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

Abstract

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.