Summability of multilinear mappings: Littlewood, Orlicz and beyond

TL;DR

This paper advances the understanding of summability in multilinear mappings between Banach spaces by extending classical theorems, establishing new summability properties, and exploring their relationships with properties like Littlewood-Orlicz.

Contribution

It introduces new summability results for multilinear mappings, including an extension of Littlewood's theorem and analysis of the Littlewood-Orlicz property.

Findings

Every continuous n-linear form on the disc algebra is (1;2,...,2)-summing.

The Littlewood-Orlicz property plays a key role in summability theory.

Connections between almost summing multilinear mappings and classical summability are established.

Abstract

In this paper we prove a plenty of new results concerning summabililty properties of multilinear mappings between Banach spaces, such as an extension of Littlewood's 4/3 Theorem. Among other features, it is shown that every continuous n-linear form on the disc algebra or the Hardy space is (1;2,...,2)-summing, the role of the Littlewood-Orlicz property in the theory is established and the interplay with almost summing multilinear mappings is explored.