Coefficients of Multilinear Forms on Sequence Spaces

Abstract

The investigation of regularity/summability properties of the coefficients of bilinear forms in sequence spaces was initiated by Littlewood in 1930. Nowadays, this topic has important connections with other fields of Pure and Applied Mathematics. In this paper we explore a regularity technique to obtain optimal parameters for several results in this framework, extending/generalizing theorems of Osikiewicz and Tonge (Linear Algebra Appl 331:1–9, 2001), Albuquerque et al. (J Funct Anal 266:3726–3740, 2016), Aron et al. (Linear Algebra Appl 531:399–422, 2017), Albuquerque and Rezende (Commun Contemp Math 20:1750087, 2018), Paulino (Rend Circ Mat Palermo 69:369–380, 2020), among others.

1 Introduction

The investigation of summability properties of the coefficients of multilinear forms defined on sequence spaces has its origins in 1930 with Littlewood’s seminal paper (Littlewood 1930). Littlewood proved that for every bilinear form $A:{\mathbb {C}}^{n}\times {\mathbb {C}}^{n}\rightarrow {\mathbb {C}}$, we have

$$\begin{aligned} \left( {\textstyle \sum \limits _{i,j=1}^{n}} \left| A\left( e_{i},e_{j}\right) \right| ^{4/3}\right) ^{3/4} \le \sqrt{2}\sup \left\{ \left| A(z^{(1)},z^{(2)})\right| :z^{(1)},z^{(2)}\in {\mathbb {D}}^{n}\right\} \text {,} \end{aligned}$$

(1.1)

where ${\mathbb {D}}^{n}$ represents the open unit polydisc in ${\mathbb {C}}^{n}$. Moreover, the exponent 4/3 cannot be improved in the sense that, if we replace 4/3 by a smaller exponent, it is not possible to change $\sqrt{2}$ by a constant not depending on n. This result is known as Littlewood’s 4/3 inequality and its original motivation was a problem posed by P.J. Daniell concerning functions of bounded variation. Note that the terms $A\left( e_{i},e_{j}\right)$ are precisely the coefficients of the bilinear form A. Nowadays it is well-known that summability properties of the coefficients of multilinear forms play an important role in Mathematics and related fields (see Núñez-Alarcón et al. (2022) and the references therein).

The natural extension of Littlewood’s 4/3 inequality to multilinear forms in ${\mathbb {K}}^{n}$ (${\mathbb {K}}={\mathbb {R}}$ or ${\mathbb {C}}$) spaces, with $\ell _{p}$ norms, was initiated for bilinear forms in (Hardy and Littlewood (1934)) and extended to multilinear forms in (Praciano-Pereira (1981)); since then several authors have investigated this subject (see Núñez-Alarcón et al. (2022) and the references therein). In this new context, for an m-linear form $A:{\mathbb {K}}^{n}\times \cdots \times {\mathbb {K}} ^{n}\rightarrow {\mathbb {K}}$, the supremum at the right-hand-side of (1.1) is replaced by

$$\begin{aligned} \left\| A\right\| :=\sup \left\{ \left| A(z^{(1)},\ldots ,z^{(m)})\right| :\left\| z^{(j)}\right\| _{\ell _{p_{j}}^{n}} \le 1\right\} \text {.} \end{aligned}$$

Above and henceforth, for the sake of simplicity, $\ell _{p}^{n}$ denotes ${\mathbb {K}}^{n}$ with the $\ell _{p}$ norm. Even if we do not explicitly mention, all inequalities along this paper hold for all positive integers n and the respective constants do not depend on n; when we write $C_{m}$ it means that the constant just depend on m (the degree of m-linearity of a multilinear form and we shall assume that $m\ge 2$). As usual, $p^{*}$ shall denote the conjugate of p, i.e., $1/p+1/p^{*}=1$ and we assume that $1/\infty =0$ and $1/0=\infty$.

There is a vast recent literature related to the Hardy–Littlewood inequalities (HL for short) for multilinear forms in sequence spaces. The main lines of research are the search of optimal exponents and optimal constants (see Núñez-Alarcón et al. (2022) and the references therein). Following the lines of the seminal paper of Hardy and Littlewood, the investigation is usually divided in two cases:

• $1/p_{1}+\cdots +1/p_{m}\le 1/2$,

• $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$.

In the first case, Praciano-Pereira in the work  (Praciano-Pereira 1981) extended the bilinear result of Hardy and Littlewood by proving that there exists a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\left| A(e_{j_{1} },\ldots ,e_{j_{m}})\right| ^{\mu }\right) ^{\frac{1}{\mu }}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

(1.2)

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, where

$$\begin{aligned} \mu :=\frac{2m}{m+1-2\left( 1/p_{1}+\cdots +1/p_{m}\right) }\text {.} \end{aligned}$$

This result was recently extended in Albuquerque et al. (2016, Theorem 2.2). In fact, if $1/p_{1}+\cdots +1/p_{m}\le 1/2$, choosing $\left( r,q\right) =\left( 1,2\right)$ and $Y={\mathbb {K}}$ in Albuquerque et al (2016, Theorem 2.2) we have: if $t_{1},\dots ,t_{m}\in \left( \left[ 1-\left( 1/p_{1}+\cdots +1/p_{m}\right) \right] ^{-1},2\right)$, there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum \limits _{j_{1}=1}^{n}\left( \cdots \left( \sum \limits _{j_{m} =1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\cdots \right) ^{\frac{t_{1}}{t_{2}} }\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

(1.3)

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ if, and only if,

$$\begin{aligned} \frac{1}{t_{1}}+\cdots +\frac{1}{t_{m}}\le \frac{m+1}{2}-\left( \dfrac{1}{p_{1}}+\cdots +\dfrac{1}{p_{m}}\right) \text {.} \end{aligned}$$

(1.4)

Notice that, due to the monotonicity of the $\ell _{p}$ norms, the interesting case in (1.4) is when the equality holds.

From now on, we say that an exponent in a HL inequality is optimal (or sharp) if it cannot be replaced by a smaller one (as, for instance, in the satement of the forthcoming Theorem 1.2). However, the equivalence (1.3)$\Leftrightarrow$(1.4) shows that, in general, there is no unique solution to the question: what is the optimal exponent at the j-th position of a certain HL inequality? In fact, at least on the aforementioned case, note that the optimal value of $t_{1}$ depends on $t_{2},\ldots ,t_{m}$ and so on. This motivates a more involved notion of optimality, introduced in Pellegrino, et al. (2017, Definition 7.1): an m-tuple of exponents $\left( t_{1},\ldots ,t_{m}\right)$ is called globally sharp if a Hardy-Littlewood type inequality holds for these exponents and, for any $\varepsilon _{j}>0$ and $j=1,\ldots ,m$, there is no HL inequality for the m-tuple of exponents $\left( t_{1},\ldots ,t_{j-1},t_{j}-\varepsilon _{j},t_{j+1},\ldots ,t_{m}\right)$.

From a different viewpoint a globally sharp m-tuple of exponents $\left( t_{1},\ldots ,t_{m}\right)$ is a border point of the set of all admissible exponents of a certain type of HL inequality (see Fig. 1).

Fig. 1

A globally sharp exponent (a, b, c) for $m=3$ in a certain HL inequality

Despite the active research in the field, several basic issues remain open and in the present paper we deal with some of these questions. Our first main result (Theorem 1.5) provides globally sharp exponents for the case $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$. We begin by recalling recent results in this line.

The first one is due to Osikiewicz and Tonge (Theorem 1.1); in 2016, Dimant and Sevilla-Peris (Theorem 1.2) obtained an optimal m-linear version for the isotropic version (choosing the same exponents in all indexes) and, more recently, anisotropic variants were obtained by Albuquerque and Rezende (Theorem 1.3) and Aron et al. (Theorem 1.4).

Theorem 1.1

(See Osikiewicz and Tonge, 2001, Theorem 5) If $p_{1},p_{2}\in (2,\infty ]\times (1,2]$ and $1/2\le 1/p_{1}+1/p_{2}<1$, then

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \sum _{j_{2}=1}^{n}\left| A(e_{j_{1} },e_{j_{2}})\right| ^{p_{2}^{*}}\right) ^{\frac{\lambda }{p_{2}^{*} }}\right) ^{\frac{1}{\lambda }}\le \left\| A\right\| \text {,} \end{aligned}$$

for all bilinear forms $A:\ell _{p_{1}}^{n}\times \ell _{p_{2}} ^{n}\rightarrow {\mathbb {K}}$, with $1/\lambda :=1-\left( 1/p_{1}+1/p_{2}\right)$.

Theorem 1.2

(See Dimant and Sevilla-Peris, 2016, Proposition 4.1) If $p_{1},\ldots ,p_{m} \in [1,\infty ]$ and $1/2\le 1/p_{1}+\cdots +1/p_{m}<1,$ then there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{\lambda }\right) ^{\frac{1}{\lambda }}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, with $1/\lambda :=1-\left( 1/p_{1} +\cdots +1/p_{m}\right)$. Moreover, $\lambda$ is optimal.

Theorem 1.3

(See Albuquerque and Rezende, 2018, Corollary 2) If $p_{1},\ldots ,p_{m} \in (1,2\,m]$ and $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$, then there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{2}}{s_{1}}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A\right\| \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{n}}^{n}\rightarrow {\mathbb {K}}$, with

$$\begin{aligned} s_{k}=\left[ \dfrac{1}{2}+\frac{m-k+1}{2m}-\left( \frac{1}{p_{k}} +\cdots +\frac{1}{p_{m}}\right) \right] ^{-1}\text {,} \end{aligned}$$

for all $k=1,\ldots ,m$.

Theorem 1.4

(See Aron et al., 2017, Theorem 3.2) If $p_{1},\ldots ,p_{m-1} \in (1,\infty ]$, $p_{m}\in (1,2]$ and $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$, then

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \sum _{j_{2}=1}^{n}\cdots \left( \sum _{j_{m}=1}^{n}\left| A(e_{j_{1}},\ldots ,e_{j_{m}})\right| ^{s_{m} }\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{1}}{s_{2}} }\right) ^{\frac{1}{s_{1}}}\le \left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, where

$$\begin{aligned} s_{k}=\left[ 1-\left( \frac{1}{p_{k}}+\cdots +\frac{1}{p_{m}}\right) \right] ^{-1}\text {,} \end{aligned}$$

for all $k=1,\ldots ,m$. Moreover the exponents $s_{1},\ldots ,s_{m}$ are optimal.

Our first main theorem, stated below, encompasses/generalizes/extends all the previous results in an essentially optimal fashion. More precisely, we extend the theorem of Osikiewicz and Tonge (Theorem 1.1) to the multilinear setting, improve the exponents provided by Theorems 1.2 and 1.3, and relax the hypothesis $1<p_{m}\le 2<p_{1},\ldots ,p_{m-1}$ of Theorem 1.4; note that we offer a different and simplified proof of Theorem 1.4. We also recover, by a completely different approach, the optimal constants from Theorem 1.1 and Theorem 1.4 (see Remark 3.1).

Theorem 1.5

Let $p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right]$ be such that $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$. There is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m}-1}{s_{m}}}\cdots \right) ^{\frac{s_{1}}{s_{2}}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

(1.5)

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, where

$$\begin{aligned} s_{k}=\left\{ \begin{array}{ll} \left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }k\le k_{0}:=\max \left\{ t:\dfrac{1}{p_{t}} +\cdots +\dfrac{1}{p_{m}}\ge \dfrac{1}{2}\right\} \text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text {.} \end{array} \right. \nonumber \\ \end{aligned}$$

(1.6)

Moreover:

(i):

The exponents $s_{1},\ldots ,s_{k_{0}}$ are optimal;

(ii):

If $p_{k_{0}}\ge 2$, the exponents $\left( s_{1},\ldots ,s_{m}\right)$ are globally sharp.

Notice that, under the hypotheses of the previous theorem, we have $p_{k}>2$ for all $k=k_{0}+1,\ldots ,m$. In fact, by the definition of $k_{0}$, we have $1/p_{k_{0}+1}+\cdots +1/p_{m}<1/2$ and hence $p_{k}>2$ for all $k=k_{0} +1,\ldots ,m$.

The improvement of Theorems 1.1, 1.2 and 1.3 is easily observed from the statement of our main theorem (see also Table 1). As to Theorem 1.4, note that an immediate corollary of Theorem 1.5 yields the following result, that recovers Theorem 1.4 for the particular case $i=m$:

Corollary 1.6

Let $p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right]$ be such that $1/2\le 1/p_{1}+\cdots +1/p_{m}<1$ with $1<p_{i}\le 2$ for a certain i. There is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \sum _{j_{2}=1}^{n}\cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{1}}{s_{2}}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, where

$$\begin{aligned} s_{k}=\left\{ \begin{array}{ll} \left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }k\le i\text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>i\text {.} \end{array} \right. \end{aligned}$$

Moreover, the exponents $s_{1},\ldots ,s_{i}$ are optimal.

The paper is organized as follows. In Section 2 we present the main tools (the Anisotropic Regularity Principle and techniques of the theory of multilinear absolutely summing operators) that will be used along the proofs of our main results. In Sections 3 and 4 we prove our main theorem. In Section 5, we obtain globally sharp exponents for the critical case $p_{1}=\cdots =p_{m}=m$. In Section 6, we show that, in general, the Anisotropic Regularity Principle is optimal. Finally, in the final section, we mention some related open problems concerning summability properties of the coefficients of multilinear forms defined on sequence spaces.

2 Regularity Principle: The Main Tool

A regularity treatment of Hardy–Littlewood inequalities was successfully launched in Pellegrino et al. (2017), where the authors investigated the following general universality problem (observe that the existence of a leeway, $\epsilon >0$, of an increment $\delta >0$, and of a corresponding bound ${\tilde{C}} _{\delta ,\epsilon }>0$ carries a regularity principle):

Problem 2.1

Let $p\ge 1$ be a real number, $X,Y,W_{1},W_{2}$ be non-void sets, $Z_{1},Z_{2},Z_{3}$ be normed spaces and $f:X\times Y\rightarrow Z_{1},\ g:X\times W_{1}\rightarrow Z_{2},\ h:Y\times W_{2}\rightarrow Z_{3}$ be particular maps. Assume there is a constant $C>0$ such that

$$\begin{aligned} { {\textstyle \sum \limits _{i=1}^{m_{1}}} {\textstyle \sum \limits _{j=1}^{m_{2}}} \left\| f(x_{i},y_{j})\right\| ^{p}\le C\left( \sup _{w\in W_{1}} {\textstyle \sum \limits _{i=1}^{m_{1}}} \left\| g(x_{i},w)\right\| ^{p}\right) \cdot \left( \sup _{w\in W_{2}} {\textstyle \sum \limits _{j=1}^{m_{2}}} \left\| h(y_{j},w)\right\| ^{p}\right) }\text {,} \end{aligned}$$

for all $x_{i}\in X$, $y_{j}\in Y$ and $m_{1},m_{2}\in {\mathbb {N}}$. Are there (universal) positive constants $\epsilon \sim \delta$, and ${\tilde{C}} _{\delta ,\epsilon }$ such that

$$\begin{aligned}{} & {} \text { }\left( { {\textstyle \sum \limits _{i=1}^{m_{1}}} {\textstyle \sum \limits _{j=1}^{m_{2}}} }\left\| f(x_{i},y_{j})\right\| ^{p+\delta }\right) ^{\frac{1}{p+\delta }}\\{} & {} \qquad \le {\tilde{C}}_{\delta ,\epsilon }\cdot \left( \sup _{w\in W_{1}} {\textstyle \sum \limits _{i=1}^{m_{1}}} \left\| g(x_{i},w)\right\| ^{p+\epsilon }\right) ^{\frac{1}{p+\epsilon } }\left( \sup _{w\in W_{2}} {\textstyle \sum \limits _{j=1}^{m_{2}}} \left\| h(y_{j},w)\right\| ^{p+\epsilon }\right) ^{\frac{1}{p+\epsilon } }\text {,} \end{aligned}$$

for all $x_{i}\in X$, $y_{j}\in Y$ and $m_{1},m_{2}\in {\mathbb {N}}$?

The answer to Problem 2.1 presented in Pellegrino et al. (2017) was obtained for a wide class of functions (note that continuity is not needed). We just need few mild assumptions. Let $\,Z_{1},\,V$ and $W_{1},\,W_{2}$ be non-void sets and $Z_{2}$ be a vector space. For $t=1,2,$ let

$$\begin{aligned} R_{t}:Z_{t}\times W_{t}\longrightarrow [0,\infty )\text { and }S:Z_{1}\times Z_{2}\times V\longrightarrow [0,\infty ) \end{aligned}$$

be map**s satisfying

$$\begin{aligned} \left\{ \begin{array}{l} R_{2}\left( \beta z,w\right) =\beta R_{2}\left( z,w\right) ,\\ S\left( z_{1},\beta z_{2},v\right) =\beta S\left( z_{1},z_{2},v\right) \end{array} \right. \end{aligned}$$

for all real scalars $\beta \ge 0.$

Theorem 2.2

[Regularity Principle Pellegrino et al. (2017)] Let $1\le p_{1}\le p_{2}:=p_{1}+\epsilon <2p_{1}$ and assume

$$\begin{aligned}{} & {} \left( \sup _{v\in V}\sum _{i=1}^{m_{1}}\sum _{j=1}^{m_{2}}S(z_{1,i},z_{2,j},v)^{p_{1}}\right) ^{\frac{1}{p_{1}}}\hspace{-0.2cm}\\{} & {} \quad \le C\left( \sup _{w\in W_{1}}\sum _{i=1}^{m_{1}}R_{1}\left( z_{1,i},w\right) ^{p_{1} }\right) ^{\frac{1}{p_{1}}}\left( \sup _{w\in W_{2}}\sum _{j=1}^{m_{2}} R_{2}\left( z_{2,j},w\right) ^{p_{1}}\right) ^{\frac{1}{p_{1}}}, \end{aligned}$$

for all $z_{1,i}\in Z_{1},z_{2,j}\in Z_{2},$ all $i=1,...,m_{1}$ and $j=1,...,m_{2}$ and $m_{1},m_{2}\in {\mathbb {N}}$. Then

$$\begin{aligned}{} & {} \left( \sup _{v\in V}\sum _{i=1}^{m_{1}}\sum _{j=1}^{m_{2}}S(z_{1,i},z_{2,j},v)^{\alpha }\right) ^{\frac{1}{\alpha }}\hspace{-0.3cm}\\{} & {} \quad \quad \le C\left( \sup _{w\in W_{1}}\sum _{i=1}^{m_{1}}R_{1}\left( z_{1,i},w\right) ^{p_{2} }\right) ^{\frac{1}{p_{2}}}\left( \sup _{w\in W_{2}}\sum _{j=1}^{m_{2}} R_{2}\left( z_{2,j},w\right) ^{p_{2}}\right) ^{\frac{1}{p_{2}}}, \end{aligned}$$

for

$$\begin{aligned} \alpha =\frac{p_{1}p_{2}}{2p_{1}-p_{2}}, \end{aligned}$$

all $z_{1,i}\in Z_{1},z_{2,j}\in Z_{2},$ all $i=1,...,m_{1}$ and $j=1,...,m_{2}$ and $m_{1},m_{2}\in {\mathbb {N}}$.

Remark 2.3

Note that the above theorem shows that, in Problem 2.1, for $\epsilon <p_{1}$, we can choose

$$\begin{aligned} \delta =\frac{2\epsilon p_{1}}{p_{1}-\epsilon }. \end{aligned}$$

(2.1)

In this section we present some preliminary notions that will be used along the paper, together with the regularity techniques originated from Pellegrino et al. (2017).

Let $E_{1},\ldots ,E_{m},F$ be Banach spaces and ${\mathcal {L}}\left( E_{1},\ldots ,E_{m};F\right)$ denote the space of all continuous m-linear operators $A:E_{1}\times \cdots \times E_{m}\rightarrow F$. If $\left( r_{1},\ldots ,r_{m}\right) ,\left( p_{1},\ldots ,p_{m}\right) \in \left[ 1,\infty \right] ^{m}$, an m-linear operator $A\in {\mathcal {L}}\left( E_{1},\ldots ,E_{m};F\right)$ is multiple $\left( r_{1},\ldots ,r_{m};p_{1},\ldots ,p_{m}\right)$-summing if there is a constant $C_{m}$ such that

$$\begin{aligned}{} & {} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left\| A\left( x_{j_{1}}^{\left( 1\right) },\ldots ,x_{j_{m}}^{\left( m\right) }\right) \right\| ^{r_{m}}\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{1}}{r_{2}}}\right) ^{\frac{1}{r_{1}}}\\{} & {} \quad \le C_{m}\prod _{k=1}^{m} \sup _{\varphi _{k}\in B_{E_{k}^{*}}}\left( \sum _{j=1}^{n}\left| \varphi _{k}\left( x_{j}^{\left( k\right) }\right) \right| ^{p_{k} }\right) ^{\frac{1}{p_{k}}} \end{aligned}$$

for all positive integers n (above, $E_{k}^{*}$ represents the topological dual of $E_{k}$ and $B_{E_{k}^{*}}$ represents its closed unit ball).

The class of all multiple $\left( r_{1},\ldots ,r_{m};p_{1},\ldots ,p_{m}\right)$-summing operators $A:E_{1}\times \cdots \times E_{m}\rightarrow F$ is denoted by $\Pi _{\left( r_{1},\ldots ,r_{m};p_{1},\ldots ,p_{m}\right) }^{m}\left( E_{1},\ldots ,E_{m};F\right)$; when $m=1$ we write $\Pi _{\left( r_{1};p_{1}\right) }$ instead of $\Pi _{\left( r_{1};p_{1}\right) }^{m}$. When $r_{1}=\cdots =r_{m}=r$ and $p_{1} =\cdots =p_{m}=p$, we simply write $\left( r;p_{1},\ldots ,p_{m}\right)$ and $\left( r_{1},\ldots ,r_{m};p\right)$, respectively. The infimum of the constants $C_{m}$ defines a complete norm for the space $\Pi _{\left( r_{1},\ldots ,r_{m};p_{1},\ldots ,p_{m}\right) }^{m}\left( E_{1},\ldots ,E_{m};F\right)$, denoted hereafter by $\pi _{(r_{1},\ldots ,r_{m};p_{1},\dots ,p_{m})}(\cdot )$. In the case of linear operators, the notion of multiple summing operators reduces to the well-known concept of absolutely summing operators, which plays a fundamental role in Banach Space Theory [see the excellent monograph Diestel et al. (1995)].

The following well-known result, that will be used several times in this paper, associates HL inequalities and multiple summing operators (see, for instance, Pellegrino et al. (2017, Theorem 3.2) and the references therein). It essentially asserts that each HL inequality corresponds to a coincidence result for multiple summing operators which holds regardless of the Banach spaces considered at the domain:

The next result also plays a fundamental role in this paper. It was ingeniously obtained by Albuquerque and Rezende (2018, Theorem 3) as a consequence of the Regularity Principle (see also Albuquerque and Rezende (2018, Theorem 4) for an anisotropic variant of Theorem 2.2):

3 Proof of Theorem 1.5: Existence

Given a positive integer n, let $A_{0}:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ be an m-linear form and $k_{0}$ be as in the statement of the theorem.

If $k_{0}=m$, then $1<p_{m}\le 2$. Denoting

$$\begin{aligned} \left( t_{1},\ldots ,t_{m-1},t_{m}\right) =\left( \infty ,\ldots ,\infty ,p_{m}\right) \text {,} \end{aligned}$$

we obtain

$$\begin{aligned} \dfrac{1}{t_{1}}+\cdots +\dfrac{1}{t_{m}}=\frac{1}{p_{m}} \end{aligned}$$

and hence

$$\begin{aligned} \frac{1}{2}\le \dfrac{1}{t_{1}}+\cdots +\dfrac{1}{t_{m}}<1\text {.} \end{aligned}$$

Thus, using Theorem 1.2, there is $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{r_{m}}\right) ^{r_{m}}\le C_{m}\left\| A\right\| \end{aligned}$$

(3.1)

for all m-linear forms $A:\ell _{t_{1}}^{n}\times \cdots \times \ell _{t_{m}}^{n}\rightarrow {\mathbb {K}}$, where $r_{m}=p_{m}^{*}$. Using the equivalence (2.2)$\Leftrightarrow$( 2.3), this means that every continuous m-linear form $A:E_{1}\times \cdots \times E_{m}\rightarrow {\mathbb {K}}$ is multiple $\left( r_{m};t_{1}^{*},\ldots ,t_{m}^{*}\right)$-summing, regardless of the Banach spaces $E_{1},\ldots ,E_{m}$, and the associated constant is $C_{m}$. In particular, $A_{0}:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ is multiple $\left( r_{m};t_{1}^{*},\ldots ,t_{m}^{*}\right)$-summing, and the associated constant is $C_{m}$. A straightforward calculation shows that

$$\begin{aligned} \frac{1}{r_{m}}-\left( \dfrac{1}{t_{1}^{*}}+\cdots +\dfrac{1}{t_{m}^{*} }\right) +\left( \dfrac{1}{p_{1}^{*}}+\cdots +\dfrac{1}{p_{m}^{*} }\right) >0\text {.} \end{aligned}$$

For each $k=1,\ldots ,m$, let

$$\begin{aligned} r_{k}=\left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {.} \end{aligned}$$

By the Anisotropic Regularity Principle we conclude that $A_{0}$ is multiple $\left( r_{1},\ldots ,r_{m};p_{1}^{*},\ldots ,p_{m}^{*}\right)$-summing with the same constant $C_{m}$, and this is equivalent to

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A_{0}\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{r_{m}}\right) ^{\frac{r_{m}-1}{r_{m}}}\cdots \right) ^{\frac{r_{1}}{r_{2}}}\right) ^{\frac{1}{r_{1}}}\le C_{m}\left\| A_{0}\right\| \text {.} \end{aligned}$$

This proves (1.5), for the case $k_{0}=m$.

So, let us consider $k_{0}<m$. It is obvious that $p_{k_{0}}<\infty$ and

$$\begin{aligned} \frac{1}{p_{k_{0}+1}}+\cdots +\frac{1}{p_{m}}<\frac{1}{2}\text {.} \end{aligned}$$

Let $\delta \ge 1$ be such that

$$\begin{aligned} \dfrac{1}{\delta p_{k_{0}}}+\dfrac{1}{p_{k_{0}+1}}+\cdots +\dfrac{1}{p_{m} }=\frac{1}{2}\text {.} \end{aligned}$$

Denoting

$$\begin{aligned} \left\{ \begin{array}{l} \left( q_{1},\ldots ,q_{k_{0}-1}\right) =\left( \infty ,\ldots ,\infty \right) \text {,}\\ \\ \left( q_{k_{0}},q_{k_{0}+1},\ldots ,q_{m}\right) =\left( \delta p_{k_{0} },p_{k_{0}+1},\ldots ,p_{m}\right) \text {,} \end{array} \right. \end{aligned}$$

we have

$$\begin{aligned} \dfrac{1}{q_{1}}+\cdots +\dfrac{1}{q_{m}}=\frac{1}{2} \end{aligned}$$

and, hence, by (1.2), there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{2}\right) ^{\frac{1}{2}}\le C_{m}\left\| A\right\| \end{aligned}$$

(3.2)

for all m-linear forms $A:\ell _{q_{1}}^{n}\times \cdots \times \ell _{q_{m}}^{n}\rightarrow {\mathbb {K}}$. Again, using the canonical association between the HL inequalities and multiple summing operators, we have that $A_{0}:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ is multiple $\left( 2;q_{1}^{*},\ldots ,q_{m}^{*}\right)$-summing, and the associated constant is $C_{m}$. Since

$$\begin{aligned} \dfrac{1}{2}-\left( \dfrac{1}{q_{1}^{*}}+\cdots +\dfrac{1}{q_{m}^{*} }\right) +\left( \dfrac{1}{p_{1}^{*}}+\cdots +\dfrac{1}{p_{m}^{*} }\right) >0\text {,} \end{aligned}$$

considering

$$\begin{aligned} s_{k}=\left\{ \begin{array}{ll} \left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }k\le k_{0}\text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text {,} \end{array} \right. \end{aligned}$$

by (2.4) we conclude that $A_{0}$ is multiple $\left( s_{1},\ldots ,s_{m};p_{1}^{*},\ldots ,p_{m}^{*}\right)$-summing with the same constant $C_{m}$, and again, this is equivalent to

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A_{0}\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m}-1}{s_{m}}}\cdots \right) ^{\frac{s_{1}}{s_{2}}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A_{0}\right\| \text {.} \end{aligned}$$

This proves (1.5) for the case $k_{0}<m$.

Remark 3.1

By Araújo and Câmara (2018, Theorem 3.3) we notice that $C_{m}$ in (3.1) and (3.2) satisfies

$$\begin{aligned} C_{m}\le 2^{\frac{m-k_{0}}{2}}\text {.} \end{aligned}$$

Thus, by our procedure, Theorem 1.5 stands with constant $C_{m}\le 2^{\frac{m-k_{0}}{2}}$ and, choosing $k_{0}=m$, we recover the optimal constants from Theorems 1.1 and 1.4.

4 Proof of Theorem 1.5: Optimality

In this section we shall prove (i) and (ii) of Theorem 1.5. The optimality of the exponents in (i) is a straightforward consequence of Aron et al. (2017, Lemma 3.1). The proof of (ii) depends on a probabilistic result that nowadays is usually called Kahane–Salem–Zygmund inequality. It has its origins in the 1970’s with independent papers of different authors ( Bennett et al. (1975); Bennett (1977); Kahane (1985); Mantero and Tonge (1980); Varopoulos (1974); see also Albuquerque and Rezende (2021); Defant and Mastylo (2023); Mastyło and Szwedek (2017); Pellegrino and Raposo (2022) for recent advances on the subject). We shall need the following variant of the Kahane–Salem–Zygmund that can be found at Albuquerque et al (2014, Lemma 6.1):

Lemma 4.1

[Kahane–Salem–Zygmund inequality] Let $p_{1},\ldots ,p_{m}\in \left[ 2,\infty \right]$. Then there exists a constant $K_{m}$ such that, for all positive integers n, there is an m-linear form $A:\ell _{p_{1}} ^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ of the form

$$\begin{aligned} A\left( z^{\left( 1\right) },\ldots ,z^{\left( m\right) }\right) =\sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\pm z_{j_{1}}^{\left( 1\right) }\cdots z_{j_{m}}^{\left( m\right) }\text {,} \end{aligned}$$

with

$$\begin{aligned} \Vert A\Vert \le K_{m}n^{\frac{1}{2}+\left( \frac{1}{2}-\frac{1}{p_{1} }\right) +\cdots +\left( \frac{1}{2}-\frac{1}{p_{m}}\right) }\text {.} \end{aligned}$$

If $p_{1},\dots ,p_{m}\in [2,\infty ]$ and $t_{1},\dots ,t_{m}\in \left[ 1,\infty \right)$, a straightforward consequence of the above inequality is that if there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{i_{1}=1}^{n}\left( \cdots \left( \sum _{i_{m}=1}^{n}\left| A\left( e_{i_{1}},\dots ,e_{i_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, then

$$\begin{aligned} \frac{1}{t_{1}}+\cdots +\frac{1}{t_{m}}\le \frac{m+1}{2}-\left( \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\right) \text {.} \end{aligned}$$

(4.1)

We shall also use the following simple lemma, which seems to be folklore:

Lemma 4.2

Let $k\in \left\{ 1,\ldots ,m-1\right\}$ and $\left( p_{1},\ldots ,p_{m}\right) \in \left[ 1,\infty \right] ^{m}$. If there exists a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A_{1}\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A_{1}\right\| \end{aligned}$$

for all m-linear forms $A_{1}:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, then

$$\begin{aligned} \left( \sum _{j_{k+1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A_{2}\left( e_{j_{k+1}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\cdots \right) ^{\frac{t_{k+1}}{t_{k+2}}}\right) ^{\frac{1}{t_{k+1}}}\le C_{m}\left\| A_{2}\right\| \end{aligned}$$

for all $\left( m-k\right)$-linear forms $A_{2}:\ell _{p_{k+1}} ^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$.

By (i) we know that the exponents $s_{1},\ldots ,s_{k_{0}}$ in (1.6) are optimal, so it is obvious that they cannot be perturbed to smaller exponents. To conclude the proof of (ii) it remains to consider the exponents $s_{k_{0}+1},\ldots ,s_{m}$.

Let us suppose that for a certain $i=k_{0}+1,\ldots ,m$ there is $\varepsilon _{i}>0$ and there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, with

$$\begin{aligned} t_{k}=\left\{ \begin{array}{ll} \left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }k\le k_{0}\text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text { and }k\not =i\\ &{} \\ 2-\varepsilon _{i}\text {,} &{} \text {if }k=i\text {.} \end{array} \right. \end{aligned}$$

We invoke Lemma 4.2 to conclude that

$$\begin{aligned} \left( \sum _{j_{k_{0}}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1} ^{n}\left| A_{2}\left( e_{j_{k_{0}}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{k_{0}} }{t_{k_{0}+1}}}\right) ^{\frac{1}{t_{k_{0}}}}\le C_{m}\left\| A_{2}\right\| \end{aligned}$$

for all $\left( m-k_{0}+1\right)$-linear forms $A_{2}:\ell _{p_{k_{0}}} ^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$. Note that

$$\begin{aligned} \frac{1}{t_{k_{0}}}+\cdots +\frac{1}{t_{m}}&=1-\left( \dfrac{1}{p_{k_{0}} }+\cdots +\dfrac{1}{p_{m}}\right) +\frac{m-k_{0}-1}{2}+\frac{1}{2-\varepsilon _{i}}\\&>\frac{\left( m-k_{0}+1\right) +1}{2}-\left( \frac{1}{p_{k_{0}}} +\cdots +\frac{1}{p_{m}}\right) \text {.} \end{aligned}$$

and it contradicts (4.1). The proof of (ii) now follows.

In order to illustrate (numerically) how Theorem 1.5 improves the previous ones, let us consider $m=9$ with in Table 1.

$$\begin{aligned} p_{1}=\cdots =p_{9}=10\text {.} \end{aligned}$$

Table 1 compares the exponents provided our first main result (Theorem 1.5) and those from Theorems 1.2 (Dimant, Sevilla-Peris 2016, Proposition 4.1), and Theorem 1.3 (Albuquerque and Rezende 2018, Corollary 2):

Table 1 HL exponents for $m=9$ and $p_{1}=\cdots =p_{9}=10$

5 The Critical Case: Globally Sharp Exponents

Until very recently, the HL inequalities were just investigated for $1/p_{1}+\cdots +1/p_{m}<1$. The reason was very simple: if we consider $1/p_{1}+\cdots +1/p_{m}\ge 1$, there does not exist a finite exponent s for which there is a constant $C_{m}$ satisfying

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\cdots \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s}\right) ^{\frac{1}{s}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$. So, at first glance, it seemed that no theory was supposed to be expected in this framework. However, this is a blurred perspective; when we consider just one exponent s at all sums, we lose information. So, in Núñez-Alarcón et al. (2023); Paulino (2020), the authors initiated the investigation of the case $1/p_{1}+\cdots +1/p_{m}\ge 1$ under an anisotropic viewpoint. In this section we follow this vein and obtain globally sharp exponents for the case $1/p_{1}+\cdots +1/p_{m}=1$. Hereafter, for the sake of simplicity, when $s=\infty$, the sum $\left( \sum _{j=1}^{\infty }\left| a_{j}\right| ^{s}\right) ^{1/s}$ denotes $\sup \left| a_{j}\right|$.

We recall that in this case some exponents in the anisotropic Hardy-Littlewood inequality are forced to be infinity. The first result dealing with this notion is the following:

Theorem 5.1

(See Paulino 2020, Theorem 1) There is a constant $C_{m}$ such that

$$\begin{aligned} \sup _{j_{1}}\left( \sum _{j_{2}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1} ^{n}\left| A\left( e_{j_{1}},\dots ,e_{j_{m}}\right) \right| ^{s_{m} }\right) ^{\frac{1}{s_{m}}s_{m-1}}\cdots \right) ^{\frac{1}{s_{3}}s_{2} }\right) ^{\frac{1}{s_{2}}}\le C_{m}\left\| A\right\| \nonumber \\ \end{aligned}$$

(5.1)

for all m-linear forms $A:\ell _{m}^{n}\times \cdots \times \ell _{m} ^{n}\rightarrow {\mathbb {K}}$, and all positive integers n, with

$$\begin{aligned} s_{k}=\frac{2m(m-1)}{mk-2k+2} \end{aligned}$$

for all $k=2,\ldots ,m$. Moreover, $s_{1}=\infty$ and $s_{2}=m$ are sharp and, for $m>2$ the optimal exponents $s_{k}$ satisfying (5.1) fulfill

$$\begin{aligned} s_{k}\ge \frac{m}{k-1}\text {.} \end{aligned}$$

As a consequence of Theorem 1.5, we have the following generalization of the previous theorem:

Proposition 5.2

Let $p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right]$ be such that $1/2\le 1/p_{2}+\cdots +1/p_{m}<1$ and

$$\begin{aligned} \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}=1\text {.} \end{aligned}$$

(5.2)

There is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{1}}{s_{2}}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A\right\| \end{aligned}$$

(5.3)

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, where

$$\begin{aligned} s_{k}=\left\{ \begin{array}{ll} \left[ 1-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }1\le k\le k_{0}:=\max \left\{ t:\dfrac{1}{p_{t} }+\cdots +\dfrac{1}{p_{m}}\ge \dfrac{1}{2}\right\} \text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text {.} \end{array} \right. \end{aligned}$$

Moreover:

(i):

The exponents $s_{1},\ldots ,s_{k_{0}}$ are optimal.

(ii):

If $p_{k_{0}}\ge 2$, all the exponents are globally sharp.

Proof

The proof of the existence is a simple consequence of Theorem 1.5; in fact, by Theorem 1.5, for any fixed vector $e_{j_{1}}$, there is a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{2}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{2}}{s_{3}}}\right) ^{\frac{1}{s_{2}}}\le C_{m}\left\| A\right\| \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ and this easily implies (5.3).

Now we shall prove (i) and (ii).

In order to prove (i), note that if $s_{1}=\infty$ could be improved, there would exist $r\in \left( 0,\infty \right)$ and $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \sum _{j_{2}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{2}}{s_{3}}}\right) ^{\frac{r}{s_{2}}}\right) ^{\frac{1}{r}}\le C_{m}\left\| A\right\| \end{aligned}$$

for all m-linear forms $A:\ell _{p_{_{1}}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$. Considering $\rho =\max \left\{ s_{2},\ldots ,s_{m},r\right\}$, by the monotonicity of the $\ell _{q}$ norms we would conclude that

$$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{n}\left| A(e_{j_{1}},\ldots ,e_{j_{m} })\right| ^{\rho }\right) ^{\frac{1}{\rho }}\le C_{m}\left\| A\right\| \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, but this is impossible due to (5.2).

On the other hand, note that if

$$\begin{aligned} \sup _{j_{1}}\left( \sum _{j_{2}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1} ^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{2}}{s_{3}} }\right) ^{\frac{1}{s_{2}}}\le C_{m}\left\| A\right\| \text {,} \end{aligned}$$

for all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, then by Lemma 4.2 we have

$$\begin{aligned} \left( \sum _{j_{2}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A_{2}\left( e_{j_{2}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{s_{m-1}}{s_{m}}}\cdots \right) ^{\frac{s_{2}}{s_{3}}}\right) ^{\frac{1}{s_{2}}}\le C_{m}\left\| A_{2}\right\| \text {,} \end{aligned}$$

for all $\left( m-1\right)$-linear forms $A_{2}:\ell _{p_{2}}^{n} \times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$. Hence, the proofs of (i) and (ii) are completed as consequence of Theorem 1.5 for $(m-1)$-linear forms. $\square$

As a consequence, we have an improvement of Theorem 5.1:

Table 2 Critical HL exponents for $m=10$ and $p_{1}=\cdots =p_{10}=10$

Corollary 5.3

There exists a constant $C_{m}$ such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\dots ,e_{j_{m}}\right) \right| ^{s_{m}}\right) ^{\frac{1}{s_{m}}s_{m-1}}\cdots \right) ^{\frac{1}{s_{2}}s_{1}}\right) ^{\frac{1}{s_{1}}}\le C_{m}\left\| A\right\| \end{aligned}$$

(5.4)

for all m-linear forms $A:\ell _{m}^{n}\times \cdots \times \ell _{m} ^{n}\rightarrow {\mathbb {K}}$, with

$$\begin{aligned} s_{k}=\left\{ \begin{array}{ll} \dfrac{m}{k-1}\text {,} &{} \text {if }1<k\le k_{0}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text {,} \end{array} \right. \end{aligned}$$

where $k_{0}:=\left\lfloor \frac{m+2}{2}\right\rfloor$ (the largest integer less than or equal to $\left( m+2\right) /2)$. Moreover, $s_{1},\ldots ,s_{k_{0}}$ are sharp, and $\left( s_{1},\ldots ,s_{m}\right)$ is globally sharp.

Proof

We shall use the previous proposition with $p_{1}=\cdots =p_{m}=m$. Observe that, if $m=2N+1$ or $m=2N$, then $\left\lfloor \frac{m+2}{2}\right\rfloor =N+1$. Let $k_{0}$ be as in the previous theorem, i.e.,

$$\begin{aligned} k_{0}:=\max \left\{ t:\dfrac{1}{p_{t}}+\cdots +\dfrac{1}{p_{m}}\ge \dfrac{1}{2}\right\} \text {.} \end{aligned}$$

Since

$$\begin{aligned} \frac{1}{p_{t}}+\cdots +\frac{1}{p_{m}}\ge \frac{1}{2}\Leftrightarrow t\le \frac{m+2}{2}\text {,} \end{aligned}$$

we conclude that $k_{0}=\left\lfloor \frac{m+2}{2}\right\rfloor$. Hence, by Proposition 5.2 we conclude that (5.4) holds as well as the optimality of the exponents. $\square$

In order to illustrate how Proposition 5.2 and its corollary improve (Paulino 2020, Theorem 1), let us consider $m=10$ in Table 2.

6 The Regularity Principle is Sharp

In this section we investigate a natural question: is the (Anisotropic) Regularity Principle sharp? Of course, for special choices of Banach spaces we can find better inclusions, as it also happens with Inclusion Theorems [see Diestel et al. (1995)]. The right question here is whether is it possible to obtain better inclusions kee** the full generality of the Regularity Principle. As we shall see, a simple consequence of what we have already proved in this paper is that the answer is no. We shall prove, for instance, that if $r\ge 2$, the parameters $s_{1},\ldots ,s_{m-1}$ in (2.4) are optimal.

Let $r\ge 2$ and suppose that for some $i\in \left\{ 1,\ldots ,m-1\right\}$ there is $\varepsilon >0$ such that

$$\begin{aligned} \Pi _{\left( r;p_{1},\ldots ,p_{m}\right) }^{m}\left( E_{1},\ldots ,E_{m};F\right) \subset \Pi _{\left( t_{1},\ldots ,t_{m};q_{1},\ldots ,q_{m}\right) }^{m}\left( E_{1},\ldots ,E_{m};F\right) \text {,} \end{aligned}$$

(6.1)

for any Banach spaces $E_{1},\ldots ,E_{m}$ and F, with $t_{i}=s_{i} -\varepsilon$. If we take $F={\mathbb {K}}$ and $E_{i}=\ell _{p_{i}}^{n}$, with $p_{m}=\frac{r}{r-1}\le 2$ and $\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<1$, by mimicking the proof of Theorem 1.5 with the inclusion (6.1) instead of (2.4), we conclude that there is a constant $C_{m}$ such that, for all positive integers n and all m-linear forms $A:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$, we have

$$\begin{aligned} \left( \sum _{j_{1}=1}^{n}\left( \cdots \left( \sum _{j_{m}=1}^{n}\left| A\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C\left\| A\right\| \text {,} \end{aligned}$$

with $t_{i}=s_{i}-\varepsilon$. But this is impossible due to the sharpness of $s_{i}$ in Theorem 1.5. Similar arguments show that the estimate of $\delta$ in (2.1) cannot be improved, in general. It is also well-known that the hypothesis $\epsilon <p_{1}$ in Theorem 2.2 cannot be relaxed to $\epsilon \le p_{1}.$

7 Related Open Problems

The main unsolved problem in this topic seems to be the determination of all admissible exponents for the Hardy–Littlewood inequalities for m-linear forms; a second (and eventually even more delicate) problem is to obtain the sharp constants. In this final section we mention a different line of investigation: what happens for non-admissible exponents? For instance, for non-admissible exponents, how does the right-hand-side of the Hardy–Littlewood inequalities is affected by the dimension n of the $\ell _{p}^{n}$ spaces involved? This problem was investigated in Araújo and Pellegrino (2015); Galicer et al. (2020); Bayart (2022). Another line of investigation in the setting of non-admissible exponents concerns the size/geometry of the set of m-linear forms $A:\ell _{p_{1} }^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$ which fail the Hardy–Littlewood inequality for a given m-uple of non-admissible exponents. This kind of problem was briefly investigated in Araújo and Pellegrino (2017, Theorem 2.1), using the notion of spaceability, but the current state of the art is far from a complete answer in this regard. We believe that this theme can be investigated from a more subtle point of view of the theory of lineability, with the notion of $\left( \alpha ,\beta \right)$-lineability introduced and explored by Fávaro et al. (see Diniz and Raposo (2021); Fávaro et al. (2019) and the references therein).