Download this article
 Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
Connectivity conditions and boundary Poincaré inequalities

Olli Tapiola and Xavier Tolsa

Vol. 17 (2024), No. 5, 1831–1870

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.

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

Open Access made possible by participating institutions via Subscribe to Open.