The Orlicz inequality for multilinear forms

Abstract

The Orlicz $\left({\ell }_2,{\ell }_1\right)$-mixed inequality states that $${\left(\sum _{j_1=1}^n{\left(\sum _{j_2=1}^n\left|A(e_{j_1},e_{j_2})\right|\right)}^2\right)}^{\frac{1}{2}}\le \sqrt{2}\Vert A\Vert$$ for all bilinear forms $A:{\mathbb{K}}^n\times {\mathbb{K}}^n\to \mathbb{K}$ and all positive integers $n$, where ${\mathbb{K}}^n$ denotes ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with ${\mathbb{K}}^n$ endowed with ${\ell }_p$ norms for all $p\in [1,\infty ].$