Keith, StephenZhong, Xiao2015-12-100003-486Xhttp://hdl.handle.net/1885/56786Let p > 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincaré inequality. Then there exists ε > 0 such that (X, d, μ) admits a (1, q)-Poincaré inequality for every q > p -ε, quantitatively.The Poincare inequality is an open ended condition20082015-12-09