Poincaré inequalities and $A_p$ weights on bow-ties

Abstract

A metric space $X$ is called a \emph{bow-tie} if it can be written as $X=X_{+}\cup X_{-}$, where $X_{+}\cap X_{-}=\{x_0\}$ and $X_{\pm }\ne \{x_0\}$ are closed subsets of $X$. We show that a doubling measure $\mu$ on $X$ supports a $(q,p)$--Poincaré inequality on $X$ if and only if $X$ satisfies a quasiconvexity-type condition, $\mu$ supports a $(q,p)$-Poincaré 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 $x_0$. In particular, we study the bow-tie $X_{{\mathbf{R}}^n}$ consisting of the positive and negative hyperquadrants in ${\mathbf{R}}^n$ equipped with a radial doubling weight and characterize the validity of the \p-Poincaré inequality on $X_{{\mathbf{R}}^n}$ in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.

References in corpus (3)

• Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions