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