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 φ {M_{\varphi}} and the fractional new maximal operator M φ , β {M_{\varphi,\beta}} in the local case. Some new pointwise gradient estimates of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {M_{\varphi,\beta,\Omega}} are given. Moreover, the boundedness of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {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.