相关论文: A note on the hypercontractivity of the polynomial…
The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$…
In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial $\sum_{|\alpha| = m} a_\alpha z^\alpha$ on $\C^n$ the $\ell^{\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a constant $C_m$ (depending only on…
The power $\frac{2m}{m+1}$ in the polynomial (and multilinear) Bohnenblust--Hille inequality is optimal. This result is well-known but its proof highly nontrivial. In this note we present a quite simple proof of this fact.
In this paper, among other results, we improve the best known estimates for the constants of the generalized Bohnenblust-Hille inequality. These enhancements are then used to improve the best known constants of the Hardy--Littlewood…
The main motivation of this paper is the following open problem: Is the hypercontractivity of the \emph{complex} polynomial Bohnenblust--Hille inequality an optimal result? We show that the solution to this problem has a close connection…
Recently, in paper published in the Annals of Mathematics, it was shown that the Bohnenblust-Hille inequality for (complex) homogeneous polynomials is hypercontractive. However, and to the best of our knowledge, there is no result providing…
A classical inequality due to Bohnenblust and Hille states that for every positive integer $m$ there is a constant $C_{m}>0$ so that $$(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\frac{2m}{m+1}})…
A classical inequality due to Bohnenblust and Hille states that for every $N \in \mathbb{N}$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$ we have…
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…
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…
The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for…
In this note, among other results, we find the optimal constants of the generalized Bohnenblust--Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $\left( 1,2,...,2\right) $, sometimes called mixed $\left(…
We provide (for both the real and complex settings) a family of constants, $% (C_{m})_{m\in \mathbb{N}}$, enjoying the Bohnenblust--Hille inequality and such that $\displaystyle\lim_{m\rightarrow \infty}\frac{C_{m}}{C_{m-1}}=1$, i.e., their…
It was recently proved by Bayart et al. that the complex polynomial Bohnenblust--Hille inequality is subexponential. We show that, for real scalars, this does no longer hold. Moreover, we show that, if $D_{\mathbb{R},m}$ stands for the real…
The Hardy--Littlewood inequality for $m$-homogeneous polynomials on $\ell_{p}$ spaces is valid for $p>m.$ In this note, among other results, we present an optimal version of this inequality for the case $p=m.$ We also show that the optimal…
We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for $m$--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong…
The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $N$ and every $m$-linear mapping $T:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow \mathbb{R}$ one…
Let $(K_{n})_{n=1}^{\infty}$ be the optimal constants satisfying the multilinear (real or complex) Bohnenblust--Hille inequality. The exact values of the constants $K_{n}$ are still waiting to be discovered since eighty years ago; recently,…
In 2015, using an innovative technique, Carando, Defant and Sevilla-Peris succeeded in proving a Bohnenblust--Hille type inequality with constants of polynomial growth in $m$ for a certain family of complex $m$-homogeneous polynomials. In…
Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r}…