## Illinois Journal of Mathematics

### Preservation of $p$-Poincaré inequality for large $p$ under sphericalization and flattening

#### 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.

#### Article information

Source
Illinois J. Math., Volume 59, Number 4 (2015), 1043-1069.

Dates
Revised: 10 October 2016
First available in Project Euclid: 27 February 2017

https://projecteuclid.org/euclid.ijm/1488186020

Digital Object Identifier
doi:10.1215/ijm/1488186020

Mathematical Reviews number (MathSciNet)
MR3628300

Zentralblatt MATH identifier
1361.31015

#### Citation

Durand-Cartagena, Estibalitz; Li, Xining. Preservation of $p$-Poincaré inequality for large $p$ under sphericalization and flattening. Illinois J. Math. 59 (2015), no. 4, 1043--1069. doi:10.1215/ijm/1488186020. https://projecteuclid.org/euclid.ijm/1488186020

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