Poincaré inequalities on hyperbolic-type spaces

Abstract

We establish global $L^p$-Poincaré inequalities, for $p\in [1,\infty )$, on a class of nondoubling hyperbolic-type metric measure spaces. The proof relies on a discretisation of the space, which gives rise to a Gromov hyperbolic graph, called spiderweb, quasi-isometric to the original space. We prove global Poincaré inequalities for spiderwebs endowed with suitable measures and develop a general transference principle from discrete graphs to metric measure spaces. Combining these results yields global Poincaré inequalities under natural geometric and measure assumptions on the base space.