The Hajłasz Capacity Density Condition is Self-improving
Abstract
We prove a self-improvement property of a capacity density condition for a nonlocal Hajłasz gradient in complete geodesic spaces with a doubling measure. The proof relates the capacity density condition with boundary Poincaré inequalities, adapts Keith–Zhong techniques for establishing local Hardy inequalities and applies Koskela–Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Hajłasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajłasz capacity density condition.
Main Authors
Format
Articles
Research article
Published
2022
Series
Subjects
Publication in research information system
Publisher
Springer Science and Business Media LLC
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202210214938Use this for linking
Review status
Peer reviewed
ISSN
1050-6926
DOI
https://doi.org/10.1007/s12220-022-00979-z
Language
English
Published in
Journal of Geometric Analysis
Citation
- Canto, J., & Vähäkangas, A. V. (2022). The Hajłasz Capacity Density Condition is Self-improving. Journal of Geometric Analysis, 32(11), Article 276. https://doi.org/10.1007/s12220-022-00979-z
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Additional information about funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. J.C. is supported by supported by the Ministerio de Economía y Competitividad (Spain) through Grant Nos. PID2020-113156GB-I00 and SEV-2017-0718, and by Basque Government through Grant Nos. IT-641-13 and BERC 2018-2021 and “Ayuda para la formación de personal investigador no doctor".
Copyright© The Author(s) 2022