Unimodular multilinear forms on sequence spaces and summability principles

Abstract

This work is divided into two parts. Initially, we investigate the existence of unimodular (complex or real) forms with relatively small norms on (Fórmula) spaces. We improve the sup norm upper estimate in Kahane–Salem–Zygmund inequality for multilinear forms: given positive (Fórmula) there exists an m-linear map (Fórmula) where (Fórmula) is a constant. This norm estimate is used to offer a definitive answer to the asymptotic behavior from infimum of norms of unimodular forms on (Fórmula). The second part concerns the summability of operators and related subjects. We apply a recent technique introduced by Pellegrino et al. [56] to obtain an improved regularity principle on sequence spaces and an inclusion theorem for summing operators. Next, we deal with a general summability notion (Λ–summing operators, see

Chapter 3) that unifies the multiple and absolutely summing operators. A general inclusion theorem that encompasses the correspondent result on each class

is provided. We also obtain applications to the multilinear Hardy–Littlewood inequality in both contexts.