In this chapter we discuss some further consequences of Poincaré inequalities in metric measure spaces. We show that many Sobolev-type inequalities follow from a basic Poincaré inequality in doubling metric measure spaces. The Lebesgue differentiation theorem tells us that every integrable function has μ-a.e. point as a Lebesgue point. We will strengthen the Lebesgue point property for Sobolev functions and show that p-capacity-a.e. point is a Lebesgue point of a function in N1, p(X : V). Finally, we also demonstrate that a metric space supporting a Poincaré inequality necessarily has the MECp property in the sense of Section 7.5.

Throughout this chapter we let X = (X, d, μ) be a metric measure space as defined in Section 3.3 and V a Banach space and suppose that X is locally compact and supports a p-Poincaré inequality. Unless otherwise stipulated, we assume that 1 ≤ p < ∞.

Sobolev–Poincaré inequalities

The Poincaré inequality (8.1.1), or its Banach-space-valued counterpart (8.1.41), gives control over the mean oscillation of a function in terms of the p-means of its upper gradient. In many classical situations, for example in Euclidean space ℝn, various Sobolev–Poincaré inequalities demonstrate that one can similarly control the q-means of the function |u − uB| for certain values of q > 1. Analogous results are valid in metric measure spaces satisfying a Poincaré inequality. This is the topic of the current section.

We recall one of the pointwise estimates (8.1.56) that follows from the p-Poincaré inequality in a doubling metric measure space X. If B is an open ball in X and if u: λB → V is integrable in B with ρ an upper gradient of u in λB then

|u(x) − uB|≤ C diam(B) (Mλdiam(B)ρp (x))1/p

for almost every x ∈ B.