Weak differentiability for fractional maximal functions of general $L^{p}$ functions on domains

Abstract Let Ω ⊂ R n be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Ω maps L p ( Ω ) → W 1 , 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.