×

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 \(Q-1<p\leq Q\).

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI