Lebesgue points via the Poincaré inequality

Abstract

We show that in a Q-doubling space (X, d, µ), Q > 1, which satisfies a chain condition, if we have a Q-Poincaré inequality for a pair of functions (u, g) where g ∈ L Q (X), then u has Lebesgue points $\mathcal{H}^h $ -a.e. for $h(t) = \log ^{1 - Q - \varepsilon } (1/t)$ . We also discuss how the existence of Lebesgue points follows for u ∈ W 1,Q (X) where (X, d, µ) is a complete Q-doubling space supporting a Q-Poincaré inequality for Q > 1.