University of Oulu

Eriksson-Bique, S. & Gong, J. (2022). Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case. Analysis and Geometry in Metric Spaces, 10(1), 373-393.

Isoperimetric and Poincaré inequalities on non-self-similar Sierpiński sponges : the borderline case

Saved in:
Author: Eriksson-Bique, Sylvester1; Gong, Jasun2
Organizations: 1Research Unit of Mathematical Sciences, University of Oulu, Oulu, Finland
2Department of Mathematics, Fordham University, Bronx, United States
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 1.7 MB)
Persistent link:
Language: English
Published: De Gruyter, 2022
Publish Date: 2023-05-03


In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that in-volves removing well behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate suc inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.

see all

Series: Analysis and geometry in metric spaces
ISSN: 2299-3274
ISSN-E: 2299-3274
ISSN-L: 2299-3274
Volume: 10
Issue: 1
Pages: 373 - 393
DOI: 10.1515/agms-2022-0144
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: S.E.-B. was partially supported by NSF grant DMS#–1704215 (U.S.) and Finnish Academy grant #–345005.
Copyright information: © 2022 Sylvester Eriksson-Bique and Jasun Gong, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License.