Poincaré inequalities on hyperbolic-type spaces

2026/08/03 by Anna Mc Court, Federico Santagati, Maria Vallarino
Mathematics · #math.CA #math.FA #msc:30L99 #msc:05C12 #msc:26D10

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arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We establish global L^p-Poincaré inequalities, for p∈[1,∞), 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.

Citations

• Calderón-Zygmund theory on some Lie groups of exponential growth
• Local and nonlocal Poincaré inequalities on Lie groups
• Sobolev spaces and hyperbolic fillings
• Discrete Approximations of Metric Measure Spaces of Controlled Geometry
• Uniform approximation of metrics by graphs
• Quasiconformal maps in metric spaces with controlled geometry
• Poincar 'e inequalities and quasiconformal structure on the boundary of\n some hyperbolic buildings