Open Access
Winter 2015 Preservation of $p$-Poincaré inequality for large $p$ under sphericalization and flattening
Estibalitz Durand-Cartagena, Xining Li
Illinois J. Math. 59(4): 1043-1069 (Winter 2015). DOI: 10.1215/ijm/1488186020

Abstract

Li and Shanmugalingam showed that annularly quasiconvex metric spaces endowed with a doubling measure preserve the property of supporting a $p$-Poincaré inequality under the sphericalization and flattening procedures. Because natural examples such as the real line or a broad class of metric trees are not annularly quasiconvex, our aim in the present paper is to study, under weaker hypotheses on the metric space, the preservation of $p$-Poincaré inequalites under those conformal deformations for sufficiently large $p$. We propose the hypotheses used in a previous paper by the same authors, where the preservation of $\infty$-Poincaré inequality has been studied under the assumption of radially star-like quasiconvexity (for sphericalization) and meridian-like quasiconvexity (for flattening). To finish, using the sphericalization procedure, we exhibit an example of a Cheeger differentiability space whose blow up at a particular point is not a PI space.

Citation

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Estibalitz Durand-Cartagena. Xining Li. "Preservation of $p$-Poincaré inequality for large $p$ under sphericalization and flattening." Illinois J. Math. 59 (4) 1043 - 1069, Winter 2015. https://doi.org/10.1215/ijm/1488186020

Information

Received: 25 April 2016; Revised: 10 October 2016; Published: Winter 2015
First available in Project Euclid: 27 February 2017

zbMATH: 1361.31015
MathSciNet: MR3628300
Digital Object Identifier: 10.1215/ijm/1488186020

Subjects:
Primary: 31E05
Secondary: 30L10 , 30L99

Rights: Copyright © 2015 University of Illinois at Urbana-Champaign

Vol.59 • No. 4 • Winter 2015
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