The Orlicz inequality for multilinear forms

The Orlicz $\left( \ell_{2},\ell_{1}\right)$-mixed inequality states that $$\left( \sum_{j_{1}=1}^{n}\left( \sum_{j_{2}=1}^{n}\left\vert A(e_{j_{1} },e_{j_{2}})\right\vert \right) ^{2}\right) ^{\frac{1}{2}}\leq\sqrt {2}\left\Vert A\right\Vert$$ for all bilinear forms $A:\mathbb{K}^{n}\times\mathbb{K}^{n}\rightarrow \mathbb{K}$ 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 $p\in\lbrack1,\infty].$