Let $\vec{ \omega}=( \omega_{ 1},...,\omega_{ m})$ be a multiple weight and $\{ \Psi_{ j}\}^{ m}_{ j=1}$ 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 $$\mathcal{ M}_{ \mathcal{ R}}^{ \vec{ \Psi}}(\vec{ f})(x)=\sup_{ R\in \mathcal{ R},R\ni x}\prod^{ m}_{ j=1}\|f_{ j}\|_{ \Psi_{ j},R}$$ 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 $\vec{ \omega}\geq 0$( 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.
Categories: math.CA