Sobolev regularity for a class of local fractional new maximal operators

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.