  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.
