Poincaré inequalities and Ap weights on bow-ties

Abstract

A metric space X is called a bow-tie if it can be written as ▪, where ▪ and ▪ are closed subsets of X. We show that a doubling measure μ on X supports a (q,p)–Poincaré inequality on X if and only if X satisfies a quasiconvexity-type condition, μ supports a (q,p)-Poincaré inequality on both ▪ and ▪, and a variational ▪-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at the point x0. In particular, we study the bow-tie XRn consisting of the positive and negative hyperquadrants in Rn equipped with a radial doubling weight and characterize the validity of the ▪-Poincaré inequality on XRn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.