Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincar\'e inequality

TL;DR

This paper proves that in doubling metric measure spaces supporting a Poincaré inequality, domains with uniformly thick boundaries have large visible boundary portions, and Sobolev function traces belong to Besov classes on these boundaries.

Contribution

It extends visibility and trace results for Sobolev functions to non-Ahlfors regular spaces with thick boundaries, under Poincaré inequality conditions.

Findings

Visible boundary has large measure under boundary thickness conditions

Sobolev function traces belong to Besov spaces on the visible boundary

Results extend to non-Ahlfors regular spaces

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 $p$-Poincar\'e inequality for $1<p<\infty$, thus extending the results of [20,2,9] 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.