# 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.

## Authors

Stephen Keith

Mathematical Sciences Institute
Australian National University
Canberra 0200
Australia

Xiao Zhong

Department of Mathematics and Statistics
University of Jyväskylä
Jyväskylä
Finland