  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.
