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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 maximalfunction 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 supremumis 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 themultilinear strong mximal function $\mathcal{M}_{\mathcal{R}}$ defined andstudied by Grafakos et al. In this paper, we first investigated theFefferman-Stein type inequality for $\mathcal{M}_{\mathcal{R}}^{\vec{\Psi}}$when $\vec{\omega}$ satisfies the $A_{\infty,\mathcal{R}}$ condition. Then, forarbitrary $\vec{\omega}\geq 0$( each $ \omega_{j}\ge 0$), the Fefferman-Steintype inequality for the multilinear strong maximal function$\mathcal{M}_{\mathcal{R}} $ associated with rectangles will be given.

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