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 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.

10 pages