Weak differentiability for fractional maximal functions of general Lp functions on domains

Abstract

Let Ω⊂Rn be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Ω maps Lp(Ω)→W1,p(Ω) for all p>1, when the smoothness index α≥1. In particular, the results are valid in the range p∈(1,n/(n−1)] that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.