Unconditionality for $m$-homogeneous polynomials on $\ell _{\infty }^{n}$

Andreas Defant¹ ; Pablo Sevilla-Peris²

¹Institut für Mathematik Universität Oldenburg D-26111 Oldenburg, Germany
²Instituto Universitario de Matemática Pura y Aplicada Universitat Politècnica de València 46022 Valencia, Spain

Studia Mathematica, Tome 232 (2016) no. 1, pp. 45-55

Résumé

Let $\chi(m,n)$ be the unconditional basis constant of the monomial basis $z^\alpha $, $\alpha \in \mathbb{N}_0^n$ with $|\alpha|=m$, of the Banach space of all $m$-homogeneous polynomials in $n$ complex variables, endowed with the supremum norm on the $n$-dimensional unit polydisc $\mathbb{D}^n$. We prove that the quotient of $\sup_m\sqrt[m]{\sup_m\chi(m,n)}$ and $\sqrt{{n/\!\log n} }$ tends to $1$ as $n\to\infty$. This reflects a quite precise dependence of $\chi(m,n)$ on the degree $m$ of the polynomials and their number $n$ of variables. Moreover, we give an analogous formula for $m$-linear forms, a reformulation of our results in terms of tensor products, and as an application a solution for a problem on Bohr radii.

Détail

DOI : 10.4064/sm8386-2-2016

Keywords: chi unconditional basis constant monomial basis nbsp alpha alpha mathbb alpha banach space m homogeneous polynomials complex variables endowed supremum norm n dimensional unit polydisc nbsp mathbb prove quotient sup sqrt sup chi sqrt log tends infty reflects quite precise dependence chi degree polynomials their number variables moreover analogous formula m linear forms reformulation results terms tensor products application solution problem bohr radii