Poincare inequalities and Ap weights on bow-ties

A metric space X is called a bow-tie if it can be written as X = X+ boolean OR X-, where X+ boolean AND X- = {x(0)} and X-+/- not equal {x(0)} are closed subsets of X. We show that a doubling measure mu on X supports a (q, p)-Poincare inequality on X if and only if X satisfies a quasiconvexity-type condition, mu supports a (q, p)-Poincare inequality on both X+ and X-, and a variational p-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at the point x(0). In particular, we study the bow-tie X-Rn consisting of the positive and negative hyperquadrants in R-n equipped with a radial doubling weight and characterize the validity of the p-Poincare inequality on X-Rn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin. (c) 2024 The Author(s). Published by Elsevier Inc.