The Fefferman-Stein type inequalities for the multilinear strong maximal functions

Abstract

Let $\vec{\omega }=({\omega }_1,...,{\omega }_m)$ be a multiple weight and $\{{\Psi }_j{\}}_{j=1}^m$ be a sequence of Young functions. Let ${\mathcal{M}}_{\mathcal{R}}^{\vec{\Psi }}$ be the multilinear strong maximal function with Orlicz norms which is defined by $${\mathcal{M}}_{\mathcal{R}}^{\vec{\Psi }}(\vec{f})(x)=\underset{R\in \mathcal{R},R\ni x}{\sup }\prod _{j=1}^m\Vert f_j{\Vert }_{{\Psi }_j,R}$$ where the supremum is taken over all rectangles with sides parallel to the coordinate axes. If ${\Psi }_j(t)=t$, then ${\mathcal{M}}_{\mathcal{R}}^{\vec{t}}$ coincides with the multilinear strong mximal function ${\mathcal{M}}_{\mathcal{R}}$ defined and studied by Grafakos et al. In this paper, we first investigated the Fefferman-Stein type inequality for ${\mathcal{M}}_{\mathcal{R}}^{\vec{\Psi }}$ when $\vec{\omega }$ satisfies the $A_{\infty ,\mathcal{R}}$ condition. Then, for arbitrary $\vec{\omega }\ge 0$( each ${\omega }_j\ge 0$), the Fefferman-Stein type inequality for the multilinear strong maximal function ${\mathcal{M}}_{\mathcal{R}}$ associated with rectangles will be given.

13 pages