Geometric characterizations of \(p\)-Poincaré inequalities in the metric setting. (English) Zbl 1334.31007

Summary: We prove that a locally complete metric space endowed with a doubling measure satisfies an \(\infty\)-Poincaré inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which “almost avoids” that set. As an application, we characterize doubling measures on \({\mathbb R}\) satisfying an \(\infty\)-Poincaré inequality. For Ahlfors \(Q\)-regular spaces, we obtain a characterization of \(p\)-Poincaré inequality for \(p>Q\) in terms of the \(p\)-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case \(Q-1<p\leq Q\).


31E05 Potential theory on fractals and metric spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
30L10 Quasiconformal mappings in metric spaces
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