  This paper is devoted to studying the regularity properties for the new
maximal operator $M_{\varphi}$ and the fractional new maximal operator
$M_{\varphi,\beta}$ in the local case. Some new pointwise gradient estimates of
$M_{\varphi,\Omega}$ and $M_{\varphi,\beta,\Omega}$ are given. Moreover, the
boundedness of $M_{\varphi,\Omega}$ and $M_{\varphi,\beta,\Omega}$ on Sobolev
space is established. As applications, we also obtain the bounds of the above
operators on Sobolev space with zero boundary values.
