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arXiv · 2308.09800

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

Sylvester Eriksson-Bique·Ryan Gibara·Riikka Korte·Nageswari Shanmugalingam
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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.

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Sylvester Eriksson-Bique, Ryan Gibara, Riikka Korte, Nageswari Shanmugalingam. 2023-08-18. Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincar\'e inequality. https://arxiv.org/abs/2308.09800

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