University of Oulu

Eriksson-Bique, S. and Gong, J. (2021), Almost uniform domains and Poincaré inequalities. Trans. London Math. Soc., 8: 243-298.

Almost uniform domains and Poincaré inequalities

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Author: Eriksson-Bique, Sylvester1; Gong, Jasun2
Organizations: 1Research Unit of Mathematical Sciences, P.O. Box 3000, Oulu, FI-90014 Finland
2Mathematics Department, Fordham University, John Mulcahy Hall Bronx, NY, 10458-5165
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.7 MB)
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Language: English
Published: London Mathematical Society, 2021
Publish Date: 2023-03-31


Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, despite the explicit constructions in our proofs, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the uniform domain condition of Martio and Sarvas. This condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components.

In doing so, our results hold true for metric spaces equipped with doubling measures and Poincaré inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any (nonabelian) Carnot group. Such subsets also give new examples of Sobolev extension domains, also in the general setting of doubling metric measure spaces.

In the Euclidean case, our construction also includes the non-self-similar Sierpiński carpets of Mackay, Tyson and Wildrick, as well as higher dimensional analogues not treated in the literature. When specialized to the plane, our results lead to new, general sufficient conditions for a planar subset to be 2-Ahlfors regular and to satisfy the Loewner condition. Two of these conditions, uniform separation and regularity of boundary components, are also necessary. The sufficiency is obtained with an additional measure density condition.

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Series: Transactions of the London Mathematical Society
ISSN: 2052-4986
ISSN-E: 2052-4986
ISSN-L: 2052-4986
Volume: 8
Issue: 1
Pages: 243 - 298
DOI: 10.1112/tlm3.12032
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: The first author was partly supported by the NSF grant DMS-1704215.
Copyright information: © 2021 The Authors. Transactions of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.