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 $${ \mathscr{ M}}_{ \mathcal{ R}}(\vec{ f})(x)=\sup_{ \substack{ R \ni x R\in\mathcal{ R}}}\prod\limits_{ i=1}^m\frac{ 1}{ |R|}\int_{ R}|f_i(y)|dy,$$ 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 $\mathscr{ M}_{ \mathcal{ R}}$ 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 ${ \mathscr{ M}}_{ \mathcal{ R}}$ 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 $B_s^{ p,q}(\mathbb{ R}^d)$, from the Triebel-Lizorkin spaces $F_{ s}^{ p_1,q}(\mathbb{ R}^d)\times\cdots\times F_s^{ p_m,q}(\mathbb{ R}^d)$ to $F_s^{ p,q}(\mathbb{ R}^d)$. As a consequence, we further showed that ${ \mathscr{ M}}_{ \mathcal{ R}}$ 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 $\mathscr{ M}_{ \mathcal{ R}}$. 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.
Categories: math.CA math.AP