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

Sylvester Eriksson-Bique, Ryan Gibara, Riikka Korte, Nageswari Shanmugalingam

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 [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.

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 -Poincaré inequality for , thus extending the results of [20,2,9] to non-Ahlfors regular spaces. We show that -codimensional thickness of the boundary for implies -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.
Paper Structure (9 sections, 10 theorems, 120 equations)

This paper contains 9 sections, 10 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Size of the visible boundary
  3. Trace result
  4. Structure of the paper
  5. Preliminaries
  6. Construction of $P_\infty$ and the Frostman measure $\nu_F$
  7. Proof of Theorem \ref{['thm:main']}
  8. Traces of Newton-Sobolev functions on the visible boundary
  9. Concluding remarks

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be a complete doubling geodesic metric measure space supporting a $p$-Poincaré inequality for some finite $p> 1$. Let $0<t<p$ and $\Omega$ be a domain in $X$, and let $c_0>0$ be such that for all $w\in\partial\Omega$ and for all $0<\rho<\mathop{\mathrm{diam}}\nolimits(\Omega)$, we ha Then for each $z_0\in\Omega$ there is a compact subset $P_\infty\subset F:=B(z_0,3d_\Omega(z_0))\ca

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 20 more