Summability of the coefficients of a multilinear form

Abstract

Let $T$ be an $m$-linear form defined on a product of ${\ell }_p$-spaces and let $\Lambda \subset {\mathbb{N}}^m$. We investigate the best exponent $s$ such that the sequence of coefficients of $T$ belongs to ${\ell }_s(\Lambda )$. The cases $\Lambda ={\mathbb{N}}^m$ and $\Lambda$ is the diagonal of ${\mathbb{N}}^m$ are already known. We study the intermediate cases using notions like combinatorial dimension of sets, multiple summing maps and random polynomials.