1 Introduction

The Hardy–Littlewood inequality for multilinear forms in \(\ell _{p}\) spaces has its origins in the seminal paper (Hardy and Littlewood 1934) and it was initially proposed for the bilinear case. It stands as a remarkable and highly intricate extension of Littlewood’s 4/3 inequality from \(\ell _{\infty }^{n}\) to \(\ell _{p}^{n}\) spaces.

To provide context, let us define \(X_{p}:=\ell _{p}\) for \(1\le p<\infty \), and set \(X_{\infty }:=c_{0}\). By convention, when \(q=\infty \), the sum \(\left( \sum _{j}\left\| x_{j}\right\| ^{q}\right) ^{1/q}\) represents the supremum of \(\left\| x_{j}\right\| \). Additionally, we introduce the notation \(f(\infty ):=\lim _{s\rightarrow \infty }f(s)\) for any function f. We also establish that \(1/0=\infty \) and \(1/\infty =0\) and moreover, for \(s\ge 1\), we use \(s^{*}\) to represent the conjugate index of s, denoted as \(\frac{1}{s}+\frac{1}{s^{*}}=1\). As usual, in this paper we denote the sequence of canonical vectors of \(\ell _{p}\) by \(\left( e_{k}\right) _{k=1}^{\infty }\).

For a pair of real numbers \(p,q\in (1,\infty ]\), verifying \(\frac{1}{p}+\frac{1}{q}<1\), let us denote

$$\begin{aligned} \rho (p,q)=\rho :=\frac{pq}{pq-p-q}, \end{aligned}$$
(1)

and

$$\begin{aligned} \mu (p,q)=\mu :=\frac{4pq}{3pq-2p-2q}. \end{aligned}$$
(2)

As convention, when \(p=q=\infty \), we take \(\mu =4/3\), that is,

$$\begin{aligned} \mu (\infty ,\infty ):=4/3. \end{aligned}$$

We are ready to state the two fundamental results proven in Hardy and Littlewood (1934), which provides the foundation and serves as the driving force behind this present manuscript.

Theorem 1.1

Let \(p,q\in [2,\infty ]\), with \(\frac{1}{p}+\frac{1}{q} \le \frac{1}{2}\). There is a constant \(C_{p,q}\ge 1\) such that

$$\begin{aligned} \left( \sum _{j_{1},j_{2}=1}^{\infty }\left| A(e_{j_{1}},e_{j_{2} })\right| ^{\mu }\right) ^{\frac{1}{\mu }}\le C_{p,q}\left\| A\right\| , \end{aligned}$$
(3)

for all continuous bilinear forms \(A:X_{p}\times X_{q}\rightarrow \mathbb {K}\).

Theorem 1.2

Let \(p,q\in [1,\infty ]\), with \(\frac{1}{2}<\frac{1}{p} +\frac{1}{q}<1\). There is a constant \(C_{p,q}\ge 1\) such that

$$\begin{aligned} \left( \sum _{j_{1},j_{2}=1}^{\infty }\left| A(e_{j_{1}},e_{j_{2} })\right| ^{\rho }\right) ^{\frac{1}{\rho }}\le C_{p,q}\left\| A\right\| , \end{aligned}$$
(4)

for all continuous bilinear forms \(A:X_{p}\times X_{q}\rightarrow \mathbb {K}\).

Hereafter \(\mathbb {K}\) denotes either the real field \(\mathbb {R}\) or the complex plane \(\mathbb {C}\). It is worth commenting that, once fixed the pair (p, q) within the range above prescribed, viz. \(\frac{1}{p} + \frac{1}{q} < 1\), the exponents \(\rho (p,q)\) and \(\mu (p,q)\) defined in (1) and (2) respectively are optimal for the inequalities displayed in Theorem 1.1 and Theorem 1.2.

Since the groundbreaking contributions of Hardy and Littlewood (1934), inequalities of this nature have continuously captivated attention throughout the years. The enduring fascination with generalizing the Hardy–Littlewood inequalities stems from both the abundant practical applications these tools offer and the inherent elegance of the problems that arise from such investigations.

Among all the possible directions of generalization, extending the theory to multilinear maps, also referred as m-linear operators, has been particularly successful. To explain this tread of research, let us convention hereafter in this paper that \(m\ge 1\) will always denote a positive integer and by \(\textbf{p}\) we represent a vector with m entries selected within the closed interval \(\left[ 1,\infty \right] \). That is:

$$\begin{aligned} \textbf{p}:=\left( p_{1},\dots ,p_{m}\right) \in \left[ 1,\infty \right] ^{m}. \end{aligned}$$

Given such a \(\textbf{p}\), we denote:

$$\begin{aligned} \left| \frac{1}{\textbf{p}}\right| :=\frac{1}{p_{1}}+\dots +\frac{1}{p_{m}}. \end{aligned}$$

Praciano and Pereira (1981, Theorems A and B), launched the investigation by showing that if \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\) with

$$\begin{aligned} \left| {\frac{1}{\textbf{p}}}\right| \le \frac{1}{2}, \end{aligned}$$

then, there exists a constant \(C>0\) such that, for every continuous m-linear form \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow \mathbb {K}\),

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }|A(e_{i_{1}},\dots ,e_{i_{m} })|^{\frac{2m}{m+1-2\left| {\frac{1}{\textbf{p}}}\right| }}\right) ^{\frac{m+1-2\left| {\frac{1}{\textbf{p}}}\right| }{2m}}\le C\Vert A\Vert . \end{aligned}$$
(5)

Several years past until Defant and Sevilla-Peris (2009, Theorem 1) proved that if \(1\le s\le q\le 2,\) then there exists a constant \(C>0\) such that, for every continuous m-linear map** \(A:X_{\infty }\times \dots \times X_{\infty }\rightarrow X_{s}\), then

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m} })\Vert _{\ell _{q}}^{\frac{2m}{m+2\left( \frac{1}{s}-\frac{1}{q}\right) } }\right) ^{\frac{m+2\left( \frac{1}{s}-\frac{1}{q}\right) }{2m}}\le C\Vert A\Vert . \end{aligned}$$

Subsequently, Albuquerque et al. (2014, Corollary 1.3) show that if \(1\le s\le q\le 2\) and \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\) such that

$$\begin{aligned} \frac{1}{s}-\frac{1}{q}-\left| {\frac{1}{\textbf{p}}}\right| \ge 0, \end{aligned}$$
(6)

then for a constant \(C>0\) and every continuous m-linear map** \(A:X_{p_{1} }\times \dots \times X_{p_{m}}\rightarrow {X_{s}}\), there holds

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m} })\Vert _{\ell _{q}}^{\frac{2m}{m+2\left( \frac{1}{s}-\frac{1}{q}-\left| {\frac{1}{\textbf{p}}}\right| \right) }}\right) ^{\frac{m+2\left( \frac{1}{s}-\frac{1}{q}-\left| {\frac{1}{\textbf{p}}}\right| \right) }{2m}}\le C\Vert A\Vert \end{aligned}$$

and the exponent is optimal.

Also relevant for this paper, let us briefly comment on the important results proven by Albuquerque et al. (2016, Theorem 1.1) and Dimant and Sevilla-Peris (2016, Proposition 4.4). We additionally cite (Albuquerque and Rezende 2021; Belacel et al. 2023; Núñez-Alarcón et al. 2022; Pellegrino et al. 2017; Raposo and Serrano-Rodríguez 2023) as relevant works in the field.

Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\), \(\rho >0\), assume either \(q\ge 2\) or \(q<2\), and \(\left| \frac{1}{\textbf{p}}\right| <\frac{1}{2}\). If one defines

$$\begin{aligned} \frac{1}{\lambda }:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}-\left| {\frac{1}{\textbf{p}}}\right| >0, \end{aligned}$$

then for every continuous m-linear map** \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), we have

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m} })\Vert _{\ell _{q}}^{\rho }\right) ^{\frac{1}{\rho }}\le C\Vert A\Vert \end{aligned}$$
(7)

for a universal constant \(C>0\) (i.e. independent of the m-linear map** A) if, and only if the following structural condition

$$\begin{aligned} \frac{m}{\rho }\le \frac{1}{\lambda }+\frac{m-1}{\max \{\lambda ,s,2\}} \end{aligned}$$

is verified. While tautological, it is worth noting that since \(\lambda \ge s\), one always has the equality \(\max \{\lambda ,s,2\}=\max \{\lambda ,2\}.\)

Another reason why the Hardy–Littlewood inequality has attract so much attention and interest is because one recovers another classical inequality, viz. the Bohnenblust–Hille inequality, Bohnenblust and Hille (1931). Despite much recent progresses and striking applications, see for instance (Bayart et al. 2014; Defant et al. 2011; Montanaro 2012), many critical open questions still remain unsolved. In particular, nuances on the optimality of the exponents seem to have been overlooked and it turns out that expressions of the form

$$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{\infty }\left\| A(e_{j_{1}},\ldots e_{j_{m} })\right\| _{\ell _{q}}^{s}\right) ^{\frac{1}{s}}\le C\left\| A\right\| \end{aligned}$$

may be considered sub-optimal, when one views it embedded into a family of (more general) anisotropic inequalities. That is, the above inequality can be understood as

$$\begin{aligned} \left( \sum _{j_{1}=1}^{\infty } \cdots \left( \sum _{j_{m-1}=1}^{\infty }\left( \sum _{j_{m}=1}^{\infty }\left\| A(e_{j_{1}},\ldots e_{j_{m}})\right\| _{\ell _{q}}^{s_{m}}\right) ^{\frac{1}{s_{m}}s_{m-1}}\right) ^{\frac{1}{s_{m-1}}}\ldots \right) ^{\frac{1}{s_{1}}}\le C\left\| A\right\| \end{aligned}$$
(8)

for the particular case when \(s_{1}=\cdots =s_{m}=s\). It turns out that anisotropic versions of the inequality allows one to investigate the optimality of the exponents much more precisely. Indeed, an extensive investigation of Hardy–Littlewood inequalities in light of multiple sums like (8) was initiated in Albuquerque et al. (2016), leading to the following deep results:

Theorem 1.3

Albuquerque et al. (2016, Theorem 2.2) Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\), X be a Banach space, Y be a cotype q space and \(1\le r\le q\), with \(\left| \frac{1}{\textbf{p}}\right| <\frac{1}{r}\). Define

$$\begin{aligned} \frac{1}{\lambda }:=\frac{1}{r}-\left| \frac{1}{\textbf{p}}\right| . \end{aligned}$$

Let us consider the following property:

There exists \(C_{m}>0\) such that, for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\), and all absolutely \(\left( r,1\right) \)-summing operators \(v:X\rightarrow Y\), we have

$$\begin{aligned} \left( \sum _{j_{1}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vA\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \pi _{r,1}\left( v\right) . \nonumber \\ \end{aligned}$$
(9)
  1. (A)

    If \(\lambda <q\) then the property is satisfied as soon as \(t_{1},\ldots ,t_{m}\in \left[ \lambda ,q\right] \) are such that

    $$\begin{aligned} \frac{1}{t_{1}}+\cdots +\frac{1}{t_{m}}\le \frac{1}{\lambda }+\frac{m-1}{q}\text {.} \end{aligned}$$
  2. (B)

    If \(\lambda \ge q\) then the property is satisfied as soon as

    $$\begin{aligned} t_{k}\ge \lambda \end{aligned}$$

    for all \(k\in \left\{ 1,\ldots ,m\right\} .\)

Theorem 1.4

Albuquerque et  al. (2016, Theorem 1.3) Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\) and \(1\le s\le q\le \infty \) be such that

$$\begin{aligned} \left| \frac{1}{\textbf{p}}\right| <\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}. \end{aligned}$$

Let

$$\begin{aligned} \frac{1}{\lambda }:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}-\left| \frac{1}{\textbf{p}}\right| . \end{aligned}$$

If \(t_{1},\ldots ,t_{m}\in \left[ \lambda ,\max \left\{ \lambda ,2\right\} \right] \) are such that

$$\begin{aligned} \frac{1}{t_{1}}+\cdots +\frac{1}{t_{m}}\le \frac{1}{\lambda }+\frac{m-1}{\max \{\lambda ,2\}}, \end{aligned}$$
(10)

then there exists \(C>0\) satisfying, for every continuous m-linear map \(A:X_{p_{1}}\times \cdots \times X_{p_{m}}\rightarrow X_{s}\),

$$\begin{aligned} \left( \sum _{i_{1}=1}^{\infty }\left( \ldots \left( \sum _{i_{m}=1}^{\infty }\left\| A\left( e_{i_{1}},\ldots ,e_{i_{m}}\right) \right\| _{\ell _{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots \right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C\Vert A\Vert . \end{aligned}$$
(11)

Moreover, the exponents are optimal except eventually if \(q<2\) and \(\left| \frac{1}{\textbf{p}}\right| >\frac{1}{2}\).

First of all it shall be clear that, due to the monotonicity of the \(\ell _{p}\) norms, the interesting case in (10) is when the equality holds. On the other hand, the condition (10) shows that, in general, there is no unique solution to the question: what is the optimal exponent at the j-th position of the Hardy–Littlewood 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 the more intricate concept of optimality, introduced in Pellegrino et al. (2017, Definition 7.1), which we reproduce below for the readers convenience:

Definition 1.5

An m-tuple of exponents \(\left( t_{1},\ldots ,t_{m}\right) \) is called “globally sharp" if it satisfies a Hardy–Littlewood type inequality, and for any \(\varepsilon _{j}>0\) (with \(j=1,\ldots ,m\)), there exists no Hardy–Littlewood inequality for the m-tuple of exponents \(\left( t_{1},\ldots ,t_{j-1},t_{j}-\varepsilon _{j},t_{j+1},\ldots ,t_{m}\right) \).

It is interesting to view globally sharp m-tuple of exponents \(\left( t_{1},\ldots ,t_{m}\right) \) as a boundary point within the set of all admissible exponents for a specific type of Hardy–Littlewood inequality.

While the preceding theorems represented significant advancements in the overall comprehension of the problem, they possess limitations when it comes to determining the complete spectrum of exponents. The isotropic form of statement (B) in Theorem 1.3 and the additional parameter conditions in Theorem 1.4 restrict the apparent generality of the anisotropic exponents \((t_{1},t_{2},\ldots ,t_{m})\). Notably, this limitation becomes evident when \(\max \left\{ \lambda ,2\right\} =\lambda \), reducing the interval \(\left[ \lambda ,\max \left\{ \lambda ,2\right\} \right] \) to a single point. Conversely, it is worth observing that Theorems 1.3 and 1.4 impose conditions such as \(t_{1},\ldots ,t_{m}\in \left[ \lambda ,\max \left\{ \lambda ,q\right\} \right] \) and \(t_{1},\ldots ,t_{m}\in \left[ \lambda ,\max \left\{ \lambda ,2\right\} \right] \), respectively. However, there exist arrangements of exponents \((t_{1},t_{2},\ldots ,t_{m})\) for which (9) and (11) remain uniformly bounded, even if some \(t_{i}\) values are strictly less than the corresponding \(\lambda \). These aspects are highlighted in the following recent result, motivating the findings explicated in this present manuscript:

Theorem 1.6

Núñez-Alarcón et al. (2022, Theorem 1.1) Let \(m\ge 2\) be an integer, Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\) and \(1\le s\le \infty \) be such that

$$\begin{aligned} \left| \frac{1}{\textbf{p}}\right| <\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{s,2\}}. \end{aligned}$$

and

$$\begin{aligned} \left( q_{1},\ldots ,q_{m}\right) \in (0,\infty ]^{m}. \end{aligned}$$

The following assertions are equivalent:

(1) There is a constant \(C_{m}\ge 1\) such that

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

for all continuous m-linear maps \(A:X_{p_{1}}\times \cdots \times X_{p_{m} }\rightarrow X_{s}\).

(2) The exponents \(q_{1},\ldots ,q_{m}\) satisfy

$$\begin{aligned} q_{k}\ge \left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{s,2\}}-\left( \frac{1}{p_{k}}+\dots +\frac{1}{p_{m}}\right) \right] ^{-1}. \end{aligned}$$

Since, in general, the inequality,

$$\begin{aligned}{} & {} \left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{s,2\}}-\left( \frac{1}{p_{k} }+\dots +\frac{1}{p_{m}}\right) \right] ^{-1}\\{} & {} \qquad <\left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{s,2\}}-\left| \frac{1}{\textbf{p}}\right| \right] ^{-1}=\lambda \end{aligned}$$

is verified, Theorem 1.6 shows that, in the anisotropic sense, the condition \(t_{1},\ldots ,t_{m}\ge \lambda \) in Theorems 1.3 and 1.4 could eventually be relaxed. Indeed, in this paper we delve into this line of investigation, obtaining a much more precise exponent control of the inequalities, and thus considerably improving both Theorem 1.3 and Theorem 1.4. In addition, we will show that our new exponents are optimal, for most of the meaningful cases. In what follows, we state the main theorems to be proven in this paper:

Theorem 1.7

Let \(p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right] \), X be a Banach space, Y be a cotype q space and \(1\le r\le q\), with

$$\begin{aligned} \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<\frac{1}{r}\text {,} \end{aligned}$$

and let, for \(k_{1}\in \left\{ 1,\ldots ,m\right\} \),

$$\begin{aligned} s_{k_{1}}^{-1}=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

Consider the following property, \(\mathbf {(P_{1})}\): there exists \(C_{m}>0\) such that, for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\), and all absolutely \(\left( r,1\right) \)-summing operators \(v:X\rightarrow Y\), we have

$$\begin{aligned} \left( \sum _{j_{1}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vA\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \pi _{r,1}\left( v\right) .\nonumber \\ \end{aligned}$$
(12)

The following assertions hold true:

  1. (A)

    If \(s_{k_{1}}<q\) then the property \(\mathbf {(P_{1})}\) is satisfied, provided that \(t_{k_{1}},\ldots ,t_{m}\in \left[ s_{k_{1}},q\right] \) are such that

    $$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}\le \frac{1}{s_{k_{1}}} +\frac{m-k_{1}}{q}\text {,} \end{aligned}$$

    and

    $$\begin{aligned} t_{i}\ge \frac{1}{\max \left\{ \frac{1}{q}-\left( \dfrac{1}{p_{i}} +\cdots +\dfrac{1}{p_{k_{1}-1}}\right) ,0\right\} } \end{aligned}$$

    for all \(i\in \left\{ 1,\ldots ,k_{1}-1\right\} \).

  2. (B)

    If \(s_{k_{1}}\ge q\) then the property \(\mathbf {(P_{1})}\) is satisfied, provided that

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

Our second main result reads as:

Theorem 1.8

Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\) and \(1\le s\le q\le \infty \) be such that

$$\begin{aligned} \left| \frac{1}{\textbf{p}}\right| <\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}. \end{aligned}$$

For each \(k_{1}\in \left\{ 1,\ldots ,m\right\} \), define

$$\begin{aligned} \frac{1}{r}:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}, \end{aligned}$$
(13)

and

$$\begin{aligned} \frac{1}{s_{k_{1}}}:=\frac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

Let us consider the following property, \(\mathbf {(P_{2})}\): there exists \(C_{m}>0\) such that, for every continuous m-linear map \(A:X_{p_{1} }\times \cdots \times X_{p_{m}}\rightarrow X_{s}\), one has

$$\begin{aligned} \left( \sum _{i_{1}=1}^{\infty }\left( \ldots \left( \sum _{i_{m}=1}^{\infty }\left\| A\left( e_{i_{1}},\ldots ,e_{i_{m}}\right) \right\| _{\ell _{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots \right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\Vert A\Vert . \end{aligned}$$
(14)
  1. (A)

    If \(s_{k_{1}}<2\), then the property \(\mathbf {(P_{2})}\) is satisfied, provided that \(t_{k_{1}},\ldots ,t_{m}\in \left[ s_{k_{1}},2\right] \) are such that

    $$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}\le \frac{1}{s_{k_{1}}} +\frac{m-k_{1}}{2}, \end{aligned}$$

    and

    $$\begin{aligned} t_{i}\ge \frac{1}{\max \left\{ \frac{1}{2}-\left( \dfrac{1}{p_{i}} +\cdots +\dfrac{1}{p_{k_{1}-1}}\right) ,0\right\} } \end{aligned}$$

    for all \(i\in \left\{ 1,\ldots ,k_{1}-1\right\} \). Moreover, the m-tuple of exponents \(\left( t_{1},\ldots ,t_{m}\right) \) is globally sharp.

  2. (B)

    If \(s_{k_{1}}\ge 2\), define

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

    Then the property \(\mathbf {(P_{2})}\) is satisfied, provided that

    $$\begin{aligned} t_{i}\ge \left\{ \begin{array}{cl} \left[ \dfrac{1}{r}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1} &{} \text {if }i\le k_{0}\\ \max \{s,2\} &{} \text {if }i>k_{0}\text {.} \end{array} \right. \end{aligned}$$

    Moreover, the following table summarizes the optimality of the exponents:

    $$\begin{aligned} \begin{array}{ll} \hbox {(I)} 2\le s\le q. &{} \left( t_{1},\ldots ,t_{m}\right) \hbox {is globally sharp}\\ \hbox {(IIa)} 1\le s\le 2\le q \hbox {and} \ p_{k_{0}}<2. &{} \left( t_{1},\ldots ,t_{k_{0} }\right) \hbox {is globally sharp}\\ \hbox {(IIb)} 1\le s\le 2\le q \hbox {and} \ p_{k_{0}}\ge 2. &{} \left( t_{1},\ldots ,t_{m}\right) \hbox {is globally sharp}\\ \hbox {(III)} 1\le s\le q<2 \hbox {and}\, k_{2}\le k_{0}, &{} \left( t_{k_{2}},\ldots ,t_{m}\right) \hbox {is globally sharp} \end{array} \ \ \ \ \end{aligned}$$

    where,

    $$\begin{aligned} k_{2}:=\min \left\{ k:\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \le \frac{1}{2}\right\} . \end{aligned}$$

Remark 1.9

(About item (A), in Theorems 1.7 and 1.8.) Observe that the case \(k_{1}=1\) in Theorem 1.7 corresponds to item (A) of Theorem 1.3. Moreover, observe that for \(k_{1}>1\) while the exponent \(s_{k_{1}}<\lambda ,\) the exponents \(t_{1},\ldots ,t_{k_{1}-1}\) in Theorem 1.7 (respectively in Theorem 1.8) are, in general, worst than the exponents in Theorem 1.3 (respectively in Theorem 1.4). In fact, if \(t_{k_{1}},\ldots ,t_{m}\in \left[ s_{k_{1} },q\right] \) are such that

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{q}, \end{aligned}$$

and

$$\begin{aligned} t_{i}=\frac{1}{\frac{1}{q}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) } \end{aligned}$$

for all \(i\in \left\{ 1,\ldots ,k_{1}-1\right\} \), we have

$$\begin{aligned} \frac{1}{t_{1}}+\cdots +\frac{1}{t_{m}}&=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{q}+\sum _{i=1}^{p_{k_{1}-1}}\frac{1}{q}-\left( \dfrac{1}{p_{i}} +\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \\&=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) \\&\quad +\frac{m-k_{1}}{q}+\sum _{i=1}^{p_{k_{1}-1}}\frac{1}{q}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \\&=\dfrac{1}{r}-\left( \dfrac{1}{p_{1}}+\cdots +\dfrac{1}{p_{m}}\right) +\frac{m-1}{q}+\sum _{i=2}^{p_{k_{1}-1}}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \\&=\frac{1}{\lambda }+\frac{m-1}{q}-\sum _{i=2}^{p_{k_{1}-1}}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \\&\le \frac{1}{\lambda }+\frac{m-1}{q}. \end{aligned}$$

in this case, it represents a sort of price to be paid for the improvement observed in the parameter \(s_{k_{1}}\).

2 Preliminaries

In this section, we present and discuss some preliminary results that will be essential throughout the paper. Initially we comment that although not explicitly mentioned, all the inequalities presented in the subsequent sections hold for all positive integers n, and the respective constants involved are independent of n. When we refer to \(C_{m}\), it implies that the constant depends solely on m, which represents the degree of m-linearity of a multilinear form; we shall always assume \(m\ge 2\).

Let \(2\le q<\infty \) and \(0<s<\infty \). We recall, see Albiac and Kalton (2005), that a Banach space X is said to have a cotype q if there exists a constant \(C>0\) such that, regardless of how we select finitely many vectors \(x_{1},\dots ,x_{n}\in X\), there holds:

$$\begin{aligned} \left( \sum _{j=1}^{n}\Vert x_{j}\Vert ^{q}\right) ^{\frac{1}{q}}\le C\left( \int _{[0,1]}\left\| \sum _{j=1}^{n}r_{j}(t)x_{j}\right\| ^{s}dt\right) ^{1/s}, \end{aligned}$$
(15)

where \(r_{j}\) denotes the j-th Rademacher function. It is well known that if inequality (15) is fulfilled for a specific \(s>0\), then it holds true for all \(s>0\). Among these constants, the smallest one is denoted as \(C_{q,s}(X)\) when considering a fixed s. Additionally, the infimum of the cotypes of X is represented by \(\cot X\).

For instance, the scalar field \(\mathbb {K}\) has cotype 2. The notion of cotype remits, in the scalar case, to Khinchin’s inequality (see Diestel et al. (1995) and for a very recent approach (Ramaré 2024)); this inequality has played a crucial role to improve the estimates for the constants of Hardy–Littlewood inequalities (see Albuquerque et al. (2014); Bayart et al. (2014); Dimant and Sevilla-Peris (2016) and the references therein).

Hereafter \(r\ge 2\) and \(p_{1},\ldots ,p_{m}\,\in [1,\infty ]\). We define \(\lambda _{r}^{p_{k},\ldots ,p_{m}}\) by

$$\begin{aligned} \lambda _{r}^{p_{k},\ldots ,p_{m}}:=\frac{1}{\max \left\{ \frac{1}{r}-\left( \frac{1}{p_{k}}+\cdots +\frac{1}{p_{m}}\right) ,0\right\} }, \end{aligned}$$

for all positive integers m and \(k=1,\ldots ,m\). We also denote

$$\begin{aligned} \lambda _{r}:=\lambda _{r}^{p_{1},\ldots ,p_{m}}\text {.} \end{aligned}$$

Vector-valued Hardy–Littlewood inequalities are in general associated to the theory of absolutely summing operators, as illustrated by the following result:

Proposition 2.1

Dimant and Sevilla-Peris (2016, Proposition 3.1) Let X be Banach space, Y be a cotype q Banach space and \(v:X\rightarrow Y\) be an absolutely (r, 1)-summing operator (with \(1\le r\le q\)). If \(p_{1},\ldots ,p_{m} \in \left[ 1,\infty \right] \) and \(1/p_{1}+\cdots +1/p_{m}\le 1/r-1/q\), then there is a constant \(C_{m}\) such that

$$\begin{aligned} \left( \sum _{j_{i}=1}^{\infty }\left( \sum _{\widehat{j_{i}}=1}^{\infty }\Vert vA(e_{j_{1}},\dots ,e_{j_{m}})\Vert _{Y}^{q}\right) ^{\frac{\lambda _{r}}{q} }\right) ^{\frac{1}{\lambda _{r}}}\le C_{m}\Vert A\Vert \pi _{r,1}(v) \end{aligned}$$

for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\) and all \(i=1,\ldots ,m\), where \(\widehat{j_{i}}\) means that the sum is over all coordinates except the i-th coordinate.

Two results will be instrumental for the goals of this paper. The first one appears in Núñez-Alarcón et al. (2022) and the second one is a slight extension of Proposition 2.1 from (Dimant and Sevilla-Peris 2016). For the readers convenience, we list both results here and discuss the proof the generalized version of Proposition 2.1 that we will need.

Theorem 2.2

Núñez-Alarcón et al. (2022, Theorem 2.2) Let \(\left( r_{1},\ldots ,r_{m}\right) \in (0,\infty ]^{m},\) \(\left( p_{1},\ldots ,p_{m}\right) \,\in [1,\infty ]^{m}\) and X be an infinite-dimensional Banach space with cotype \(\cot X=q.\) The following assertions are equivalent:

(a) There is a constant \(C_{m}\ge 1\) such that

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

for all continuous m-linear operators \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\).

(b) The exponents \(q_{1},\ldots ,q_{m}\) satisfy

$$\begin{aligned} r_{k}\ge \lambda _{q}^{p_{k},\ldots ,p_{m}}\text {,} \end{aligned}$$

for all \(k=1,\ldots ,m\).

Next we state and proof the extension of the Proposition 2.1.

Proposition 2.3

Let X be Banach space, Y be a cotype q Banach space and \(v:X\rightarrow Y\) be an absolutely (r, 1)–summing operator (with \(1\le r\le q\)). If \(p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right] \) and \(1/p_{1}+\cdots +1/p_{m}<1/r\) are such that

$$\begin{aligned} \sum _{k\ne i}\frac{1}{p_{k}}\le \frac{1}{r}-\frac{1}{q} \end{aligned}$$
(16)

for some \(i\in \left\{ 1,\dots ,m\right\} \), then there is a constant \(C_{m}\), such that

$$\begin{aligned} \left( \sum _{j_{i}=1}^{\infty }\left( \sum _{\widehat{j_{i}}=1}^{\infty }\Vert vA(e_{j_{1}},\dots ,e_{j_{m}})\Vert _{Y}^{q}\right) ^{\frac{\lambda _{r}}{q} }\right) ^{\frac{1}{\lambda _{r}}}\le C_{m}\Vert A\Vert \pi _{r,1}(v) \end{aligned}$$

for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\).

Proof

For all \(n\in \mathbb {N} \) let \(A:\ell _{p_{1}}^{n}\times \dots \times \ell _{p_{m}}^{n}\rightarrow X\) be an m-linear operator. Choose an index i satisfying (16) and fix \(x\in \ell _{p_{i}}^{n}\). Consider

$$\begin{aligned} A_{i}:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{i-1}}^{n}\times \ell _{\infty }^{n}\times \ell _{p_{i+1}}^{n}\times \cdots \times \ell _{p_{m}} ^{n}\rightarrow X \end{aligned}$$

defined by

$$\begin{aligned} A_{i}(z^{(1)},\dots ,z^{(m)})=A(z^{(1)},\dots ,z^{\left( i-1\right) },xz^{(i)},z^{\left( i+1\right) },\dots ,z^{(m)})\text {,} \end{aligned}$$

where \(xz^{(i)}=\left( x_{j}z_{j}^{(i)}\right) _{j=1}^{n}\). Let \(1/\lambda _{r}^{\prime }:=1/\lambda _{r}+1/p_{i}\). Applying Proposition 2.1 to \(A_{i}\) yields

$$\begin{aligned}&\left( \sum _{j_{i}=1}^{n}\left| x_{j_{i}}\right| ^{\lambda _{r}^{\prime }}\left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1} },\dots ,e_{j_{i-1}},e_{j_{i}},e_{j_{i+1}},\dots ,e_{j_{m}})\right\| _{Y} ^{q}\right) ^{\frac{1}{q}\times \lambda _{r}^{\prime }}\right) ^{\frac{1}{\lambda _{r}^{\prime }}}\nonumber \\&\quad =\left( \sum _{j_{i}=1}^{n}\left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{i-1}},x_{j_{i}}e_{j_{i}},e_{j_{i+1}},\dots ,e_{j_{m} })\right\| _{Y}^{q}\right) ^{\frac{1}{q}\times \lambda _{r}^{\prime } }\right) ^{\frac{1}{\lambda _{r}^{\prime }}}\nonumber \\&\quad =\left( \sum _{j_{i}=1}^{n}\left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA_{i}(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{1}{q}\times \lambda _{r}^{\prime }}\right) ^{\frac{1}{\lambda _{r}^{\prime }} }\nonumber \\&\quad \le C_{m}\Vert A\Vert \Vert x\Vert _{\ell _{p_{i}}^{n}}\pi _{r,1} (v)\text {.} \end{aligned}$$
(17)

Since \(\left( p_{i}/\lambda _{r}^{\prime }\right) ^{*}=\lambda _{r} /\lambda _{r}^{\prime }\), we can further estimate

$$\begin{aligned}&\left( \sum _{j_{i}=1}^{n}\left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{1}{q} \times \lambda _{r}}\right) ^{\frac{1}{\lambda _{r}}}\\&\quad =\left( \sum _{j_{i}=1}^{n}\left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{1}{q} \times \lambda _{r}^{\prime }\times \left( \frac{p_{i}}{\lambda _{r}^{\prime } }\right) ^{*}}\right) ^{\frac{1}{\lambda _{r}^{\prime }}\times \frac{1}{\left( \frac{p_{i}}{\lambda _{r}^{\prime }}\right) ^{*}}}\\&\quad =\left( \left\| \left( \left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{1}{q} \times \lambda _{r}^{\prime }}\right) _{j_{i}=1}^{n}\right\| _{\left( \frac{p_{i}}{\lambda _{r}^{\prime }}\right) ^{*}}\right) ^{\frac{1}{\lambda _{r}^{\prime }}}\\&\quad =\left( \sup _{y\in B_{\ell _{\frac{p_{i}}{\lambda _{r}^{\prime }}}^{n}}} \sum _{j_{i}=1}^{n}\left| y_{j_{i}}\right| \left( \sum _{\widehat{j_{i}}=1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y} ^{q}\right) ^{\frac{1}{q}\times \lambda _{r}^{\prime }}\right) ^{\frac{1}{\lambda _{r}^{\prime }}}\\&\quad =\left( \sup _{x\in B_{\ell _{p_{i}}^{n}}}\sum _{j_{i}=1}^{n}\left| x_{j_{i}}\right| ^{\lambda _{r}^{\prime }}\left( \sum _{\widehat{j_{i}} =1}^{n}\left\| vA(e_{j_{1}},\dots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{1}{q}\times \lambda _{r}^{\prime }}\right) ^{\frac{1}{\lambda _{r}^{\prime }}}\\&\quad \le C_{m}\Vert A\Vert \pi _{r,1}(v)\text {,} \end{aligned}$$

where the last inequality holds in view of (17). \(\square \)

As a direct consequence, we obtain a cotype q version of Albuquerque et al. (2014, Proposition 4.1), which is of independent interest, and thus we state here:

Corollary 2.4

Let X, Y be Banach spaces where Y has cotype q. Let \(v:X\rightarrow Y\) be an absolutely (r, 1)-summing operator with \(1\le r\le q\). If \(p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right] \) and \(1/p_{1}+\cdots +1/p_{m}\le 1/r\) are such that

$$\begin{aligned} \sum _{k\ne i}\frac{1}{p_{k}}\le \frac{1}{r}-\frac{1}{q} \end{aligned}$$

for all \(i\in \{1,\dots ,m\}\), then there is a constant \(C_{m}\), such that

$$\begin{aligned} \left( \sum _{j_{i}=1}^{\infty }\left( \sum _{\widehat{j_{i}}=1}^{\infty }\Vert vA(e_{j_{1}},\dots ,e_{j_{m}})\Vert _{Y}^{q}\right) ^{\frac{\lambda _{r}}{q} }\right) ^{\frac{1}{\lambda _{r}}}\le C_{m}\Vert A\Vert \pi _{r,1}(v) \end{aligned}$$

for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\) for all \(i\in \{1,\dots ,m\}\).

In what follows, we will need to derive exponents for an m-linear vector-type Hardy–Littlewood inequality based on previously known exponents for a M-linear vector-type Hardy–Littlewood inequality, with \(M<m\). This is the contents of the following instrumental result:

Theorem 2.5

Let \(m\ge 2\) and \(p_{1},\ldots ,p_{m}\in \left[ 1,\infty \right] \), X be a Banach space, Y be a cotype q space and \(1\le r\le q\). If \(1<k\le m\) and there is a constant \(C_{m,k}\) such that

$$\begin{aligned} \left( \sum _{j_{k}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vA\left( e_{j_{k}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{r_{m}}\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{k} }{r_{k+1}}}\right) ^{\frac{1}{r_{k}}}\le C_{m,k}\left\| A\right\| \pi _{r,1}\left( v\right) \nonumber \\ \end{aligned}$$
(18)

for all continuous \(\left( m-k+1\right) \)-linear operator \(A:X_{p_{k}}\times \cdots \times X_{p_{m}}\rightarrow X\) and all absolutely \(\left( r,1\right) \)-summing operator \(v:X\rightarrow Y\), then there is a constant \(C_{m}\) such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vB\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{r_{m}}\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{1} }{r_{2}}}\right) ^{\frac{1}{r_{1}}}\le C_{m}\left\| B\right\| \pi _{r,1}\left( v\right) \end{aligned}$$

for all continuous m-linear operator \(B:X_{p_{1}}\times \cdots \times X_{p_{m}}\rightarrow X\) and all absolutely \(\left( r,1\right) \)-summing operator \(v:X\rightarrow Y\), with

$$\begin{aligned} r_{1}\ge \lambda _{\mu }^{p_{1},\ldots ,p_{k-1}},r_{2}\ge \lambda _{\mu } ^{p_{2},\ldots ,p_{k-1}},\ldots ,r_{k-1}\ge \lambda _{\mu }^{p_{k-1}}, \end{aligned}$$

where \(\mu =\max \left\{ r_{k},r_{k+1},\ldots ,r_{m},q\right\} .\)

Proof

We start off by noting that, given a continuous m-linear operator \(B:X_{p_{1}}\times \cdots \times X_{p_{m}}\rightarrow X\), we can generate a new continuous \(\left( k-1\right) \)-linear operator

$$\begin{aligned} vB_{e}:X_{p_{1}}\times \cdots \times X_{p_{k-1}}\rightarrow \ell _{r_{k}}\left( \ell _{r_{k+1}}\left( \cdots \ell _{r_{m}}\left( Y\right) \cdots \right) \right) \end{aligned}$$

as follows:

$$\begin{aligned} vB_{e}(x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) })=\left( vB\left( x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) },e_{j_{k} },\ldots ,e_{j_{m}}\right) \right) _{j_{k},\ldots ,j_{m}=1}^{\infty }. \end{aligned}$$

Above, and henceforth, the notation \(\ell _{r_{k}}\left( \ell _{r_{k+1}}\left( \cdots \ell _{r_{m}}\left( Y\right) \cdots \right) \right) \) refers to vector valued sequence spaces. We claim that in the construction above, one can obtain a constant \(C_{m.k}>0\) such that

$$\begin{aligned} \left\| vB_{e}\right\| \le C_{m.k}\left\| B\right\| \pi _{r,1}\left( v\right) . \end{aligned}$$

Indeed, for fixed \(x^{(1)},\ldots ,x^{(k-1)},\) by (18) (for the respective continuous \(\left( m-k+1\right) \)-linear operator) and obtain

$$\begin{aligned}&\left( \sum _{j_{k}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1} ^{\infty }\left\| vB\left( x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) },e_{j_{k}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{r_{m} }\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{k}}{r_{k+1}} }\right) ^{\frac{1}{r_{k}}}\\&\quad \le C_{m,k}\left\| B\left( x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) },\cdot ,\cdots ,\cdot \right) \right\| \pi _{r,1}\left( v\right) \\&\quad \le C_{m,k}\left\| B\right\| \pi _{r,1}\left( v\right) \left\| x^{\left( 1\right) }\right\| \cdot \cdots \cdot \left\| x^{\left( k-1\right) }\right\| . \end{aligned}$$

Thus

$$\begin{aligned}&\left\| vB_{e}\right\| =\sup _{\left\| x^{(1)}\right\| ,\cdots ,\left\| x^{(k-1)}\right\| \le 1}\left\| vB_{e}\left( x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) }\right) \right\| \\&=\sup _{\left\| x^{\left( 1\right) }\right\| ,\cdots ,\left\| x^{\left( k-1\right) }\right\| \le 1}\\&\quad \left( \sum _{j_{k}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vB\left( x^{\left( 1\right) },\ldots ,x^{\left( k-1\right) },e_{j_{k}},\ldots ,e_{j_{m}}\right) \right\| _{Y}\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{k} }{r_{k+1}}}\right) ^{\frac{1}{r_{k}}}\\&\le C_{m,k}\left\| B\right\| \pi _{r,1}\left( v\right) \text {.} \end{aligned}$$

However, since \(\ell _{r_{k}}\left( \ell _{r_{k+1}}\left( \cdots \ell _{r_{m} }\left( Y\right) \cdots \right) \right) \) is infinite dimensional and it has cotype \(\mu \), with

$$\begin{aligned} \mu =\max \left\{ r_{k},r_{k+1},\ldots ,r_{m},q\right\} \text {,} \end{aligned}$$

(see Diestel et al. (1995, Theorem 11.12)), it follows from Theorem 2.2, applied to the continuous \(\left( k-1\right) \)-linear operator \(vB_{e}\), that there is a constant \(C_{k-1}\) such that

$$\begin{aligned}&\left( \sum _{j_{1}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1} ^{\infty }\left\| vB\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{r_{m}}\right) ^{\frac{r_{m-1}}{r_{m}}}\cdots \right) ^{\frac{r_{1} }{r_{2}}}\right) ^{\frac{1}{r_{1}}}\\&\quad =\left( \sum _{j_{1}=1}^{\infty }\left( \sum _{j_{2}=1}^{\infty } \cdots \left( \sum _{j_{k-1}=1}^{\infty }\left\| vB_{e}(e_{j_{1}} ,\ldots ,e_{j_{k-1}})\right\| _{\ell _{r_{k}}\left( \ell _{r_{k+1}}\left( \cdots \ell _{r_{m}}\left( Y\right) \cdots \right) \right) }^{r_{k-1} }\right) ^{\frac{r_{k-2}}{r_{k-1}}}\cdots \right) ^{\frac{r_{1}}{r_{2}} }\right) ^{\frac{1}{r_{1}}}\\&\quad \le C_{k-1}\left\| vB_{e}\right\| \\&\quad \le C_{k-1}C_{m-k}\left\| B\right\| \pi _{r,1}\left( v\right) \text {,} \end{aligned}$$

with

$$\begin{aligned} r_{1}\ge \lambda _{\mu }^{p_{1},\ldots ,p_{k-1}},r_{2}\ge \lambda _{\mu } ^{p_{2},\ldots ,p_{k-1}},\ldots ,r_{k-1}\ge \lambda _{\mu }^{p_{k-1}}, \end{aligned}$$

and thus the Theorem is proven. \(\square \)

3 Proof of Theorem 1.7

In this section, we discuss the proof of Theorem 1.7. It will be obtained as an application of Theorem 2.5. It is worth emphasizing that Theorem 1.7 represents a significant advancement beyond the results of Theorems 1.3. Here are the details:

Proof

We recall that the case \(k_{1}=1\) corresponds to the assertion of Theorem 1.3. Consequently, for the remainder of the proof, our attention shall be devoted to the case \(k_{1}>1\).

(A) We start off by noting that the inequality \(s_{k_{1}}<q\) is equivalent to

$$\begin{aligned} \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}<\dfrac{1}{r}-\frac{1}{q}. \end{aligned}$$
(19)

Hence, taking \(t_{i}=q\), for each \(i\in \left\{ k_{1}+1,\ldots ,m\right\} \) and \(t_{k_{1}}=s_{k_{1}}\), kee** in mind that,

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

we readily obtain \(t_{k_{1}},\ldots ,t_{m}\in \left[ s_{k_{1}},q\right] \) with

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}&=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) +\frac{m-k_{1}}{q}\\&=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) +\frac{\left( m-k_{1}+1\right) -1}{\max \left\{ s_{k_{1}},q\right\} }\text {.} \end{aligned}$$

Theorem 1.3 then yields the existence of a constant \(C_{m-k_{1}+1}>0\) such that

$$\begin{aligned}{} & {} \left( \sum _{j_{k_{1}}=1}^{\infty }\left( \cdots \left( \sum _{j_{m} =1}^{\infty }\left\| vB\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{k_{1}}}{t_{k_{1}+1}}}\right) ^{\frac{1}{t_{k_{1}}}}\nonumber \\{} & {} \qquad \le C_{m-k_{1}+1}\left\| B\right\| \pi _{r,1}\left( v\right) \end{aligned}$$
(20)

for all continuous \(\left( m-k_{1}+1\right) \)-linear operator \(B:X_{p_{k_{1}}}\times \dots \times X_{p_{m}}\rightarrow X\) and all absolutely \(\left( r,1\right) \)-summing operators \(v:X\rightarrow Y\). Applying the previous argument, we obtain (20) for the exponents

$$\begin{aligned} \left( t_{k_{1}},\ldots ,t_{m}\right) =\left( q,\ldots ,q,s_{k_{1}},q,\ldots ,q\right) \end{aligned}$$

with \(s_{k_{1}}\) in the k-th position, for all \(k=k_{1},\ldots ,m\). The idea now is to interpolate

$$\begin{aligned} \left( s_{k_{1}},q,\ldots ,q\right) ,\left( q,s_{k_{1}},q,\ldots ,q\right) ,\ldots ,\left( q,\ldots ,q,s_{k_{1}}\right) \end{aligned}$$

in the sense of Albuquerque et al. (2014), as to attain (20) for all \(t_{k_{1} },\ldots ,t_{m}\in [s_{k_{1}},q]\) such that

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{q}. \end{aligned}$$

Now, since \(s_{k_{1}}<q\), we have \(\cot \left( \ell _{s_{k_{1}}}\left( \ell _{q}\left( \cdots \ell _{q}\left( Y\right) \cdots \right) \right) \right) =q\) (see Diestel et al. (1995, Theorem 11.12)), and thus Theorem 2.5 assures the existence of another constant \(C_{m}>0\) such that

$$\begin{aligned} \left( \sum _{j_{1}=1}^{\infty }\left( \cdots \left( \sum _{j_{m}=1}^{\infty }\left\| vA\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\| _{Y}^{t_{m}}\right) ^{\frac{t_{m}-1}{t_{m}}}\cdots \right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\le C_{m}\left\| A\right\| \pi _{r,1}\left( v\right) \end{aligned}$$

for all continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\) and all absolutely \(\left( r,1\right) \)-summing operators \(v:X\rightarrow Y\), with \(t_{k_{1}},\ldots ,t_{m}\in [s_{k_{1}},q]\) such that

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{q}, \end{aligned}$$

and

$$\begin{aligned} t_{k}\ge \lambda _{q}^{p_{k},\ldots ,p_{k_{1}-1}}, \end{aligned}$$

for all \(k\in \left\{ 1,\ldots ,k_{1}-1\right\} \). The result (A) now follows by the inclusion of the \(\ell _{r}\) spaces.

(B) We now turn our attention to the proof of statement (B). Initially, we note that \(s_{k_{1}}\ge q\) is equivalent to

$$\begin{aligned} \frac{1}{r}-\frac{1}{q}\le \frac{1}{p_{k_{1}}}+\cdots +\frac{1}{p_{m}} \end{aligned}$$

and, in this case, we also have

$$\begin{aligned} k_{1}\le k_{0}:=\max \left\{ t:\dfrac{1}{p_{t}}+\cdots +\dfrac{1}{p_{m}} \ge \dfrac{1}{r}-\dfrac{1}{q}\right\} . \end{aligned}$$

If \(k_{0}=1\), then

$$\begin{aligned} \frac{1}{p_{2}}+\cdots +\frac{1}{p_{m}}<\frac{1}{r}-\frac{1}{q}, \end{aligned}$$

and thus taking

$$\begin{aligned} s_{1}=\left[ \dfrac{1}{r}-\left( \dfrac{1}{p_{1}}+\cdots +\dfrac{1}{p_{m} }\right) \right] ^{-1} \end{aligned}$$

and \(s_{k}=q\) for all \(k>1\), the aimed conclusion follows directly by Proposition 2.3.

The more interesting case is when \(k_{0}>1\). Since

$$\begin{aligned} \frac{1}{r}-\frac{1}{q}\le \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}, \end{aligned}$$
(21)

there holds

$$\begin{aligned} \frac{1}{R}:=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m} }\right) \le \frac{1}{q}\text {.} \end{aligned}$$
(22)

Hence, by the inequality

$$\begin{aligned} \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<\frac{1}{r}\text {,} \end{aligned}$$

we conclude

$$\begin{aligned} \frac{1}{p_{1}}+\cdots +\frac{1}{p_{k_{0}-1}}<\frac{1}{R}. \end{aligned}$$
(23)

It then follows by the very definition of \(k_{0}\) that

$$\begin{aligned} \dfrac{1}{p_{k_{0}+1}}+\cdots +\dfrac{1}{p_{m}}<\dfrac{1}{r}-\dfrac{1}{q}\text {.} \end{aligned}$$
(24)

Therefore,

$$\begin{aligned} \frac{1}{p_{k_{0}}}+\cdots +\frac{1}{p_{m}}<\frac{1}{r}, \end{aligned}$$

are such that

$$\begin{aligned} \sum _{k\ne k_{0}}\frac{1}{p_{k}}\le \frac{1}{r}-\frac{1}{q}. \end{aligned}$$

Proposition 2.3 then yields the existence of a constant \(C_{k_{0}-1}\), such that

$$\begin{aligned} \left( \sum _{j_{k_{0}}=1}^{\infty }\left( \sum _{j_{k_{0}+1},\ldots ,j_{m} =1}^{\infty }\Vert vT(e_{j_{k_{0}}},\dots ,e_{j_{m}})\Vert _{Y}^{q}\right) ^{\frac{R}{q}}\right) ^{\frac{1}{R}}\le C_{k_{0}-1}\Vert T\Vert \pi _{r,1}(v), \end{aligned}$$

for every continuous \(\left( m-k_{0}+1\right) \)-linear operator \(T:X_{p_{k_{0}}}\times \dots \times X_{p_{m}}\rightarrow X\).

Now, since \(r<R\), we have that \(\ell _{R}\left( \ell _{q}\cdots \left( \ell _{q}\left( Y\right) \cdots \right) \right) \) has cotype R, and Theorem 2.5 assures the existence of a constant \(C_{m}\) such that

$$\begin{aligned}&\left( \sum _{j_{1}=1}^{\infty }\left( \sum _{j_{2}=1}^{\infty }\cdots \left( \sum _{j_{k_{0}-1}=1}^{\infty }\left( \sum _{j_{k_{0}}=1}^{\infty }\left( \sum _{j_{k_{0}+1},\ldots ,j_{m}=1}^{\infty }\left\| vA(e_{j_{1}} ,\ldots ,e_{j_{m}})\right\| _{Y}^{q}\right) ^{\frac{R}{q}}\right) ^{\frac{r_{k_{0}-1}}{R}}\right) ^{\frac{r_{k_{0}-2}}{r_{k_{0}-1}}} \cdots \right) ^{\frac{r_{1}}{r_{2}}}\right) ^{\frac{1}{r_{1}}}\\&=\left( \sum _{j_{1}=1}^{\infty }\left( \sum _{j_{2}=1}^{\infty } \cdots \left( \sum _{j_{k_{0}-1}=1}^{\infty }\left\| vA_{e}(e_{j_{1}} ,\ldots ,e_{j_{k_{0}-1}})\right\| _{\ell _{R}^{n}\left( \ell _{q}^{n} \cdots \left( \ell _{q}^{n}\left( Y\right) \cdots \right) \right) }^{r_{k_{0}-1}}\right) ^{\frac{r_{k_{0}-2}}{r_{k_{0}-1}}}\cdots \right) ^{\frac{r_{1}}{r_{2}}}\right) ^{\frac{1}{r_{1}}}\\&\le C_{k_{0}-1}\left\| vA_{e}\right\| \\&\le C_{k_{0}-1}C_{m-k_{0}+1}\left\| A\right\| \pi _{r,1}\left( v\right) \text {,} \end{aligned}$$

for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X\), with

$$\begin{aligned} r_{k}\ge \begin{array}{ll} \left[ \dfrac{1}{R}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{0}-1} }\right) \right] ^{-1}\text {,}&\text {if }k\le k_{0}-1\text {.} \end{array} \end{aligned}$$

Finally, we note that

$$\begin{aligned} \frac{1}{R}=\dfrac{1}{r}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m} }\right) \end{aligned}$$

and for \(k<k_{0}\), one has:

$$\begin{aligned} \dfrac{1}{R}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{0}-1}}\right)= & {} \dfrac{1}{r}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}\right) -\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{0}-1}}\right) \\= & {} \dfrac{1}{r}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \text {,} \end{aligned}$$

which completes the proof. \(\square \)

To demonstrate the numerical enhancement provided by Theorem 1.7 over Theorem 1.3, consider the following scenario: let \(X=\ell _{1}\), and \(Y=\ell _{2}\), \(m=9\), and \(p_{1}=\cdots =p_{9}=10\). In this case we have \(q=2\) and by Grothendieck’s theorem we can choose \(r=1\) for all \(v:\ell _{1}\rightarrow \ell _{2}\). Since

$$\begin{aligned} 1-\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{9}}=\frac{9}{10}<1\text {,} \end{aligned}$$

we have \(k_{0}=5\), and also \(s_{k_{1}}\ge 2\) for all \(k_{1}\in \left\{ 1,2,3,4,5\right\} \) and \(s_{k_{1}}<2\) for all \(k_{1}\in \left\{ 6,7,8,9\right\} \). We present the following table, which contrasts the exponents derived from Theorems 1.7 (for each \(k_{1}\in \left\{ 1,\ldots ,9\right\} \)) and Theorem 1.3 for \(\left( X,Y,m,r,p_{j}\right) =\left( \ell _{1},\ell _{2},9,1,10\right) \) for all \(j=1,\ldots ,9\):

Result

\(s_{1}\)

\(s_{2}\)

\(s_{3}\)

\(s_{4}\)

\(s_{5}\)

\(s_{6}\)

\(s_{7}\)

\(s_{8}\)

\(s_{9}\)

Theorem 1.3

10

10

10

10

10

10

10

10

10

Theorem 1.7, \(k_{1}=1\)

10

5

10/3

10/4

2

2

2

2

2

Theorem 1.7, \(k_{1}=2\)

10

5

10/3

10/4

2

2

2

2

2

Theorem 1.7, \(k_{1}=3\)

10

5

10/3

10/4

2

2

2

2

2

Theorem 1.7, \(k_{1}=4\)

10

5

10/3

10/4

2

2

2

2

2

Theorem 1.7, \(k_{1}=5\)

10

5

10/3

10/4

2

2

2

2

2

Theorem 1.7, \(k_{1}=6\)

\(\infty \)

10

5

10/3

10/4

10/6

2

2

2

Theorem 1.7, \(k_{1}=7\)

\(\infty \)

\(\infty \)

10

5

10/3

10/4

10/7

2

2

Theorem 1.7, \(k_{1}=8\)

\(\infty \)

\(\infty \)

\(\infty \)

10

5

10/3

10/4

10/8

2

Theorem 1.7, \(k_{1}=9\)

\(\infty \)

\(\infty \)

\(\infty \)

\(\infty \)

10

5

10/3

10/4

10/9

4 Proof of Theorem 1.8

4.1 Proof of the Existence of the Exponents in Theorem 1.8

Proof

The main idea is to combine the conclusions of Theorem 1.7 with the Bennett-Carl inequalities (Bennett 1977; Carl 1974): for \(1\le s\le q\le \) \(\infty \), the inclusion map \(\ell _{s}\hookrightarrow \ell _{q}\) is \(\left( r,1\right) \)-summing, where the optimal r is given by (13).

We have

$$\begin{aligned} \frac{1}{s_{k_{1}}} :&=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{2,q\}} -\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) \\&=\frac{1}{r}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

(A) s\(_{k_{1}}<2\) (here obviously \(\max \{s,2\}=2\) and then \(\cot \left( \ell _{s}\right) =2\)): if \(t_{k_{1}},\ldots ,t_{m}\in \) \(\left[ s_{k_{1}},2\right] \) are such that

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}\le \frac{1}{s_{k_{1}}} +\frac{m-k_{1}}{2}, \end{aligned}$$

and

$$\begin{aligned} t_{i}\ge \lambda _{2}^{p_{i},\ldots ,p_{k_{1}-1}}, \end{aligned}$$

for all \(i\in \left\{ 1,\ldots ,k_{1}-1\right\} \), we obtain the conclusion as a direct application of Theorem 1.7.

(B) If \(s_{k_{1}}\ge 2\), then clearly

$$\begin{aligned} s_{k_{1}}\ge \cot \left( \ell _{s}\right) =\max \{s,2\}. \end{aligned}$$

Hence, kee** in mind the definition of \(k_{0}\) as stated in the Theorem, when

$$\begin{aligned} u_{i}\ge \left\{ \begin{array}{cl} \left[ \dfrac{1}{r}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1} &{} \text {if }i\le k_{0}\\ \max \{s,2\} &{} \text {if }i>k_{0}\text {,} \end{array} \right. \end{aligned}$$

the aimed conclusion follows by Theorem 1.7. \(\square \)

4.2 Proof of Theorem 1.8: The Optimality of the Exponents

We will restore to an important tool from Albuquerque et al. (2014, Lemma 6.2), which we reproduce below for the readers convenience:

Lemma 4.1

Albuquerque et al. (2014, Lemma 6.2) Let \(d,n\ge 1\), \(q_{1},\dots ,q_{d+1} \in [1,\infty ]^{d+1}\) and let, for \(q\ge 1\),

$$\begin{aligned} \alpha (q)=\left\{ \begin{array}{ll} \frac{1}{2}-\frac{1}{q} &{} \text {if }q\ge 2\\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

Then there exists a d-linear map** \(A:\ell _{q_{1}}^{n}\times \dots \times \ell _{q_{d}}^{n}\rightarrow \ell _{q_{d+1}}^{n}\) which may be written

$$\begin{aligned} A\big (x^{(1)},\dots ,x^{(d)}\big )=\sum _{i_{1},\dots ,i_{d+1}=1}^{n}\pm x_{i_{1} }^{(1)}\cdots x_{i_{d}}^{(d)}e_{i_{d+1}} \end{aligned}$$

such that

$$\begin{aligned} \Vert A\Vert \le C_{d}n^{\frac{1}{2}+\alpha (q_{1})+\dots +\alpha (q_{d} )+\alpha (q_{d+1}^{*})}. \end{aligned}$$

In what follows, the isotropic result proven by Albuquerque et al. (2016), and Dimant and Sevilla-Peris (2016), already commented in the introduction, will be instrumental to our analysis. We state it as we shall need it, viz. for \(\lambda \ge 2\):

Theorem 4.2

Albuquerque et al. (2016, Theorem 1.1) Dimant and Sevilla-Peris (2016, Proposition 4.4) Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\), and \(s,q\in [1,\infty ]\) be such that \(s\le q\). Assume either \(q\ge 2\) or \(q<2\), and \(\left| \frac{1}{\textbf{p}}\right| \le \frac{1}{2}\). If

$$\begin{aligned} \frac{1}{\lambda }:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}-\left| {\frac{1}{\textbf{p}}}\right| \in \left( 0,\frac{1}{2}\right] , \end{aligned}$$

then for every continuous m-linear operator \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), we have

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m} })\Vert _{\ell _{q}}^{\rho }\right) ^{\frac{1}{\rho }}\le C\Vert A\Vert \end{aligned}$$

for a universal constant \(C>0\) (i.e. independent of the m-linear operator A) if, and only if

$$\begin{aligned} \rho \ge \lambda \text {.} \end{aligned}$$

For the next two instrumental lemmas, we shall use the following notation:

$$\begin{aligned} \frac{1}{s_{k}}:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

Lemma 4.3

Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\), and \(s,q\in [1,\infty ]\) be such that \(s\le q\). Assume either \(q\ge 2\) or \(q<2\), and \(\left| \frac{1}{\textbf{p}}\right| \le \frac{1}{2}\). If

$$\begin{aligned} \frac{1}{s_{k_{1}}}\in \left( 0,\frac{1}{2}\right] , \end{aligned}$$

and for every continuous m-linear map \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), we have

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{k_{1}-1}=1}^{\infty }\left( \sum _{i_{k_{1}},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q}} ^{t}\right) ^{\frac{u}{t}}\right) ^{\frac{1}{u}}\le C\Vert A\Vert \end{aligned}$$

for a universal constant \(C>0\) (i.e. independent of the m-linear map A), then

$$\begin{aligned} t\ge s_{k_{1}}\text {,} \end{aligned}$$

and, if \(t=s_{k_{1}}\) we have

$$\begin{aligned} u\ge s_{1}. \end{aligned}$$

Proof

Initially we note that the estimate for the parameter t follows directly by Theorem 4.2, for \(\left( m-k_{1}+1\right) \)-linear maps. For the parameter u, if \(t=s_{k_{1}}\), observe that since \(\frac{1}{s_{1}}\le \frac{1}{s_{k_{1}}}\le \frac{1}{2}\). If \(s_{1}=s_{k_{1}}\), the estimate for the parameter u follows, again, by Theorem 4.2 for m-linear maps. If \(s_{1}>s_{k_{1}}\), and we suppose that there exist \(u<s_{1}\) and \(C\ge 1\) such that

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{k_{1}-1}=1}^{\infty }\left( \sum _{i_{k_{1}},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q} }^{s_{k_{1}}}\right) ^{\frac{u}{s_{k_{1}}}}\right) ^{\frac{1}{u}}\le C\Vert A\Vert \end{aligned}$$

for all m-linear maps \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), by the monotonicity of the \(\ell _{q}~\)norms we conclude that there is a constant \(C\ge 1\) such that

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m} })\Vert _{\ell _{q}}^{w}\right) ^{\frac{1}{w}}\le C\left\| A\right\| \end{aligned}$$

for all m-linear maps \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), with \(w=\max \left\{ s_{k_{1}},u\right\} <s_{1}\). But this contradicts Theorem 4.2, for m-linear maps, because \(w<s_{1}\). \(\square \)

Lemma 4.4

Let \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\), and \(s,q\in [1,\infty ]\) be such that \(s\le q\). Assume either \(q\ge 2\) or \(q<2\) and \(\left| \frac{1}{\textbf{p}}\right| \le \frac{1}{2}\). If

$$\begin{aligned} \frac{1}{s_{k_{1}}}\in \left( 0,\frac{1}{2}\right] , \end{aligned}$$

and for every continuous m-linear map \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), we have

$$\begin{aligned} \left( \sum _{i_{1}=1}^{\infty }\left( \cdots \sum _{i_{k_{1}-1}=1}^{\infty }\left( \sum _{i_{k_{1}},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q}}^{t_{k_{1}}}\right) ^{\frac{t_{k_{1}-1} }{s_{k_{1}}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}} }\le C\Vert A\Vert \end{aligned}$$

for a universal constant \(C>0\) (i.e. independent of the m-linear map A), then

$$\begin{aligned} t_{k_{1}}\ge s_{k_{1}}\text {,} \end{aligned}$$

and, if \(j\in \left\{ 2,\ldots ,k_{1}\right\} \) is such that \(t_{k}=s_{k}\) for all \(k\in \left\{ j,\ldots ,k_{1}\right\} \), then \(t_{j-1}\ge s_{j-1}\).

Proof

Initially we note that the estimate for the parameter t follows directly by Theorem 4.2, for \(\left( m-k_{1}+1\right) \)-linear maps. For the parameter \(j=k_{1}-1\), if \(t_{k_{1}}=s_{k_{1}}\), observe that since \(s_{k_{1}-1}\ge s_{k_{1}}\ge 2.\) If \(s_{k_{1}-1}=s_{k_{1}}\), the estimate for the parameter \(t_{k_{1}-1}\) follows, by Lemma 4.3 for \(\left( m-k_{1}\right) \)-linear maps. If \(s_{k_{1}-1}>s_{k_{1}}\), and we suppose that there exist \(t_{k_{1}-1}<s_{k_{1}-1}\) and \(C\ge 1\) such that

$$\begin{aligned} \left( \sum _{i_{1}=1}^{\infty }\left( \cdots \sum _{i_{k_{1}-1}=1}^{\infty }\left( \sum _{i_{k_{1}},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q}}^{s_{k_{1}}}\right) ^{\frac{t_{k_{1}-1} }{s_{k_{1}}}}\cdots \right) ^{\frac{t_{1}}{t_{2}}}\right) ^{\frac{1}{t_{1}} }\le C\Vert A\Vert \end{aligned}$$

for all m-linear maps \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\), by the monotonicity of the \(\ell _{q}~\)norms we conclude that there is a constant \(C\ge 1\) such that

$$\begin{aligned} \left( \sum _{i_{1},\dots ,i_{k_{1}-2}=1}^{\infty }\left( \sum _{i_{k_{1} -1},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q} }^{w}\right) ^{\frac{t}{w}}\right) ^{\frac{1}{t}}\le C\left\| A\right\| \end{aligned}$$

for all m-linear maps \(A:X_{p_{1}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\),with \(w=\max \left\{ t_{k_{1}-1},s_{k_{1}}\right\} <s_{k_{1}-1}\) and \(t=\max \left\{ t_{1},t_{2},\dots t_{k_{1}-2}\right\} \). But this contradicts 4.3, for m-linear maps, because \(w<s_{k_{1}-1}\). Thus, \(t_{k_{1}-1}\ge s_{k_{1}-1}\). Arguing inductively we conclude the proof of the Lemma. \(\square \)

Proof of the optimality

We begin by examining item (A). Consider

$$\begin{aligned} \frac{1}{s_{k_{1}}}=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min \{q,2\}}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

If \(s_{k_{1}}>0\), take \(t_{k_{1}},\ldots ,t_{m}\in \left[ s_{k_{1}},2\right] \) with

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{2}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{t_{i}}=\frac{1}{2}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \end{aligned}$$

for all \(i\in \left\{ 1,\ldots ,k_{1}-1\right\} \). The goal is to prove that the m-tuple \(\left( t_{1},\ldots ,t_{m}\right) \) of exponents is globally sharp. Initially we note that the optimality of the \(\left( m-k_{1}+1\right) \)-tuple \(\left( t_{k_{1}},\ldots ,t_{m}\right) \in \) \(\left[ s_{k_{1} },2\right] ^{\left( m-s_{k_{1}}+1\right) }\) when

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{2}, \end{aligned}$$

follows directly by Theorem 1.4.

Let us now investigate the optimality of

$$\begin{aligned} \frac{1}{t_{k}}=\frac{1}{2}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) \end{aligned}$$

for all \(k\in \left\{ 1,\ldots ,k_{1}-1\right\} \). Define, for each \(k\in \left\{ 1,\ldots ,k_{1}-1\right\} \),

$$\begin{aligned} \frac{1}{\rho _{k}}=\dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{1}-1}}, \end{aligned}$$

and in the sequel let \(t_{k_{1}},\dots ,t_{m}\in [s_{k_{1}},2]\) be such that

$$\begin{aligned} \frac{1}{t_{k_{1}}}+\cdots +\frac{1}{t_{m}}=\frac{1}{s_{k_{1}}}+\frac{m-k_{1} }{2}. \end{aligned}$$

Let \(A:\ell _{\rho _{k}}^{n}\times \ell _{p_{k_{1}}}^{n}\dots \times \ell _{p_{m}}^{n}\rightarrow \ell _{s}^{n}\) be the \(\left( m-k_{1}+2\right) \) be the linear map given by Lemma 4.1, that is:

$$\begin{aligned} A\big (x^{(k)},x^{(k_{1})},\dots ,x^{(m)}\big )=\sum _{i_{k},i_{k_{1}} \dots ,i_{m+1}=1}^{n}\pm x_{i_{k}}^{(k)}x_{i_{k_{1}}}^{(k_{1})}\cdots x_{i_{m} }^{(m)}e_{i_{m+1}}. \end{aligned}$$

We have

$$\begin{aligned} \Vert A\Vert&\le C_{m,k_{1}}n^{\frac{1}{2}+\frac{m-k_{1}+2}{2} +\alpha \left( \rho _{k}\right) -\frac{1}{p_{k_{1}}}-\dots -\frac{1}{p_{m} }-\frac{1}{s^{*}}}\\&=C_{m,k_{1}}n^{\frac{m-k_{1}+1}{2}+\alpha \left( \rho _{k}\right) -\frac{1}{p_{k_{1}}}-\dots -\frac{1}{p_{m}}+\frac{1}{s}}\\&=C_{m,k_{1}}n^{\frac{1}{s_{k_{1}}}+\frac{m-k_{1}}{2}+\frac{1}{q} +\alpha \left( \rho _{k}\right) }. \end{aligned}$$

Next, define the \(\left( m-k+1\right) \)-linear map \(T:X_{p_{k}}\times \dots \times X_{p_{m}}\rightarrow X_{s}\) given by

$$\begin{aligned} T\big (x^{(k)},\dots ,x^{(m)}\big )=\sum _{i_{l},i_{k_{1}}\dots ,i_{m+1}=1}^{n}\pm x_{i_{l}}^{(k)}\cdots x_{i_{l}}^{(k_{1}-1)}x_{i_{k_{1}}}^{(k_{1})}\cdots x_{i_{m}}^{(m)}e_{i_{m+1}}. \end{aligned}$$

By Hölder’s inequality, it readily follows that

$$\begin{aligned} \Vert T\Vert \le \Vert A\Vert \le C_{m,k_{1}}n^{\frac{1}{s_{k_{1}}} +\frac{m-k_{1}}{2}+\frac{1}{q}+\alpha \left( \rho _{k}\right) }. \end{aligned}$$

On the other hand,

$$\begin{aligned}&\left( \sum _{i_{k}=1}^{n}\left( \ldots \left( \sum _{i_{m}=1} ^{n}\left\| T\left( e_{i_{k}},\ldots ,e_{i_{m}}\right) \right\| _{\ell _{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots \right) ^{\frac{t_{k}}{t_{k+1}}}\right) ^{\frac{1}{t_{k+1}}}\\&=\left( \sum _{i_{k}=1}^{n}\left( \sum _{i_{k_{1}}=1}^{n}\ldots \left( \sum _{i_{m}=1}^{n}\left\| T\left( e_{i_{k}},\ldots ,e_{i_{k}},e_{i_{k_{1}} },\ldots ,e_{i_{m}}\right) \right\| _{\ell _{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots \right) ^{\frac{t_{k}}{t_{k_{1}}}}\right) ^{\frac{1}{t_{k}}}\\&=n^{\frac{1}{q}+\frac{1}{t_{k}}+\frac{1}{t_{k_{1}}}+\dots +\frac{1}{t_{m}} }=n^{\frac{1}{q}+\frac{1}{t_{k}}+\frac{1}{s_{k_{1}}}+\frac{m-k_{1}}{2}}. \end{aligned}$$

Thus, for the property to be satisfied, one needs

$$\begin{aligned} \frac{1}{t_{k}}\le \alpha \left( \rho _{k}\right) =\left\{ \begin{array}{ll} \frac{1}{2}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\right) &{} \text {if }\dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{k_{1}-1}}\le \frac{1}{2}\\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

This concludes the proof of item (A).

Now we turn to the case (I) of item (B). In this case \(k_{0}=m\) and thus the exponent is

$$\begin{aligned} t_{i}=\left\{ \begin{array}{cl} \left[ \dfrac{1}{s}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}&\text {if }i\le m. \end{array} \right. \end{aligned}$$

Thus, the optimality of the exponents \(t_{i}\) follows from Lemma 4.4.

In what follows, let us discuss (III) of item (B). In this case, \(s_{k_{1} }\ge 2\), and

$$\begin{aligned} k_{0}:= & {} \max \left\{ k:\dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\ge \frac{1}{s}-\frac{1}{q}\right\} \quad \text {and}\\ k_{2}= & {} \min \left\{ k:\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \le \frac{1}{2}\right\} , \end{aligned}$$

Note that, for (III), we also have

$$\begin{aligned} \cot \left( \ell _{s}\right) =2\quad \text { and }\quad s_{k_{1}}=\left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{q}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}. \end{aligned}$$

We want to show that the exponents

$$\begin{aligned} t_{k}=\left\{ \begin{array}{ll} \left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{q}-\left( \dfrac{1}{p_{k}} +\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\text {,} &{} \text {if }k_{2}\le k\le k_{0}\text {,}\\ &{} \\ 2\text {,} &{} \text {if }k>k_{0}\text {.} \end{array} \right. \end{aligned}$$
(25)

are globally sharp. Since \(q<2\),

$$\begin{aligned} \left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \le \frac{1}{2}, \end{aligned}$$

and

$$\begin{aligned} s_{k_{0}}=\left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{q}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}\in (0,\frac{1}{2}], \end{aligned}$$

it follows by Lemma 4.4 that, if for every continuous \(\left( m-k_{2}+1\right) \)-linear map \(A:X_{p_{k_{2}}}\times \dots \times X_{p_{m} }\rightarrow X_{s}\), there holds:

$$\begin{aligned} \left( \sum _{i_{k_{2}}=1}^{\infty }\left( \cdots \sum _{i_{k_{0}-1}=1}^{\infty }\left( \sum _{i_{k_{0}},\dots ,i_{m}=1}^{\infty }\Vert A(e_{i_{1}},\dots ,e_{i_{m}})\Vert _{\ell _{q}}^{t_{k_{0}}}\right) ^{\frac{t_{k_{0}-1} }{t_{k_{0}}}}\cdots \right) ^{\frac{t_{k_{2}}}{t_{k_{2}+1}}}\right) ^{\frac{1}{t_{k_{2}}}}\le C\Vert A\Vert \end{aligned}$$

for a universal constant \(C>0\), then

$$\begin{aligned} t_{k_{0}}\ge s_{k_{0}}\text {,} \end{aligned}$$

and, if \(j\in \left\{ k_{2}+1,\ldots ,k_{0}\right\} \) is such that \(t_{k}=s_{k}\) for all \(k\in \left\{ k_{2}+1,\ldots ,k_{0}\right\} \), then \(t_{j-1}\ge s_{j-1}\). This implies that the exponents \(\left( t_{k_{2}},\ldots ,t_{k_{0} }\right) \) in Theorem 1.8 are optimal.

Next, by the definition of \(k_{2}\), one has \(p_{k_{2}},\dots ,p_{m}\in \left[ 2,\infty \right] \), and in particular \(p_{k_{0}},\dots ,p_{m}\in \left[ 2,\infty \right] \). Thus, the exponents

$$\begin{aligned} \left( t_{k_{0}},t_{k_{0}+1},\ldots ,t_{m}\right) =\left( \left[ \frac{1}{2}+\frac{1}{s}-\frac{1}{q}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1},2,\ldots ,2\right) \end{aligned}$$

are optimal as a consequence of Lemma 4.1.

For the proofs of (IIa) and (IIb), it suffices to note that in these cases \(\cot \left( \ell _{s}\right) =2\) and

$$\begin{aligned} s_{k_{1}}=\left[ \frac{1}{s}-\left( \dfrac{1}{p_{k_{1}}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1}. \end{aligned}$$

The optimality of the exponents

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

follows, since \(\frac{1}{s_{k_{0}}}\in (0,\frac{1}{2}]\), from Lemma 4.4, which is the statement of (IIa). For (IIb), the optimality of the \(\left( m-k_{0}+1\right) \)-tuple

$$\begin{aligned} \left( t_{k_{0}},\ldots ,t_{m}\right) =\left( \left[ \frac{1}{s}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1},2,\ldots ,2\right) \end{aligned}$$

follows from Lemma 4.1. \(\square \)

To demonstrate the numerical improvement provided by Theorem 1.8 over Theorem 1.4, let’s consider the following: let \(s=q=1\), \(m=9\), and \(p_{1}=\cdots =p_{9}=20\). In this case, for all \(k_{1}\in \left\{ 1,\ldots ,9\right\} \) we have

$$\begin{aligned} \frac{1}{s_{k_{1}}}=\frac{k_{1}}{20}<\frac{1}{2}\text {,} \end{aligned}$$

thus \(s_{k_{1}}>2\) for all \(k_{1}\), and \(k_{0}=9\). We obtain the table below, which compares the exponents \(t_{j}\) provided by Theorem 1.8 and Theorem 1.4 for \(\left( s,q,m,p_{j}\right) =\left( 1,1,9,20\right) \) for all \(j=1,\ldots ,9\):

Result

\(t_{1}\)

\(t_{2}\)

\(t_{3}\)

\(t_{4}\)

\(t_{5}\)

\(t_{6}\)

\(t_{7}\)

\(t_{8}\)

\(t_{9}\)

Theorem 1.4

20

20

20

20

20

20

20

20

20

Theorem 1.8

20

10

20/3

5

4

20/6

20/7

20/8

20/9

Moreover, since \(k_{2}=1<k_{0}\), we have that the \(9-\)uple \(\left( 20,10,\frac{20}{3},5,4,\frac{20}{6},\frac{20}{7},\frac{20}{8},\frac{20}{9}\right) \) is globally sharp.

Remark 4.5

Despite the great generality of Theorem 1.8, the problem of determining the entire range of possible exponents for which an Hardy–Littlewood inequality is valid still remains open. A recent line of investigation, in the setting of non-admissible exponents, concerns the size/geometry of the set of continuous m-linear operators \(A:X_{p_{1}} \times \dots \times X_{p_{m}}\rightarrow X_{s}\) 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. Maybe this theme can be investigated from a more subtle point of view of the theory of lineability, with the notion of \((\alpha ,\beta )\)-lineability introduced and explored by Fávaro et al. (see Araújo et al. (2023); Diniz and Raposo (2021); Fávaro et al. (2019) and the references therein). We also recommend the work (Núñez-Alarcón et al. 2024) where a different technique is used to check optimality of exponents in inequalities that involve summability of continuous m-linear operators.

Remark 4.6

We emphasize that in Theorem 1.8, when \(q\ge \max \left\{ s,2\right\} \), the exponents of the form \(\frac{1}{t_{k}}=\frac{1}{s}-\left( \dfrac{1}{p_{k}}+\cdots +\dfrac{1}{p_{m}}\right) \) are in fact the best possible. This is a direct consequence of Aron et al. (2017, Lemma 3.1).

As an application of Theorem 1.8, we recover the following recent result:

Corollary 4.7

Raposo and Serrano-Rodríguez (2023, Theorem 1.5) Let \(1\le s\), \(q:=\max \left\{ s,2\right\} \) and \(\textbf{p}\in \left[ 1,\infty \right] ^{m}\)be such that

$$\begin{aligned} \frac{1}{s}-\frac{1}{2}\le \left| \frac{1}{\textbf{p}}\right| <\frac{1}{s}. \end{aligned}$$

Let us define

$$\begin{aligned} \frac{1}{s_{k_{0}}}:=\frac{1}{s}-\left( \dfrac{1}{p_{k_{0}}}+\cdots +\dfrac{1}{p_{m}}\right) . \end{aligned}$$

where

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

There is a constant \(C_{m}\) such that

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

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

$$\begin{aligned} t_{i}\ge \left\{ \begin{array}{cl} \left[ \frac{1}{s}-\left( \dfrac{1}{p_{i}}+\cdots +\dfrac{1}{p_{m}}\right) \right] ^{-1} &{} \text {if }i\le k_{0}\\ q &{} \text {if }i>k_{0}\text {.} \end{array} \right. \end{aligned}$$

Moreover:

  1. (i)

    The exponents \(t_{1},\ldots ,t_{k_{0}}\) are optimal;

  2. (ii)

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

Proof

Since in this case \(s_{k_{0}}\ge 2\), we apply Theorem 1.8 to obtain the existence of the exponents. For the optimality of the exponents, observe that, if \(q=\max \left\{ s,2\right\} =2\), the optimality of the exponents in (i) follows by the above remark, and if \(p_{k_{0}}\ge 2\), by the case (IIb) in Theorem 1.8, the exponents \(\left( t_{1},\ldots ,t_{m}\right) \) are globally sharp. Finally, if \(q=\max \left\{ s,2\right\} =s\), we have \(k_{0}=m\). The above remark yields, in this case, the optimality for the exponents \(t_{1},\ldots ,t_{m}\). \(\square \)