Unified Grothendieck’s and Kwapień’s Theorems for Multilinear Operators

Abstract

Kwapień’s theorem asserts that every continuous linear operator from \(\ell _{1}\) to \(\ell _{p}\) is absolutely \(\left( r,1\right) \)-summing for \(1/r=1-\left| 1/p-1/2\right| .\) When \(p=2\) it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators.