The focus of this chapter is the Keith–Zhong theorem on the self-improvement of p-Poincaré inequalities for 1 < p < ∞. In [153], Keith and Zhong proved that whenever X is a complete metric space equipped with a doubling measure and supporting a p-Poincaré inequality for some 1 < p < ∞ then X also supports a q-Poincaré inequality for some q ≥ 1 with q < p. Stated another way, for complete and doubling metric measure spaces the Poincaré inequality is an open-ended condition, that is, the collection of p for which X supports a p-Poincaré inequality is a relatively open subset of [1, ∞). This result has numerous applications and corollaries; for a sample of these see Theorems 12.3.13 and 12.3.14.

Throughout this chapter our standing assumptions are that X = (X, d) is a complete metric space, that μ is a doubling measure on X, and that the metric measure space (X, d, μ) supports a p-Poincaré inequality for some 1 < p < ∞.

As discussed in Corollary 8.3.16 and Lemma 8.3.18 we may, and will, also assume without loss of generality that X is a geodesic space. Finally, in view of Theorem 9.1.15(i) and Hölder's inequality, we may assume that the integrals on both sides of the Poincaré inequality are taken over the same ball, i.e., that the parameter λ in (8.1.1) is equal to 1. We occasionally repeat these assumptions for emphasis.

For a positive real number x, we write ⌈x⌉ for the smallest integer greater than or equal to x.

Geometric properties of geodesic doubling metric measure spaces

We begin with a few miscellaneous facts about metric measure spaces.

In many arguments in this chapter we will need to consider inclusions between dilations of balls. Note that if B and B′ are balls in X with B ⊂ B′ and λ > 0, it is not necessarily the case that λB ⊂ λB′.