Unconditionality for $m$-homogeneous polynomials on $\ell _{\infty }^{n}$
Volume 232 / 2016
Studia Mathematica 232 (2016), 45-55
MSC: Primary 46G25, 46B04; Secondary 46B28.
DOI: 10.4064/sm8386-2-2016
Published online: 10 March 2016
Abstract
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.
