Super-critical Hardy--Littlewood inequalities for multilinear forms

Abstract

The multilinear Hardy--Littlewood inequalities provide estimates for the sum of the coefficients of multilinear forms $T:{\ell }_{p_1}^n\times \cdots \times {\ell }_{p_m}^n\to \mathbb{R}$ (or $\mathbb{C}$) when $1/p_1+\cdots +1/p_m<1.$ In this paper we investigate the critical and super-critical cases; i.e., when $1/p_1+\cdots +1/p_m\ge 1.$