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The real anisotropic Littlewood's $4 / 3$ inequality is an extension of a famous result obtained in 1930 by J. E. Littlewood. It asserts that, for $a , b \in ( 0 , \infty )$, the following conditions are equivalent: $\bullet$ There is an optimal constant $\mathsf{L}_{ a , b }^{ \mathbb{R} } \in [ 1 , \infty )$ such that \[ \Biggl ( \, \sum_{ k = 1 }^{ \infty } \biggl ( \, \sum_{ j = 1 }^{ \infty } \bigl \lvert A \bigl ( \boldsymbol{e}^{ (k) } , \boldsymbol{e}^{ (j) } \bigr ) \bigr \rvert^a \biggr )^{ \frac{b}{a} } \Biggr )^{ \frac{1}{b} } \leq \mathsf{L}_{ a , b }^{ \mathbb{R} } \cdot \lVert A \rVert \] for every continuous bilinear form $A \colon c_0 \times c_0 \to \mathbb{R}$. $\bullet$ The values $a , b$ satisfy $a , b \geq 1$ and $\frac{1}{a} + \frac{1}{b} \leq \frac{3}{2}$. Several authors have obtained the values of $\mathsf{L}_{ a , b }^{ \mathbb{R} }$ for diverse pairs $( a , b )$. In this paper we provide the complete list of such optimal values, as well as new estimates for $\mathsf{L}_{ a , b }^{ \mathbb{C} }$ (the analog for continuous $\mathbb{C}$-bilinear forms), which are exact in several cases. As an application we prove, in terms of the values $\mathsf{L}_{ 1 , r }^{ \mathbb{C} }$, a variant of Khinchin's inequality for Steinhaus variables, and we determine all the optimal $( q , s )$-cotype constants of the spaces $\ell_1 ( \mathbb{K} )$ (with $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$) in terms of the values $\mathsf{L}_{ 1 , q }^{ \mathbb{R} }$., Comment: Corrected typos and improved presentation and formatting. Added the section about cotype

Kwapie\'{n}'s theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{p}$ is absolutely $\left( r,1\right) $-summing for $1/r=1-\left\vert 1/p-1/2\right\vert .$ When $p=2$ it recovers the famous Grothendieck's theorem. In this paper investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapie\'{n}'s and Grothendieck's results that encompasses the cases of multiple summing and absolutely summing multilinear operators., Comment: 10 pages

The investigation of regularity/summability properties of the coefficients of bilinear forms in sequence spaces was initiated by Littlewood in $1930$. Nowadays, this topic has important connections with other fields of Pure and Applied Mathematics as Complex Analysis, Quantum Information Theory, Theoretical Computer Science and Combinatorial Games. In this paper we explore a regularity technique to obtain optimal parameters for several results in this framework, extending/generalizing theorems of Osikiewicz and Tonge ($2001$), Albuquerque \textit{et al.} ($2016$), Aron \textit{et al.} ($2017$), Albuquerque and Rezende ($2018$), Paulino ($2020$), among others.

The Orlicz $\left( \ell_{2},\ell_{1}\right) $-mixed inequality states that $$ \left( \sum_{j_{1}=1}^{n}\left( \sum_{j_{2}=1}^{n}\left\vert A(e_{j_{1} },e_{j_{2}})\right\vert \right) ^{2}\right) ^{\frac{1}{2}}\leq\sqrt {2}\left\Vert A\right\Vert $$ for all bilinear forms $A:\mathbb{K}^{n}\times\mathbb{K}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$, where $\mathbb{K}^{n}$ denotes $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with $\mathbb{K}^{n}$ endowed with $\ell_{p}$ norms for all $p\in\lbrack1,\infty].$

The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $\left\{ p_{1},...,p_{m}\right\} $ intercepts both $[2,\infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors., Comment: 7 pages

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}., Comment: 16 pages

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type $(\ell_{1},\ell_{2})$ theorems for multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy--Littlewood inequality., Comment: 32 pages, 1 figure

The Hardy--Littlewood inequalities on $\ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $\ell _{p}$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for $q_{1},...,q_{m}>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $\cot Y$, if \begin{equation*} \frac{1}{p_{1}}+...+\frac{1}{p_{m}}<\frac{1}{\cot Y}, \end{equation*} then the following assertions are equivalent: (a) There is a constant $C_{p_{1},...,p_{m}}^{Y}\geq 1$ such that \begin{equation*} \left( \sum_{j_{1}=1}^{\infty }\left( \sum_{j_{2}=1}^{\infty }\cdots \left( \sum_{j_{m}=1}^{\infty }\left\Vert A(e_{j_{1}},...,e_{j_{m}})\right\Vert ^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}}\cdots \right) ^{\frac{q_{1}}{q_{2}} }\right) ^{\frac{1}{q_{1}}}\leq C_{p_{1},...,p_{m}}^{Y}\left\Vert A\right\Vert \end{equation*} for all continuous $m-$linear operators $A:\ell _{p_{1}}\times \cdots \times \ell _{p_{m}}\rightarrow Y.$ (b) The exponents $q_{1},...,q_{m}$ satisfy \begin{equation*} q_{1}\geq \lambda _{m,\cot Y}^{p_{1},...,p_{m}},q_{2}\geq \lambda _{m-1,\cot Y}^{p_{2},...,p_{m}},...,q_{m}\geq \lambda _{1,\cot Y}^{p_{m}}, \end{equation*} where, for $k=1,...,m,$ \begin{equation*} \lambda _{m-k+1,\cot Y}^{p_{k},...,p_{m}}:=\frac{\cot Y}{1-\left( \frac{1}{ p_{k}}+...+\frac{1}{p_{m}}\right) \cot Y}. \end{equation*} As an application of the above result we generalize to the $m$-linear setting one of the classical Hardy--Littlewood inequalities for bilinear forms. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums., Comment: 20 pages

In this paper we obtain the sharp estimates for the mixed $\left( \ell_{1},\ell _{2}\right) $-Littlewood inequality for real scalars with exponents $\left(2,1,2,2....,2\right) .$ These results are applied to find sharp estimates for the constants of a family of $3$-linear Bohnenblust--Hille inequalities with multiple exponents.

We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araujo et al. We also obtain the optimal constants of the generalized Hardy--Littlewood inequality in some special cases., Comment: The title was modified. The computer-assisted part of the previous version is now fixed and this new version has a new section which corresponds to the arXiv submission 1504.04207. The arXiv submission 1504.04207 does not exist as an independent paper and now is part of the present paper

We investigate the growth of the constants of the polynomial Hardy-Littlewood inequality.

Bayart, Pellegrino and Seoane recently proved that the polynomial Bohnenblust--Hille inequality for complex scalars is subexponential. We show that a vector valued polynomial Bohnenblust-Hille inequality on complex Banach lattices is also subexponential for some special cases. Our main result result recovers the best known constants of the classical polynomial inequality provided in \cite{bps}.

We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sep\'ulveda. More precisely, as a consequence of our main result, for $1\leq t<2$ and $m\in \mathbb{N}$ we prove that $$ \left(\sum_{i_{1},\dots,i_{m}=1}^{\infty}\left\vert U\left(e_{i_{1}},\dots,e_{i_{m}}\right) \right\vert^{\frac{2tm}{2+(m-1)t}}\right)^{\frac{2+(m-1)t}{2tm}} \leq \left[\prod_{j=2}^{m}\Gamma \left(2-\frac{2-t}{jt-2t+2}\right) ^{\frac{t(j-2)+2}{2t-2jt}}\right] \left\Vert U\right\Vert $$ for all complex $m$-linear forms $U:c_{0}\times \cdot \cdot \cdot \times c_{0}\rightarrow \mathbb{C}$.

In this paper we introduce an abstract approach to the notion of absolutely summing multilinear operators. We show that several previous results on different contexts (absolutely summing, almost summing, Cohen summing) are particular cases of our general results.

The upper estimates for the optimal constants of the multilinear Bohnenblust--Hille inequality obtained in [J. Funct. Anal. 264 (2013), 429--463] are here improved to: {0.1cm} {enumerate} For real scalars: $K_{n}\leq\sqrt{2}(n-1)^{0.526322}$. For complex scalars: $K_{n}\leq\frac{2}{\sqrt{\pi}}(n-1)^{0.304975}$.{enumerate} {0.1cm} \noindent We also obtain sharper estimates for higher values of $n$. For instance, \[ K_{n}<1.30379(n-1) ^{0.526322}\] for real scalars and $n>2^{8}$ and \[ K_{n}<0.99137(n-1) ^{0.304975}\] for complex scalars and $n > 2^{15}.$

The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form $U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and the positive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponential growth. Very recently, a new panorama emerged and the importance, for many applications, of the knowledge of the optimal constants (denoted by $(K_{n})_{n=1}^{\infty}$) was stressed. The title of this paper is part of our Fundamental Lemma, one of the novelties presented here. It brings surprising new (and precise) information on the optimal constants (for both real and complex scalars). For instance, [K_{n+1}-K_{n}<\frac{0.87}{n^{0.473}}] for infinitely many $n$'s. In the case of complex scalars we present a curious formula, where $\pi,e$ and the famous Euler--Mascheroni constant $\gamma$ appear together: [K_{n}<1+(\frac{4}{\sqrt{\pi}}(1-e^{\gamma/2-1/2}) {\sum\limits_{j=1}^{n-1}}j^{^{\log_{2}(e^{-\gamma/2+1/2}) -1}%})] for all $n\geq2$. Numerically, the above formula shows a surprising low growth, [K_{n}<1.41(n-1)^{0.305}-0.04] for every integer $n \geq2$. We also provide a brief discussion on the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities., Comment: An Appendix was added

For the scalar field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, the multilinear Bohnenblust--Hille inequality asserts that there exists a sequence of positive scalars $(C_{\mathbb{K},m})_{m=1}^{\infty}$ such that %[(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}}%,...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{\mathbb{K},m}\sup_{z_{1},...,z_{m}\in\mathbb{D}^{N}}|U(z_{1},...,z_{m})|] for all $m$-linear form $U:\mathbb{K}^{N}\times...\times\mathbb{K}% ^{N}\rightarrow\mathbb{K}$ and every positive integer $N$, where $(e_{i})_{i=1}^{N}$ denotes the canonical basis of $\mathbb{K}^{N}$ and $\mathbb{D}^{N}$ represents the open unit polydisk in $\mathbb{K}^{N}$. Since its proof in 1931, the estimates for $C_{\mathbb{K},m}$ have been improved in various papers. In 2012 it was shown that there exist constants $(C_{\mathbb{K},m})_{m=1}^{\infty}$ with subexponential growth satisfying the Bohnenblust-Hille inequality. However, these constants were obtained via a complicated recursive formula. In this paper, among other results, we obtain a closed (non-recursive) formula for these constants with subexponential growth.