Optimal estimates for summing multilinear operators

Abstract

We show that given a positive integer m, a real number p ∈ [2, ∞) and 1 ≤ s < p * the set of non-multiple (r; s)-summing m-linear forms on p × · · · × p contains, except for the null vector, a closed subspace of maximal dimension whenever r < 2ms s+2m−ms . This result is optimal since for r ≥ 2ms s+2m−ms all m-linear forms on p × · · · × p are multiple (r; s)-summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botelho et al.The classical Bohnenblust-Hille inequality [7], in the modern terminology, can be stated in terms of multiple summing operators, as remarked in [23] (see also [10]):

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

“…(ℓ p * 0 , ℓ p * 1 ); (X 0 , X 1 ) and (3) (ℓ p * 0 , ℓ p * 1 ); (ℓ 1 , ℓ 1 ) satisfy the injective θ-property for any 0 < θ < 1. For the family (1) we have…”

“…, X m and all r ≥ 1. In [3], it was observed that this is optimal in the following sense: L( m c 0 , . .…”

A note on multiple summing operators and applications

“…Theorem 1 is in some sense optimal. In fact it was recently proved in [4] that the Defant-Voigt Theorem is optimal in the following sense: every continuous m-linear form is absolutely (1; 1, ..., 1)summing and this result can not be improved to (p; 1, ..., 1)-summing with p < 1.…”

An index of summability for pairs of Banach spaces

“…We shall need the following coincidence theorem that can be essentially found in [2,4] to obtain estimates of the indices of summabilitity along this paper.…”