Poincaré inequalities and Ap weights on bow-ties

A metric space X is called a bow-tie if it can be written as Image 1, where Image 2 and Image 3 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 Image 4 and Image 5, and a variational Image 6-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 Image 6-Poincaré inequality on XRn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.