Abstract

The main motivation of this paper is the following general problem: under what (nontrivial) conditions a vector-valued $m$-linear operator $T:{\ell }_p\times \cdots \times {\ell }_p\to {\ell }_q$ satisfies a summability property which originally holds for all scalar-valued $m$-linear forms $T:{\ell }_p\times \cdots \times {\ell }_p\to ℂ$? For instance, under what conditions on $T:{\ell }_p\times \cdots \times {\ell }_p\to {\ell }_q$, the famous Bohnenblust–Hille inequality and Hardy–Littlewood inequalities for $m$-linear forms are lifted to $T$? We prove a general result for nonlinear operators which solves this problem as a very particular case. Our methods encompass Lipschitz operators, $m$-linear operators and nonlinear operators under mild assumptions. We show that, even in a very nonlinear environment, if the adjoint of $T$ is almost $q$-summing, then $T$ has the desired property. A straightforward application of our main result provides a generalization of a theorem of S. Kwapień, stated originally for linear operators.

Keywords

Mathematical Subject Classification

Milestones

Authors