Let $m\ge 1$, in this paper, our object of investigation is the regularity and continuity properties of the following multilinear strong maximal operator${\mathcal{M}}_{\mathcal{R}}(\overrightarrow{f})(x)=\underset{\begin{array}{c}\hfill R\ni x\hfill \\ \hfill R\in \mathcal{R}\hfill \end{array}}{\sup }\prod _{i=1}^m\frac{1}{|R|}\underset{R}{\int }|f_i(y)|dy,$ 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 ${\mathcal{M}}_{\mathcal{R}}$ 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 product 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 product 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 product 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 ${\mathcal{M}}_{\mathcal{R}}$ is bounded and continuous from the product fractional Sobolev spaces to fractional Sobolev space. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of ${\mathcal{M}}_{\mathcal{R}}$. In addition, we proved that ${\mathcal{M}}_{\mathcal{R}}(\overrightarrow{f})$ is approximately differentiable a.e. when $\overrightarrow{f}=(f_1,\cdots ,f_m)$ with each $f_j\in L^1({\mathbb{R}}^d)$ being approximately differentiable a.e.

让 $米\ge \mathrm{1个}$，在本文中，我们的研究目标是以下多线性强极大算子的正则和连续性${\mathcal{中号}}_{\mathcal{[R}}（\overrightarrow{F}）（X）=\underset{\begin{array}{c}\hfill \mathrm{[R}\ni X\hfill \\ \hfill \mathrm{[R}\in \mathcal{[R}\hfill \end{array}}{\mathrm{SUP}}\prod _{\mathrm{一世}=\mathrm{1个}}^{米}\frac{\mathrm{1个}}{|\mathrm{[R}|}\underset{\mathrm{[R}}{\int }|F_{\mathrm{一世}}（ÿ）|dÿ，$ 哪里 $X\in {\mathbb{[R}}^d$ 和 $\mathcal{[R}$ 表示中的所有矩形的族 ${\mathbb{[R}}^d$侧面与轴平行。什么时候$米=\mathrm{1个}$，表示 ${\mathcal{中号}}_{\mathcal{[R}}$ 通过 ${\mathcal{中号}}_{\mathcal{[R}}$。然后，${\mathcal{中号}}_{\mathcal{[R}}$与Jessen，Marcinkiewicz和Zygmund最初研究的经典强大的最大函数相吻合。我们证明了${\mathcal{中号}}_{\mathcal{[R}}$ 与产品Sobolev空间有界且连续 ${\mathrm{w\; ^}}^{\mathrm{1个}，p_{\mathrm{1个}}}（{\mathbb{[R}}^d）\times \cdots \times {\mathrm{w\; ^}}^{\mathrm{1个}，p_{米}}（{\mathbb{[R}}^d）$ 至 ${\mathrm{w\; ^}}^{\mathrm{1个}，p}（{\mathbb{[R}}^d）$，来自产品Besov space ${乙}_s^{p_{\mathrm{1个}}，q}（{\mathbb{[R}}^d）\times \cdots \times {乙}_s^{p_{米}，q}（{\mathbb{[R}}^d）$ 至 ${乙}_s^{p，q}（{\mathbb{[R}}^d）$，来自产品Triebel-Lizorkin空间 $F_s^{p_{\mathrm{1个}}，q}（{\mathbb{[R}}^d）\times \cdots \times F_s^{p_{米}，q}（{\mathbb{[R}}^d）$ 至 $F_s^{p，q}（{\mathbb{[R}}^d）$。结果，我们进一步表明${\mathcal{中号}}_{\mathcal{[R}}$从乘积Sobolev空间到分数Sobolev空间是有界且连续的。作为应用，我们获得了Sobolev容量的弱类型不等式，可用于证明p的p-拟连续性${\mathcal{中号}}_{\mathcal{[R}}$。此外，我们证明了${\mathcal{中号}}_{\mathcal{[R}}（\overrightarrow{F}）$ 大约可微分 $\overrightarrow{F}=（F_{\mathrm{1个}}，\cdots ，F_{米}）$ 与每个 $F_{Ĵ}\in {\mathrm{大号}}^{\mathrm{1个}}（{\mathbb{[R}}^d）$ 近似可微