This paper is devoted to studying the regularity properties for the new maximal operator Mϕ and the fractional new maximal operator Mϕ,β in the local case. Some new pointwise gradient estimates of Mϕ,Ω and Mϕ,β,Ω are given. Moreover, the boundedness of Mϕ,Ω and Mϕ,β,Ω on Sobolev space is established. As applications, we also obtained the bounds of the above operators on Sobolev space with zero boundary values
MR(2010) Subject Classification 42B25, 46E35
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