Regularity and continuity of local Multilinear Maximal type operator

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{\alpha,\Omega}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,\Omega^c)}\frac{r^\alpha}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leq\alpha<mn,$$ where $\Omega$ is a subdomain in $\mathbb{R}^n$, $\Omega^c=\mathbb{R}^n\backslash\Omega$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,\Omega}$ and the fractional maximal functions $\mathfrak{M}_{\alpha,\Omega}$ $(0<\alpha< mn)$ will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.