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

created by durandcar on 12 Sep 2015
modified on 23 Feb 2018

[BibTeX]

Published Paper

Inserted: 12 sep 2015
Last Updated: 23 feb 2018

Journal: Illinois Journal of Mathematics
Volume: 59
Number: 4
Pages: 1043-1069
Year: 2015

Abstract:

Li and Shanmugalingam showed in \cite{LS} 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 \cite{DL}, 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.