Traces of Newton–Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincaré inequality

We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, “visibility” indicates the existence of John curves connecting the interior point to the points on the “visible boundary”. In this paper, we provide an affirmative answer in the setting of a doubling metric measure space supporting a p -Poincaré inequality for 1<p<\infty , thus extending the results of Koskela–Nandi–Nicolau (2018), Azzam (2019), and Gibara–Korte (2022) to non-Ahlfors regular spaces. We show that t -codimensional thickness of the boundary for 0<t<p implies p -codimensional thickness of the visible boundary. For such domains, we prove that traces of Sobolev functions on the domain belong to the Besov class of the visible boundary.