Continuity and regularity for local multi-fractional new maximal operators

Abstract

In this paper, it is proved that the following form of local multilinear fractional new maximal operatorMφ,β,Ω(f→)(x)=sup0<r<dist(x,Ωc)⁡rβ(1+rn)γβnφ(|B(x,r)|)m|B(x,r)|m×∏i=1m∫B(x,r)|fi(y)|dy is bounded from W1,p1(Ω)×W1,p2(Ω)×⋯×W1,pm(Ω) into W1,q(Ω) by applying the pointwise gradient inequalities for Mφ,β,Ω, and it is also bounded on Sobolev space W1,q(Ω) as fi∈Lpi(Ω). Meanwhile, the pointwise gradient estimate and the Sobolev boundedness for the local multilinear new maximal operator Mφ,Ω are given. Moreover, the authors also investigate the continuity of Mφ,β,Ω and Mφ,Ω on Sobolev space W1,q(Ω). As applications, the bounds for Mφ,Ω and Mφ,β,Ω on Sobolev space with zero boundary values W01,q(Ω) will be showed.