The Orlicz inequality for multilinear forms

The Orlicz $({?}_2,{?}_1)$-mixed inequality states that${\left(\sum _{j_1=1}^n{\left(\sum _{j_2=1}^n|A(e_{j_1},e_{j_2})|\right)}^2\right)}^{\frac{1}{2}}\le \sqrt{2}6A6$ 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 ${?}_p$ norms for all $p\in 1,\infty ]$.