Let Omega subset of R-n be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Omega maps L-p(Omega) -> W-1,W-p (Omega) for all p > 1, when the smoothness index alpha >= 1. In particular, the results are valid in the range p is an element of (1, n/(n - 1)] that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting. (C) 2020 Elsevier Inc. All rights reserved.