Lower bounds for the complex polynomial Hardy--Littlewood inequality

Abstract

The Hardy--Littlewood inequality for complex homogeneous polynomials asserts that given positive integers $m\ge 2$ and $n\ge 1$, if $P$ is a complex homogeneous polynomial of degree $m$ on ${\ell }_p^n$ with $2m\le p\le \infty$ given by $P(x_1,\dots ,x_n)={\sum }_{|\alpha |=m}{a}_{\alpha }{\mathbf{x}}^{\alpha }$, then there exists a constant $C_{\mathbb{C},m,p}^{\mathrm{pol}}\ge 1$ (which is does not depend on $n$) such that \[ \left( {\sum\limits_{\left\vert α\right\vert =m}}\left\vert a_{α}\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq C_{\mathbb{C},m,p}^{\mathrm{pol}}\left\Vert P\right\Vert , \] with $\Vert P\Vert :={\sup }_{z\in B_{{\ell }_p^n}}|P(z)|$. In this short note, among other results, we provide nontrivial lower bounds for the constants $C_{\mathbb{C},m,p}^{\mathrm{pol}}$. For instance we prove that, for $m\ge 2$ and $2m\le p<\infty$, \[ C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m}{p}}% \] for $m$ even, and \[ C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m-1}{p}}% \] for $m$ odd. Estimates for the case $p=\infty$ (this is the particular case of the complex polynomial Bohnenblust--Hille inequality) were recently obtained by D. Nuñez-Alarcón in 2013.

References in corpus (1)

• On the upper bounds for the constants of the Hardy-Littlewood inequality

Cited by in corpus (2)

• A new estimate for the constants of an inequality due to Hardy and Littlewood
• Some applications of the Regularity Principle in sequence spaces