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

Let $Ω\subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $Ω$ maps $L^{p}(Ω) \to W^{1,p}(Ω)$ for all $p > 1$, when the smoothness index $α\geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

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