is absolutely $(s;q)$-summing

Throughout the development of the theory, inclusion theorems reveals as challenging problems (see, for instance, [23]). In [21, Proposition 3.4], the authors proved the followimg multilinear inclusion result:

Theorem 1** (Pellegrino, Santos, Serrano and Teixeira).**

Let $m$ be a positive integer, $r,p,q\in[1,+\infty)$ be such that $q\geq p$ and

[TABLE]

Then

[TABLE]

for any Banach spaces $X_{1},\dots,X_{m}$, with

[TABLE]

and the inclusion operator has norm $1$.

Independently, F. Bayart in [5, Theorem 1.2] obtained a more general version. For $\mathbf{p}\in[1,+\infty]^{m}$ and each $k\in\{1,\dots,m\}$ , we define $\left|\frac{1}{\mathbf{p}}\right|_{\geq k}:=\frac{1}{p_{k}}+\cdots+\frac{1}{p_{m}}$. When $k=1$ we write $\left|\frac{1}{\mathbf{p}}\right|$ instead of $\left|\frac{1}{\mathbf{p}}\right|_{\geq 1}$

Theorem 2** (Bayart).**

Let $m$ be a positive integer, $r,s\in[1,+\infty),\,\mathbf{p},\mathbf{q}\in[1,+\infty)^{m}$ are such that $q_{k}\geq p_{k},\,k=1,\dots,m$ and

[TABLE]

Then

[TABLE]

for any Banach spaces $X_{1},\dots,X_{m}$, with

[TABLE]

In the next section we prove the following inclusion theorem; the techniques are inspired by [21] and contained in the proof of the forthcoming Regularity Principle, in Section 3:

Theorem 3**.**

Let $m$ be a positive integer, $r\geq 1,\,\mathbf{s},\mathbf{p},\mathbf{q}\in[1,+\infty)^{m}$ are such that $q_{k}\geq p_{k}$, for $k=1,\dots,m$ and

[TABLE]

Then

[TABLE]

for any Banach spaces $X_{1},\dots,X_{m}$, with

[TABLE]

for each $k\in\{1,\dots,m\}$, and the inclusion operator has norm $1$.

2. The new Inclusion Theorem

Despite of the general status of the result, only basic facts are used along its proof. The first one is the classical linear inclusion. We need other standard inclusion type result that we write for future reference.

Inclusion on $\ell_{p}$ spaces**.**

For $q\geq p>0,\,\|\cdot\|_{q}\leq\|\cdot\|_{p}$.

The last ingredient is a corollary of one of the many versions of Minkowski’s inequality (see [14, Corollary 5.4.2]):

Minkowski’s inequality**.**

For any $0<p\leq q<\infty$ and for any scalar matrix $(a_{ij})_{i,j\in\mathbb{N}}$,

[TABLE]

2.1. The proof of Theorem 3

The argument is inspired on the Regularity Principle of [21, Theorem 2.1]. We will proceed by induction on $m$. The initial case bilinear is a straightforward application of classical inclusion of linear operators and $\ell_{p}$ spaces. The ideas used are revealed in the case $m=3$, thus we it discuss in details. Let $T\in\Pi_{(r;\mathbf{p})}^{3}\left(X_{1},X_{2},X_{3};Y\right)$. Then there exists a constant $C>0$ such that

[TABLE]

for all sequences $x^{k}=(x_{j}^{k})_{j\in\mathbb{N}}\in\ell_{p_{k}}^{w}(X_{k}),\,k=1,2,3$. Let $x^{k}\in\ell_{p_{k}}^{w}(X_{k})$ with $k=1,2$ fixed. Defining $v_{3}:X_{3}\longrightarrow\ell_{(r,r)}(Y)$ by

[TABLE]

for all $x_{3}\in X_{3}$. By (2.1) we obtain, for all $x^{3}\in\ell_{p_{3}}^{w}(X_{3})$,

[TABLE]

with $C_{3}=C\prod_{k=1}^{2}\left\|x^{k}\right\|_{w,p_{k}}$, i.e., $v_{3}\in\Pi_{(r;p_{3})}\left(X_{3};\ell_{(r,r)}(Y)\right)$. The linear inclusion [12, Theorem 10.4] lead us to $v_{3}\in\Pi_{(s_{3};q_{3})}\left(X_{3};\ell_{(r,r)}(Y)\right)$ with $q_{3}\geq p_{3},\,s_{3}\geq r$ such that

[TABLE]

Let us take $\frac{1}{s_{3}}=\frac{1}{r}-\frac{1}{p_{3}}+\frac{1}{q_{3}}>0$. Applying norm inclusion on $\ell_{p}$ we obtain

[TABLE]

Fixing $x^{1}\in\ell_{p_{1}}^{w}(X_{1}),\,x^{3}\in\ell_{q_{3}}^{w}(X_{3})$ and defining $v_{2}:X_{2}\longrightarrow\ell_{(s_{3},s_{3})}(Y)$ by

[TABLE]

we observe that (2.2) leads us to

[TABLE]

for all $x^{2}\in\ell_{p_{2}}^{w}(X_{2})$ with $C_{2}=C\left\|x^{3}\right\|_{w,q_{3}}\left\|x^{1}\right\|_{w,p_{1}}$, i.e., $v_{2}\in\Pi_{(s_{3};p_{2})}\left(X_{2};\ell_{(s_{3},s_{3})}(Y)\right)$. By the linear inclusion theorem, $v_{2}\in\Pi_{(s_{2};q_{2})}\left(X_{2};\ell_{(s_{3},s_{3})}(Y)\right)$ with $q_{2}\geq p_{2},\,s_{2}\geq s_{3}$ and $\frac{1}{p_{2}}-\frac{1}{s_{3}}\leq\frac{1}{q_{2}}-\frac{1}{s_{2}}$, we get

[TABLE]

Taking

[TABLE]

since $s_{3}\leq s_{2}$, by using norm inclusion on $\ell_{p}$, we have

[TABLE]

Now let us fix $x^{k}\in\ell_{q_{k}}^{w}(X_{k})$ with $k=2,3$ and let us define, for all $x_{1}\in X_{1}$,

[TABLE]

Thus $v_{1}\in\Pi_{(s_{2};p_{1})}\left(X_{1};\ell_{(s_{2},s_{3})}(Y)\right)$. By combining (2.3) and the linear inclusion theorem, we get that $v_{1}\in\Pi_{(s_{1};q_{1})}\left(X_{1};\ell_{(s_{2},s_{3})}(Y)\right)$ with $q_{1}\geq p_{1}$ and $s_{1}\geq s_{2}$ such that

[TABLE]

By choosing $\frac{1}{s_{1}}=\frac{1}{s_{2}}-\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{r}-\sum_{k=1}^{3}\frac{1}{p_{k}}+\sum_{k=1}^{3}\frac{1}{q_{k}}>0$, we have

[TABLE]

once that $v_{1}\in\Pi_{(s_{1};q_{1})}\left(X_{1};\ell_{(s_{2},s_{3})}(Y)\right)$. Therefore, $T\in\Pi_{(\mathbf{s};\mathbf{q})}^{3}\left(X_{1},X_{2},X_{3};Y\right)$.

Now we shall conclude the proof by an induction argument. Let us suppose the result is true for $m-1$ and let $T\in\Pi_{(r;\mathbf{p})}^{m}\left(X_{1},\dots,X_{m};Y\right)$, i.e.,

[TABLE]

for all sequences $x^{k}\in\ell_{p_{k}}^{w}(X_{k})$. For a fixed $x^{1}\in\ell_{p_{1}}^{w}(X_{1})$, $v:X_{2}\times\cdots\times X_{m}\rightarrow\ell_{r}(Y)$ given by

[TABLE]

belongs to $\Pi_{(r;p_{2},\dots,p_{m})}^{m-1}\left(X_{2},\dots,X_{m};\ell_{r}(Y)\right)$. Consequently, by induction hypothesis, norm inclusion and the Minkowski inequality,

[TABLE]

with $r\leq s_{m}\leq\dots\leq s_{2}$ and

[TABLE]

Fixing $x^{k}\in\ell_{q_{k}}^{w}(X_{k}),\,k=2,\dots,m$ and defining, for all $x_{1}\in X_{1}$,

[TABLE]

we have that $u\in\Pi_{(s_{2};p_{1})}\left(X_{1};\ell_{(s_{2},\dots,s_{m})}(Y)\right)$. Applying the classical linear inclusion on (2.4), with $q_{1}\geq p_{1}$ and $s_{1}\geq s_{2}$ such that

[TABLE]

we gain $u\in\Pi_{(s_{1};q_{1})}\left(X_{1};\ell_{(s_{2},\dots,s_{m})}(Y)\right)$. Taking $\frac{1}{s_{1}}=\frac{1}{s_{2}}-\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{r}-\left|\frac{1}{\mathbf{p}}\right|+\left|\frac{1}{\mathbf{q}}\right|>0$, since $u\in\Pi_{(s_{1};q_{1})}\left(X_{1};\ell_{(s_{2},\dots,s_{m})}(Y)\right)$, we have

[TABLE]

Therefore, $T\in\Pi_{(\mathbf{s};\mathbf{q})}^{m}\left(X_{1},\dots,X_{m};Y\right)$. Also note that the inclusion operator has norm 1, since the constant C is preserved. This concludes the proof. ∎

It is important to highlight the difference between Theorems 1, 2 and 3. Under the hypothesis of Theorem 3, by using the usual inclusion of $\ell_{p}$ spaces and Theorem 2 with $\frac{1}{s_{1}}:=\frac{1}{r}-\left|\frac{1}{\mathbf{p}}\right|+\left|\frac{1}{\mathbf{q}}\right|$ one may get that

[TABLE]

But if $r\leq s_{m}\leq\cdots\leq s_{1}$,

[TABLE]

Nevertheless we may not conclude by Theorem 2 that

[TABLE]

However this is provided by Theorem 3. For instance, let us illustrate with a numerical example: let $r:=3,\,\mathbf{s}:=(5,3),\,\mathbf{p}:=(3,2)$ and $\mathbf{q}:=(5,2)$. From Theorem 2 we have

[TABLE]

while Theorem 3 provides

[TABLE]

The same can also be done when $m=3$: let $r:=2,\,\mathbf{s}:=(6,3,2),\,\mathbf{p}:=(2,2,1)$ and $\mathbf{q}:=(3,3,1)$. Then

[TABLE]

where the first inclusion is assured by Theorem 3.

3. A new Regularity Principle for sequence spaces

The investigation of regularity-type results in this setting was initiated in [20] and expanded in [21]. In this short section we present a stronger version of these results.

Let $m\geq 2$ and $Z_{1}$, $V$ and $w_{1},\dots,W_{m}$ be arbitrary non-empty sets and $Z_{2},\dots,Z_{m}$ be vector spaces. Let also

[TABLE]

be arbitrary maps, with $k=1,\dots,m,$ satisfying

[TABLE]

for all scalars $\lambda\geq 0$ and $j,k\in\{2,\dots,m\}$. We shall work with each $p_{k}\geq 1$ and also assuming that

[TABLE]

Despite the abstract context, the proof is similar to the the proof of Theorem 3, and we omit the details.

Theorem 4** (Anisotropic Regularity Principle).**

Let $m$ be a positive integer, $r\geq 1,\,\mathbf{s},\mathbf{p},\mathbf{q}\in[1,+\infty)^{m}$ be such that $q_{k}\geq p_{k}$, for $k=1,\dots,m$ and

[TABLE]

Assume that there exists a constant $C>0$ such that

[TABLE]

for all $z_{j}^{(k)}\in Z_{k}$ and $n_{k}\in\mathbb{N}$ with $k=1,\dots,m$. Then

[TABLE]

for all $z_{j}^{(k)}\in Z_{k}$ and $n_{k}\in\mathbb{N},\,k=1,\dots,m$, with

[TABLE]

4. Applications: Hardy–Littlewood’s inequalities

The Hardy–Littlewood inequalities have been investigated in depth in the recent years (see, for instance, [1, 2, 3, 4, 10, 11, 13, 17, 19, 21, 22]). Here $\mathcal{X}_{p}$ stands for $\ell_{p}$ if $p<\infty$ and $\mathcal{X}_{\infty}:=c_{0}$.

Theorem 5** (Albuquerque, Bayart, Pellegrino, Seoane [2], 2014).**

Let $\mathbf{p},\mathbf{s}\in[1,+\infty]^{m}$ such that $\left|1/\mathbf{p}\right|\leq\frac{1}{2}$ and $\mathbf{s}\in\left[\left(1-\left|1/\mathbf{p}\right|\right)^{-1},2\right]^{m}$. There is a constant $C_{m,\mathbf{p},\mathbf{s}}^{\mathbb{K}}\geq 1$ such that

[TABLE]

for every continuous $m$-linear form $A:\mathcal{X}_{p_{1}}\times\cdots\times\mathcal{X}_{p_{m}}\rightarrow\mathbb{K}$ if, and only if,

[TABLE]

Theorem 6** (Dimant, Sevilla-Peris [13], 2016).**

Let $\mathbf{p}\in[1,+\infty]^{m}$ such that $1/2\leq\left|1/\mathbf{p}\right|<1$. There exists a (optimal) constant $D_{m,\mathbf{p}}^{\mathbb{K}}\geq 1$ such that

[TABLE]

For every continuous $m$-linear form $A:\mathcal{X}_{p_{1}}\times\cdots\times\mathcal{X}_{p_{m}}\rightarrow\mathbb{K}$. Moreover, the exponent is optimal.

The above exponent $\frac{1}{1-\left|1/\mathbf{p}\right|}$ is optimal, but not in the anisotropic sense. In [4, Theorem 3.2] the authors improved Theorem 6, under some restriction over $\mathbf{p}$. The following recent result is a kind of anisotropic extension of it:

Theorem 7** (Aron, Núñez, Pellegrino, Serrano [4], 2017).**

Let $m\geq 2,q_{1},\dots,q_{m}>0$, and $1<p_{m}\leq 2<p_{1},\dots,p_{m-1}$, with $\left|1/\mathbf{p}\right|<1$. There is a (optimal) constant $C_{\mathbf{p}}\geq 1$ such that

[TABLE]

for every continuous $m$-linear form $A:\mathcal{X}_{p_{1}}\times\cdots\times\mathcal{X}_{p_{m}}\rightarrow\mathbb{K}$ if, and only if,

[TABLE]

Recently, W.V. Cavalcante has shown that (4.1) is a consequence of the inclusion result for multiple summing operators due to Pellegrino et al. combined with Theorem 5 (see [10]). The standard isometries between $\mathcal{L}(\mathcal{X}_{p},X)$ and $\ell_{p^{\ast}}^{w}(X)$, for $1<p\leq\infty$, allow us to read the previous Theorems 5, 6, 7 as coincidence results (see [12]). The key point is to begin with the coincidence below, obtained by revisiting Theorem 5 as a coincidence result with $s_{1}=\cdots=s_{m}=2$ and $p_{1}=\cdots=p_{m}=2m$,

[TABLE]

for all Banach spaces $X_{1},\dots,X_{m}$, and use an inclusion-type result. We shall combine these isometries with the inclusion result Theorem 3 to gain refined inclusions and coincidences.

Theorem 8**.**

Let $m$ be a positive integer and $\mathbf{s},\,\mathbf{p}\in[1,+\infty]^{m}$. If $\left|1/\mathbf{p}\right|<1$ and $p_{1},\dots,p_{m}\leq 2m$, then

[TABLE]

for any Banach spaces $X_{1},\dots,X_{m}$, with

[TABLE]

Proof.

Since each $p_{k}\leq 2m$ and

[TABLE]

applying Inclusion Theorem 3, it is obtained the stated inclusion with

[TABLE]

for each $k=1,\dots,m$. ∎

Corollary 1**.**

If $\left|1/\mathbf{p}\right|<1$ and $p_{1},\dots,p_{m}\leq 2m$, then

[TABLE]

for any Banach spaces $X_{1},\dots,X_{m}$ and

[TABLE]

Bringing Theorem 8 to the context of sequence spaces, the announced anisotropic result will be achieved. The main result of this section reads as follows.

Corollary 2**.**

Let $m$ be a positive integer and $\mathbf{p}\in[1,+\infty)^{m}$ such that $\left|1/\mathbf{p}\right|<1$ and $p_{1},\dots,p_{m}\leq 2m$. Then, for all continuous $m$-linear forms $A:\ell_{p_{1}}\times\cdots\times\ell_{p_{m}}\rightarrow\mathbb{K}$

[TABLE]

with

[TABLE]

In order to clarify the new result, we illustrate the simpler case: when dealing with $m<p_{1}=\cdots=p_{m}=p\leq 2m$, we get (4.2) with exponents

[TABLE]

that is,

[TABLE]

It is obvious that the above exponents are better than the estimates of Theorem 6 that provides

[TABLE]

The following example is illustrative:

Example 1**.**

Suppose $m=3$ and $p=4$. By Theorem 6 we know that (4.2) holds with

[TABLE]

whereas by Corollary 2 we have

[TABLE]