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

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