The Poincaré inequality is an open ended condition

Abstract

Let $p >1$ and let $(X,d,\mu)$ be a complete metric measure space with $\mu$ Borel and doubling that admits a $(1,p)$-Poincaré inequality. Then there exists $\varepsilon >0$ such that $(X,d,\mu)$ admits a $(1,q)$-Poincaré inequality for every $q >p – \varepsilon$, quantitatively.