Regularity and continuity of the multilinear strong maximal operators

Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathcal{M}}_{\mathcal{R}}(\vec{f})(x)=\underset{\begin{array}{c}R\ni xR\in \mathcal{R}\end{array}}{\sup }\prod _{i=1}^m\frac{1}{\mathrm{\mid }R\mathrm{\mid }}{\int }_R\mathrm{\mid }{f}_i(y)\mathrm{\mid }dy,$$ where $x\in {\mathbb{R}}^d$ and $\mathcal{R}$ denotes the family of all rectangles in ${\mathbb{R}}^d$ with sides parallel to the axes. When $m=1$, denote ${\mathcal{M}}_{\mathcal{R}}$ by ${\mathcal{M}}_{\mathcal{R}}$.Then, $\mathcal {M}{$\mathcal{R}$}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathcal{M}}_{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W^{1,p_1}({\mathbb{R}}^d)\times \cdots \times W^{1,p_m}({\mathbb{R}}^d)$ to $W^{1,p}({\mathbb{R}}^d)$, from the Besov spaces $B{s}^{p_1,q} ($\mathbb{R}$^d)\times\cdots\times B_s^{p_m,q}($\mathbb{R}$^d)$to$B_s^{p,q}($\mathbb{R}$^d)$,fromtheTriebel-Lizorkinspaces$F_{s}^{p_1,q}($\mathbb{R}$^d)\times\cdots\times F_s^{p_m,q}($\mathbb{R}$^d)$to$F_s^{p,q}($\mathbb{R}$^d)$\mathrm{.}Asaconsequence,wefurthershowedthat${\mathscr{M}}_{$\mathcal{R}$}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p_1}($\mathbb{R}$^d)\times \cdots\times W^{s,p_m}($\mathbb{R}$^d)$to$W^{s,p}($\mathbb{R}$^d)$for$0<s\leq 1$and$1<p<\infty$. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the $p$-quasicontinuity of ${\mathcal{M}}_{\mathcal{R}}$. The discrete type of the strong maximal operators has also been considered. We showed that this discrete type of the maximal operators enjoys somewhat unexpected regularity properties.