Optimal estimates for summing multilinear operators

Abstract

We show that given a positive integer $m$, a real number $p\in \left[2,\infty \right)$ and $1\le s<p^{\ast }$ 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 $r\ge \frac{2ms}{s+2m-ms}$ 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.}

References in corpus (1)

• Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators

Cited by in corpus (2)

• Optimal Hardy-Littlewood type inequalities for $m$-linear forms on ${\ell }_p$ spaces with $1\le p\le m$
• A note on multiple summing operators and applications