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Connectivity conditions and boundary Poincaré inequalities

Olli Tapiola and Xavier Tolsa

Vol. 17 (2024), No. 5, 1831–1870
Abstract

Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets Ω n+1 , with codimension-1 Ahlfors–David regular boundaries. First, we prove that if Ω satisfies both the local John condition and the exterior corkscrew condition, then Ω also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if Ω is a 2-sided chord-arc domain, then the boundary Ω supports a Heinonen–Koskela-type weak 1-Poincaré inequality. We also construct an example of a set Ω n+1 such that the boundary Ω is Ahlfors–David regular and supports a weak boundary 1-Poincaré inequality but Ω is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.

Keywords
John-type conditions, Harnack chains, boundary Poincaré inequalities, chord-arc domains, Sobolev spaces
Mathematical Subject Classification
Primary: 28A75, 46E35
Secondary: 35J25
Milestones
Received: 19 June 2022
Accepted: 21 February 2023
Published: 20 June 2024
Authors
Olli Tapiola
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Barcelona
Catalonia
Xavier Tolsa
ICREA, Departament de Matemàtiques
Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica
Barcelona
Catalonia

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