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

Li and Shanmugalingam showed in [20] that annularly quasiconvex metric spaces endowed with a doubling measure preserve the property of supporting a p-Poincare 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 hypothesis on the metric space, the preservation of p-Poincare inequalites under those conformal deformations for sufficiently large p. We propose similar hypothesis to the ones used in [9], where the preservation of∞-Poincare 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 point is not a PI space.