Optimal estimates for summing multilinear operators

TL;DR

This paper characterizes the maximal dimension of certain non-multiple summing multilinear forms on ℓₚ spaces, establishing optimal bounds and generalizing previous results related to cotype.

Contribution

It provides an optimal characterization of non-multiple summing multilinear forms on ℓₚ spaces, extending known results and identifying precise thresholds for summing properties.

Findings

Identifies the maximal dimension of non-multiple summing forms for given parameters.

Establishes the optimality of the summing bounds.

Generalizes a 2010 result related to cotype.

Abstract

We show that given a positive integer $m$, a real number $p\in\left[ 2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left( r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$ contains, except for the null vector, a closed subspace of maximal dimension whenever $r<\frac{2ms}{s+2m-ms}$. This result is optimal since for $r\geq\frac{2ms}{s+2m-ms}$ all $m$--linear forms on $\ell_{p}\times \cdots\times\ell_{p}$ are multiple $\left( r;s\right)$--summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botelho \textit{et al.}