The best constants in the Multiple Khintchine Inequality

TL;DR

This paper determines the optimal constants in the multiple Khintchine inequality, enabling precise bounds in related inequalities and completing a line of research initiated by Pellegrino.

Contribution

It provides the exact best constants for the multiple Khintchine inequality, advancing the understanding of related inequalities in analysis.

Findings

Established the optimal constants for the multiple Khintchine inequality.

Derived the best constants for the mixed (ℓ_{p/(p-1)}, ℓ_2)-Littlewood inequality.

Resolved a problem previously studied by Pellegrino.

Abstract

In this work we provide the best constants of the multiple Khintchine inequality. This allows us, among other results, to obtain the best constants of the mixed $\left( \ell_{\frac{p}{p-1}},\ell_{2}\right) $-Littlewood inequality, thus ending completely a work started by Pellegrino in \cite{pell}.

Equations

$${\left(\sum _{j=1}^n|a_j{|}^2\right)}^{\frac{1}{2}}\le {\mathrm{A}}_p{\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^{p}dt\right)}^{\frac{1}{p}},$$

$$r_j$$

$$r_j\left(t\right)$$

$${\int }_0^1{\left|\sum _{j=1}^{\infty }{a}_j{r}_j\left(t\right)\right|}^{2}dt=\sum _{j=1}^{\infty }{\left|a_j\right|}^2$$

$${\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j\left(t\right)\right|}^{p}dt={\int }_0^1{\left|\sum _{j=1}^n{\delta }_j{a}_j{r}_j\left(t\right)\right|}^{p}dt,$$

$${\int }_{I^m}{\left|\sum _{i_1,...,i_m=1}^n{y}_{i_1...i_m}{r}_{i_1}(t_1)\cdots r_{i_m}(t_m)\right|}^{p}d{t}_1\cdots d{t}_m$$

$$={\int }_{I^m}{\left|\sum _{i_1,...,i_m=1}^n{y}_{i_1...i_m}{r}_{i_1}(t_1)\cdots r_{i_m}(t_m){\epsilon }_{i_1}\cdots {\epsilon }_{i_m}\right|}^{p}d{t}_1\cdots d{t}_m$$

$${\mathrm{A}}_p=\frac{1}{\sqrt{2}}{\left(\frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi }}\right)}^{-\frac{1}{p}},\text{ for }1.85\approx p_0<p<2$$

$${\mathrm{A}}_p=2^{\frac{1}{p}-\frac{1}{2}},\text{ for }0<p\le p_0\approx \mathrm{1.85.}$$

$$\Gamma \left(\frac{p_0+1}{2}\right)=\frac{\sqrt{\pi }}{2}.$$

$$\Vert T\Vert$$

$$=\sup \left\{\left|\sum _{i,j}{a}_{ij}{x}_i{y}_j\right|:{\Vert x\Vert }_{X_p}\le 1,{\Vert y\Vert }_{X_q}\le 1\right\},$$

$$\frac{1}{2}<\frac{1}{p}+\frac{1}{q}<1$$

$${\left(\sum _{j,k=1}^{\infty }{\left|A(e_j,e_k)\right|}^{\frac{pq}{pq-p-q}}\right)}^{\frac{pq-q-p}{pq}}\le C_{p,q}\Vert A\Vert$$

$$\frac{1}{p}+\frac{1}{q}\le \frac{1}{2}$$

$${\left(\sum _{j,k=1}^{\infty }{\left|A(e_j,e_k)\right|}^{\frac{4pq}{3pq-2p-2q}}\right)}^{\frac{3pq-2p-2q}{4pq}}\le C_{p,q}\Vert A\Vert$$

$${\left(\sum _{j=1}^{\infty }{\left(\sum _{k=1}^{\infty }{\left|A(e_j,e_k)\right|}^2\right)}^{\frac{\lambda }{2}}\right)}^{\frac{1}{\lambda }}\le C_{p,q}\Vert A\Vert$$

$${\left(\sum _{i=1}^{\infty }{\left(\sum _{j=1}^{\infty }|T(e_i,e_j){|}^2\right)}^{\frac{1}{2}\frac{p}{p-1}}\right)}^{\frac{p-1}{p}}\le C_{p,\infty }\Vert T\Vert ,$$

$$\frac{{\left(2^{-N}{\sum }_{\beta \in {\left\{-1,1\right\}}^N}{\left|{\sum }_{j=1}^N{a}_j{\beta }_j\right|}^{\rho }\right)}^{\frac{1}{\rho }}}{{\left({\sum }_{j=1}^N|a_j{|}^2\right)}^{\frac{1}{2}}}$$

$${\left(\sum _{i=1}^{\infty }{\left(\sum _{j=1}^{\infty }|T(e_i,e_j){|}^2\right)}^{\frac{1}{2}\frac{p}{p-1}}\right)}^{\frac{p-1}{p}}\le C_{p,\infty }\Vert T\Vert ,$$

$${\left(\sum _{i=1}^n{\left(\sum _{j=1}^n{\left|T\left(e_i,e_j\right)\right|}^2\right)}^{\frac{1}{2}\frac{p}{p-1}}\right)}^{\frac{p-1}{p}}$$

$$={\mathrm{A}}_{\frac{p}{p-1}}{\left({\int }_0^1\sum _{i=1}^n{\left|T\left(e_i,\sum _{j=1}^n{r}_j(t)e_j\right)\right|}^{\frac{p}{p-1}}dt\right)}^{\frac{p-1}{p}}$$

$$\le {\mathrm{A}}_{\frac{p}{p-1}}{\left({\int }_0^1{\Vert T\left(\cdot ,\sum _{j=1}^n{r}_j(t)e_j\right)\Vert }^{\frac{p}{p-1}}dt\right)}^{\frac{p-1}{p}}$$

$$\le {\mathrm{A}}_{\frac{p}{p-1}}\underset{t\in [0,1]}{\sup }\Vert T\left(\cdot ,\sum _{j=1}^n{r}_j(t)e_j\right)\Vert$$

$$\le {\mathrm{A}}_{\frac{p}{p-1}}\Vert T\Vert .$$

$${\left(\sum _i{\left(\sum _j|a_{ij}{|}^p\right)}^{\frac{q}{p}}\right)}^{\frac{1}{q}}\le {\left(\sum _j{\left(\sum _i|a_{ij}{|}^q\right)}^{\frac{p}{q}}\right)}^{\frac{1}{p}}.$$

$${\left(\sum _{j=1}^{\infty }{\left(\sum _{i=1}^{\infty }|T(e_i,e_j){|}^{\frac{p}{p-1}}\right)}^{2\frac{p-1}{p}}\right)}^{\frac{1}{2}}\le D_{p,\infty }\Vert T\Vert ,$$

$${\left(\sum _{j=1}^{\infty }{\left(\sum _{i=1}^{\infty }|T(e_i,e_j){|}^{\frac{p}{p-1}}\right)}^{2\frac{p-1}{p}}\right)}^{\frac{1}{2}}$$

$$\le C_{p,\infty }\Vert T\Vert .$$

$${\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^{p}dt\right)}^{\frac{1}{p}}={\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t){\epsilon }_j\right|}^{p}dt\right)}^{\frac{1}{p}}.$$

$${\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^p\right)}^{\frac{1}{p}}$$

$$={\left({\left(\frac{1}{2}\right)}^n\sum _{i=1}^{2^n}{\left|\sum _{j=1}^n{a}_j{\delta }_j^{\left(i\right)}{\epsilon }_j\right|}^p\right)}^{\frac{1}{p}}$$

$$\left|{\left(\frac{1}{2}\right)}^{\frac{n}{p}}\sum _{i=1}^{2^n}\sum _{j=1}^n{a}_j{\delta }_j^{\left(i\right)}{\epsilon }_j{x}_i\right|$$

$$={\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^p\right)}^{\frac{1}{p}}{\Vert {\left(x_i\right)}_{i=1}^{2^n}\Vert }_{p^{\ast }}$$

$$\le {\left({\int }_0^1{\left|\sum _{j=1}^n{a}_j{r}_j(t)\right|}^p\right)}^{\frac{1}{p}}.$$

$${\left(\sum _{j=1}^n{\left|a_j\right|}^2\right)}^{\frac{1}{2}}$$

Full text

1. Introduction

Let $\left(a_{i}\right)_{i=1}^{\infty}$ be a scalar sequence. In 1922 Hans Rademacher showed that if, for almost all choice of signs $\delta_{i}$, the series $\sum_{i=1}^{\infty}a_{i}\delta_{i}$ converges, $\sum_{i=1}^{\infty}\left|a_{i}\right|^{2}$ also does it. A few months later, Aleksandr Khintchine showed the following more general result (see [16]):

Theorem 1.1** (Khintchine inequality).**

For any $0<p<\infty$, there is a positive constant $\mathrm{A}_{p}$ (depends only on $p$) such that regardless of the scalar sequence $(a_{j})_{j=1}^{n}\in\mathbb{K}^{n}$,we have

[TABLE]

where $r_{j}$ denotes the $j-$th Rademacher function.

Above and henceforth, $\mathbb{K}$ denotes the scalar field $\mathbb{R}$ or $\mathbb{C}$.

The Rademacher functions are defined as follows

[TABLE]

In probabilistic terms, the system $\left(r_{j}\left(t\right)\right)_{j=1}^{\infty}$ is a sequence of independent identically distributed random variables defined on the closed interval $\left[0,1\right]$ with a Lebesgue measure on its Borel subsets. Up to a linear transformation, Rademacher functions describe binomial trials with probability of success equal to $\frac{1}{2}$.

These functions have, among others, the following important properties: The Rademacher functions are an orthonormal sequence in $L_{2}\left[0,1\right]$, and then

[TABLE]

for all $a=\left(a_{j}\right)_{j=1}^{\infty}\in\ell_{2}$.

On the other hand, for every $n\in\mathbb{N}$ and $\delta_{1},\delta_{2},...,\delta_{n}=\pm 1$, $\sum_{j=1}^{n}a_{j}r_{j}\left(t\right)$ and $\sum_{j=1}^{n}\delta_{j}a_{j}r_{j}\left(t\right)$ have the same distribution in $L_{p}$, for all $p>0$. That means, for every $n\in\mathbb{N}$ and $\delta_{1},\delta_{2},...,\delta_{n}=\pm 1$,

[TABLE]

for all $p>0$.

Inductively, for every $m\in\mathbb{N}$, if $n\in\mathbb{N}$ and $\left(y_{i_{1}...i_{m}}\right)_{i_{1,}...,i_{m}=1}^{n}$ is an array of scalars, then for any choice of signs $\left(\varepsilon_{i_{1}}\right)_{i_{1}=1}^{n},\left(\varepsilon_{i_{2}}\right)_{i_{2}=1}^{n},...,\left(\varepsilon_{i_{m}}\right)_{i_{m}=1}^{n}\in\{-1,1\}^{n}$ we have

[TABLE]

for all $p>0$. Above and henceforth $I$ denotes the closed interval $[0,1]$.

Using duality and (2) we also get a similar upper bound in (1). In other words, Khintchine inequality shows that we can control the sum $\sum_{j=1}^{\infty}a_{j}r_{j}$ in any $L_{p}$ norm by the $\ell_{2}-$norm of the scalar sequence $\left(a_{j}\right)_{j=1}^{\infty}$.

Khintchine inequality is originally a result from probability, but it is also frequently used in Analysis and Topology (see [5, 7, 10, 11, 12, 13, 21]). The importance of the Rademacher functions and the Khintchine inequality in Functional Analysis lies mainly on the fact of its utility in the study of the geometry of Banach spaces (see [5, 10, 12]). Furthermore, the concern of the Rademacher functions in the theory of functional and trigonometric series and in the theory of Banach spaces is well known and it is commonly attributable to stochastic independence of the Rademacher functions. One of the main manifestations of this stochastic independence is, namely, the Khintchine inequality. Moreover, the Khintchine inequality (and also related results and its variants) is an important auxiliary result frequently used to prove results concerning to summability, specially the case in which $1\leq p\leq 2$ (see [10]).

The optimal constants $\mathrm{A}_{p}$ are known; obviously $\mathrm{A}_{p}=1,$ for $2\leq p<\infty,$ and by the other hand, Uffe Haagerup ([14]) proved that

[TABLE]

and

[TABLE]

The exact definition of $p_{0}$ is the following: $p_{0}\in(1,2)$ is the unique real number satisfying

[TABLE]

In this work we will show, among other results, an interesting connection between the Khintchine inequality and a famous inequality of Hardy and Littlewood.

For $p\geq 1$, we introduce the following notation: $X_{p}:=\ell_{p}$ and $X_{\infty}:=c_{0}$. From now on, $\left(e_{k}\right)_{k=1}^{\infty}$ denotes the sequence of canonical vectors in $X_{p}$. For any function $f$ we shall consider $f(\infty):=\lim_{s\rightarrow\infty}f(s)$ and for any $s>1$ we denote the conjugate index of $s$ by $s^{\ast},$ i.e., $\frac{1}{s}+\frac{1}{s^{\ast}}=1$. Furthermore, we denote $1^{\ast}$ by $\infty$.

Let $p,q\in[1,\infty]$. We recall that for a continuous bilinear form $T:X_{p}\times X_{q}\rightarrow\mathbb{K}$:

[TABLE]

where $T\left(e_{i},e_{j}\right)=a_{ij}$, for all $i,j\in\mathbb{N}.$

The Hardy–Littlewood inequality ([15], 1934) is a continuation of a famous work of Littlewood ([17], 1930) and can be stated as follows:

• •

[15, Theorems 2 and 4] If $p,q\in[2,\infty]$ are such that

[TABLE]

then there is a constant $C_{p,q}\geq 1$ such that

[TABLE]

for all continuous bilinear forms $A:X_{p}\times X_{q}\rightarrow\mathbb{K}$. Moreover the exponent $\frac{pq}{pq-p-q}$ is optimal.

• •

[15, Theorems 1 and 4] If $p,q\in[2,\infty]$ are such that

[TABLE]

then there is a constant $C_{p,q}\geq 1$ such that

[TABLE]

for all continuous bilinear forms $A:X_{p}\times X_{q}\rightarrow\mathbb{K}$. Moreover the exponent $\frac{4pq}{3pq-2p-2q}$ is optimal, where, for the case that $p$ and $q$ are simultaneously $\infty$, the optimal exponent is $4/3$.

As mentioned in [8], an unified version of the above two results of Hardy and Littlewood asserts that:

Theorem 1.2**.**

Let $p,q\in[2,\infty]$ be such that $\lambda:=\frac{pq}{pq-p-q}>0.$ There is a constant $C_{p,q}\geq 1$ such that

[TABLE]

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

Moreover the exponents $2$ and $\lambda$ are optimal, where, for the case that $p$ and $q$ are simultaneously $\infty$, the optimal exponent $\lambda$ is $1$.

The optimal constants of the previous inequalities are essentially unknown; these depend of the chosen scalar field. One of the few cases in which the optimal constants are known is the case of the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality (see [19, 20, 22]):

Theorem 1.3** **(Mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood

inequality).

Let $p\geq 2$. There is a constant $C_{p,\infty}$ such that

[TABLE]

for all continuous bilinear forms $T:X_{p}\times X_{\infty}\rightarrow\mathbb{R}$. Moreover, $2^{\frac{1}{2}-\frac{1}{p}}\leq C_{p,\infty}\leq\mathrm{A}_{\frac{p}{p-1}}$. Particularly, if $p_{0}$ is the number defined in (5) then the optimal constant $C_{p,\infty}$ is $2^{\frac{1}{2}-\frac{1}{p}}$, for all $p\geq\frac{p_{0}}{p_{0}-1}\approx 2.18006$.

The Hardy–Littlewood inequalities for bilinear forms in $\ell_{p}$ spaces were proved in 1934 [15]; the original proofs of Hardy and Littlewood rely on a result proved by Littlewood that, in general terms, is none other than the Kintchine inequality. The result, in modern mathematical notation, given by Littlewood is the following:

Theorem 1.4** ([17], pag. 170).**

Let $0<\rho<\infty$ be a real number. Then

[TABLE]

lies between two constants $A_{\rho}$ and $B_{\rho}$ (depending only on $\rho$), whatever is the scalar sequence $(a_{j})_{j=1}^{N}\in\mathbb{K}^{N}$ and whatever is the value of $N$.

The Hardy–Littlewood inequalities consist in optimal extensions of Littlewood’s $4/3$ inequality [17] (originally stated for $c_{0}$ spaces). In the last years the interest in this subject (which can be considered part of the theory of multiple summing and absolutely summing operators) was renewed and several authors became interested in this field ([1, 2, 3, 6, 7, 20, 22, 23]).

Our first aim in this work is to show that the Khintchine inequality and the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality are equivalent; it means that one can be obtained from the other one. This assertion not only works in the way we have stated, but as we will see later, the multiple Khintchine inequality and the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality (see [19]) are also equivalent.

The Khintchine inequality and the multiple Khintchine inequality are very useful tools in the theory of absolutely summing operators (even in its non linear extensions) and related classical inequalities. We stress, for instance, the striking advances in the estimates of the Hardy–Littlewood constants (see [3]). Our second aim in this work is to estimate the optimal constants in the multiple Khintchine inequality. We completely solve this issue in the Section 3 and as application, in the final section, we obtain the optimal constants of the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality, completing the recent estimates in [19, 20].

2. The Khintchine inequality is equivalent to the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality

In this section we prove the equivalence between two classical inequalities which have been mentioned on the last section. Moreover, we extract from this equivalence the equality $C_{p,\infty}=\mathrm{A}_{\frac{p}{p-1}}$, for all $p\in[2,\infty].$ This estimate complements, in the bilinear case, the paper [19]. We emphasize which the seminal ideas of this general equivalency was given in the book [4, Chapter 1].

Let us start showing how a proof of the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality can be made from the Kintchine inequality. This fact is known in this field and we shall include a short proof (following the ideas of [19, Theorem 1.2]) for the sake of completeness.

Henceforth, for all $n\in\mathbb{N}$, $X_{p}^{n}$ will denote the finite dimensional Banach space $\mathbb{R}^{n}$ endowed with the $\ell_{p}$ norm.

Theorem 2.1** **(Mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood

inequality).

Let $p\geq 2$. There is a constant $C_{p,\infty}$ such that

[TABLE]

for all continuous bilinear forms $T:X_{p}\times X_{\infty}\rightarrow\mathbb{R}$. Moreover, $C_{p,\infty}\leq\mathrm{A}_{p^{\ast}}$.

Proof.

Let $n$ be a positive integer and $T:X_{p}^{n}\times X_{\infty}^{n}\rightarrow\mathbb{R}$ be a bilinear form. Then, invoking the Khintchine inequality, we have

[TABLE]

As follows, the theorem is proved and $C_{p,\infty}\leq\mathrm{A}_{p^{\ast}}.$ ∎

The following well-known lemma, credited to Hermann Minkowski, will be useful along this paper (see [12, Corollary 5.4.2]):

Lemma 2.2**.**

For $0<p<q<+\infty$, and any sequence of scalars $\left(a_{ij}\right)_{i,j\in\mathbb{N}}$ we have

[TABLE]

Now, let us recall that using Lemma 2.2, the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality implies the next result:

Theorem 2.3**.**

Let $p\geq 2$. There is a constant $D_{p,\infty}$ such that

[TABLE]

for all continuous bilinear forms $T:X_{p}\times X_{\infty}\rightarrow\mathbb{R}$. Moreover, $D_{p,\infty}\leq C_{p,\infty}.$

Proof.

It is enough to note that $\frac{p}{p-1}\leq 2$, thus by Lemma 2.2 and the Theorem 2.1 we obtain

[TABLE]

Then, the theorem is proved with the estimate $D_{p,\infty}\leq C_{p,\infty}.$ ∎

In this way, we have proved that Khintchine inequality implies Theorem 2.1 as well as the Theorem 2.1 implies Theorem 2.3. Furthermore, $D_{p,\infty}\leq C_{p,\infty}\leq\mathrm{A}_{p^{\ast}}.$ Then, if Theorem 2.3 implies the Khintchine inequality, we will have that the mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality and the Khintchine inequality are equivalent. That is the assertion in the following:

Theorem 2.4**.**

Theorem (2.3) implies Khintchine inequality (the case in which $1\leq p\leq 2$). Moreover, $\mathrm{A}_{p}\leq D_{p^{\ast},\infty}$.

Proof.

Let $p\in\left[1,2\right]$, $n\in\mathbb{N}$, and $\left(a_{j}\right)_{j=1}^{n}$ a scalar sequence.

By (3), we know that for any choice of signs $\left(\varepsilon_{j}\right)_{j=1}^{n}\in\{-1,1\}^{n}$ we have

[TABLE]

Now, solving the integral on the right hand we obtain

[TABLE]

where each $\delta_{j}^{\left(i\right)}$ is $1$ or $-1$, for all $i,j$.

Hence, if $\left(x_{i}\right)_{i=1}^{\infty}\in B_{X_{p^{\ast}}}$, by Hölder’s inequality and (6) we have

[TABLE]

Now, define the bilinear form $A:X_{p^{\ast}}^{2^{n}}\times X_{\infty}^{n}\rightarrow\mathbb{R}$, such that $A\left(e_{i},e_{j}\right)=\left(\frac{1}{2}\right)^{\frac{n}{p}}a_{j}\delta_{j}^{\left(i\right)}$, for $1\leq j\leq n$, $1\leq i\leq 2^{n}$. Clearly $A$ is bounded and thus

[TABLE]

where, the last inequality stands by (7).

As follows, the theorem is proved and $\mathrm{A}_{p}\leq D_{p^{\ast},\infty}$. ∎

Note that we have proved that for all $p\in\left[1,2\right]$

[TABLE]

or, equivalently, for all $p\in\left[2,\infty\right]$

[TABLE]

As announced, this recovers and completes the estimates of the papers [19, 20] for the bilinear case.

3.

Optimal constants in the multiple Khintchine inequality

There are several extensions of the Khintchine inequality, some of them deal with higher dimensions. In this section we will deal with a very important extension, the so-called Khintchine inequality for multiple sums, or multiple Khintchine inequality, (see [9, pag. 455] and the references therein):

Theorem 3.1**.**

Let $0<r<\infty$, $m\in\mathbb{N}$, and $\left(y_{i_{1}...i_{m}}\right)_{i_{1,}...,i_{m}=1}^{N}$ an array of scalars. There is a constant $K_{m,r}\geq 1$, such that

[TABLE]

for all $N\in\mathbb{N}$, where $r_{i_{j}}\left(t_{j}\right)$ are denoting the Rademacher functions, for all $j\in\left\{1,...,m\right\}$ and $i_{j}\in\left\{1,...,N\right\}.$ Moreover, $K_{m,r}\leq\left(\mathrm{A}_{r}\right)^{m}$.

In [9] was given the details for $m=2$, but these arguments contain all the elements of the general case. For the sake of completeness, we give an elementary proof:

Proof.

The proof will be obtained by induction in $m$. The case $m=1$ is exactly the Khintchine inequality. Let us start the proof for the case $0<r\leq 2$. Assume inductively the result holds for $m-1$, then

[TABLE]

Since $\frac{2}{r}\geq 1$, using the Minkowski integral inequality we get

[TABLE]

From (9) and (10), we have

[TABLE]

Now, Khintchine inequality furnishes us the inequality

[TABLE]

thus

[TABLE]

and by Fubini’s theorem

[TABLE]

On the other hand, since particularly we have proved

[TABLE]

the assertion in the case $2<r<\infty$ follows trivially by the norm inclusion between the $L_{p}$ spaces. ∎

Theorem 3.1 has important applications, for example, in multiple summing operator theory. In fact, a frequent application of the Theorem 3.1 can be traced in modern proofs of the Hardy–Littlewood inequalities (see [1, 3, 20]).

Recently, Pellegrino et.al. (see [21, Proposition 1]) showed that for all $m\in\mathbb{N}$, $K_{m,1}=\left(\sqrt{2}\right)^{m}=\left(\mathrm{A}_{1}\right)^{m}$. Our goal in this section is to prove that, in fact, the optimal constants $K_{m,r}$ are $\left(\mathrm{A}_{r}\right)^{m}$, for all $m\in\mathbb{N}$ and for all $0<r<\infty.$ In order to achieve it, we need an auxiliary result proved in [24, Proposition 1]:

Proposition 3.2**.**

Let $0<r<\infty$, $Z$ be a Banach space, $m,N_{1},...,N_{m}\in\mathbb{N}$, and $\left(z_{i_{1}...i_{m}}\right)_{i_{1,}...,i_{m}=1}^{N_{1},...,N_{m}}\subset Z$. Then

[TABLE]

where $\eta^{\left(j\right)}=\left(\eta_{1}^{\left(j\right)},...,\eta_{N_{j}}^{\left(j\right)}\right)\in D_{N_{j}}$ for all $j\in\left\{1,...,m\right\}$, and for all $n\in\mathbb{N}$ the set $D_{n}:=\left\{-1,1\right\}^{n}$.

Another result we will use, due to Haagerup in ([14]), involving the Central Limit Theorem is:

Lemma 3.3**.**

Let $r\in\left(0,2\right)$. For each $n\in\mathbb{N}$ consider $\left(a_{1},...,a_{n}\right)=\frac{1}{\sqrt{n}}\left(1,...,1\right)$. Then

[TABLE]

where $r_{j}\left(t\right)$ denotes the $j-$th Rademacher function for all $j$.

Theorem 3.4**.**

For all $0<r<\infty$ and $m\in\mathbb{N},$ the optimal constant $K_{m,r}$ in (8) is $\left(\mathrm{A}_{r}\right)^{m}$.

Proof.

From Theorem 3.1 we already know that $K_{m,r}\leq\left(\mathrm{A}_{r}\right)^{m}$ for all $0<r<\infty$ and $m\in\mathbb{N}$, so we only need to check the other inequality.

Obviously $K_{m,r}\geq\left(\mathrm{A}_{r}\right)^{m}=\left(1\right)^{m}=1,$ for all $2\leq r<\infty$. Let us to prove the case $0<r<2$. Firstly, let $0<r\leq p_{0}$, where $p_{0}$ is the number defined in (5). We start the proof by showing the case $m=2$, i.e., $K_{2,r}\geq\left(2^{\frac{1}{r}-\frac{1}{2}}\right)^{2}=2^{\frac{2-r}{r}}$.

If we take

[TABLE]

we have

[TABLE]

On the other side, from Proposition 3.2 we have

[TABLE]

In that case

[TABLE]

For the general case $m$, we consider

[TABLE]

Then,

[TABLE]

and

[TABLE]

In this way

[TABLE]

i.e., the assertion is proved for $0<r\leq p_{0}$ and $m\in\mathbb{N}$.

On the other hand, let $r\in\left(p_{0},2\right)$ and $m\in\mathbb{N}$. For each $N\in\mathbb{N}$ consider

[TABLE]

We have

[TABLE]

and, at the same time, from Proposition 3.2 we have

[TABLE]

where $\left(a_{1},...,a_{N}\right)=\frac{1}{\sqrt{N}}\left(1,...,1\right)$. Using the Lemma 3.3, we have

[TABLE]

Thus, asymptotically we obtain

[TABLE]

for $r\in\left(p_{0},2\right)$ and $m\in\mathbb{N}$. ∎

4. Application: Optimal constants in the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality

The objective in this section is to prove that the multiple Khintchine inequality is equivalent to the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality (see [19]). As application of the Section 3, we obtain the optimal constants of the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality, recovering and ending completely the recent estimates obtained in [19, 20].

Let us start this section using the Khintchine inequality for multiple sums in order to prove the following result, also called mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality (or multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality) (see [19]). Mimicking the proof of Theorem 2.1 we obtain:

Theorem 4.1** **(Multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood

inequality).

Let $M\geq 3$ be a positive integer and $p\geq 2$. There is an optimal constant $C_{(M),p}$ such that

[TABLE]

for all continuous $M$-linear forms $T:X_{p}\times X_{\infty}\times\dots\times X_{\infty}\rightarrow\mathbb{R}$. Moreover, $C_{(M),p}\leq\left(\mathrm{A}_{p^{\ast}}\right)^{M-1}$.

Proof.

Let $N$ be a positive integer and $T:X_{p}^{N}\times X_{\infty}^{N}\times\dots\times X_{\infty}^{N}\rightarrow\mathbb{R}$ be a continuous $M$-linear form. By the Theorem 3.1 we know that

[TABLE]

and the proof is done, with estimates $C_{(M),p}\leq\left(\mathrm{A}_{\frac{p}{p-1}}\right)^{M-1}$. ∎

In [19], the authors obtain the estimate $\left(2^{\frac{1}{2}-\frac{1}{p}}\right)^{M-1}\leq C_{(M),p}\leq\left(\mathrm{A}_{\frac{p}{p-1}}\right)^{M-1}$, for all $p\geq 2$. Particularly, for all $p\geq\frac{p_{0}}{p_{0}-1}\approx 2.18006$, $C_{(M),p}=\left(\mathrm{A}_{\frac{p}{p-1}}\right)^{M-1}.$

It occurs to us to ask about the relation between the Theorem 3.1 and Theorem 4.1. We are going to see that, as occurs in the classical inequalities in the Section 2, these two theorems are also equivalent. Moreover, we will prove that for all $p\geq 2$, $C_{(M),p}=\left(\mathrm{A}_{\frac{p}{p-1}}\right)^{M-1},$ recovering and completing the estimates of the papers [19, 20] for the multilinear case.

We saw that multiple Khintchine inequality implies Theorem 4.1. On the other hand, proceeding as in Theorem 2.3 and using Lemma 2.2 several times, from Theorem 4.1 it is not difficult to prove that in fact we have:

Theorem 4.2**.**

Let $M\geq 3$ be a positive integer and $p\geq 2$. There is an optimal constant $D_{(M),p}$ such that

[TABLE]

for all continuous $M$-linear forms $T:X_{p}\times X_{\infty}\times\dots\times X_{\infty}\rightarrow\mathbb{R}$. Moreover, $D_{(M),p}\leq C_{(M),p}$.

Our goal is to prove that this theorem implies the multiple Khintchine inequality. In order to achieve it, let us recall that for a continuous $m-$linear form $T:X_{p_{1}}\times\cdots\times X_{p_{m}}\rightarrow\mathbb{R},$

[TABLE]

where $p_{1},...,p_{m}\in[1,\infty]$ and $T\left(e_{i_{1}},...,e_{i_{m}}\right)=a_{i_{1}\cdots i_{m}}$, for all $i_{j}\in\mathbb{N}.$

Theorem 4.3**.**

Theorem 4.2 implies multiple Khintchine inequality for $p\in[1,2]$. Moreover, $\left(\mathrm{A}_{p}\right)^{m-1}\leq D_{(m),p^{\ast}}$.

Proof.

The proof uses the ideas of the proof of the Theorem 2.1. We are going to give the details for the case $m=2$. The proof is not different in the general case, merely more notationally complicated.

Let $p\in\left[1,2\right]$, $n\in\mathbb{N}$, and $\left(y_{jk}\right)_{j,k=1}^{n}$ an array of scalars. By (4), for any choice of signs $\left(\varepsilon_{j}\right)_{j=1}^{n},\left(\lambda_{k}\right)_{k=1}^{n}\in\{-1,1\}^{n}$ we have

[TABLE]

Now, by Proposition 3.2, solving the integral on the right hand and rewriting and ordering the indexes we have

[TABLE]

where each $a_{jk}^{\left(i\right)}$ is $1$ or $-1.$

Hence, if $\left(x_{i}\right)_{i=1}^{\infty}\in B_{X_{p^{\ast}}}$ by Hölder’s inequality it holds

[TABLE]

Define the $3-$linear form $A:X_{p^{\ast}}^{2^{2n}}\times X_{\infty}^{n}\times X_{\infty}^{n}\rightarrow\mathbb{R}$, such that $A\left(e_{i},e_{j},e_{k}\right)=\left(\frac{1}{2}\right)^{\frac{2n}{p}}y_{jk}a_{jk}^{\left(i\right)}$, for $1\leq j,k\leq n$, $1\leq i\leq 2^{2n}$. Clearly $A$ is bounded, then by Theorem 4.2 ($M=3$) and 12:

[TABLE]

As follows, the multiple Khintchine inequality is proved for $m=2$, and $\left(\mathrm{A}_{p}\right)^{2}\leq D_{(3),p^{\ast}}$.

The proof of the general case, $m>2$, on the multiple Khintchine inequality will follow from Theorem 4.2, considering the case $M=m+1,$ with estimate

[TABLE]

as asserted. ∎

In this way, note that we have proved that the multiple Khintchine inequality is equivalent to the multilinear mixed $\left(\ell_{\frac{p}{p-1}},\ell_{2}\right)$-Littlewood inequality and, in general, for all $p\in\left[1,2\right]$ and $m\geq 2$

[TABLE]

or, equivalently, for all $p\in\left[2,\infty\right]$ and $m\geq 2$

[TABLE]

As announced, this recovers and completes the estimates of the papers [19, 20] for the general case.