# Lebesgue points via the Poincaré inequality

@article{Karak2014LebesguePV, title={Lebesgue points via the Poincar{\'e} inequality}, author={Nijjwal Karak and Pekka Koskela}, journal={Science China Mathematics}, year={2014}, volume={58}, pages={1697-1706} }

- Published 2014
DOI:10.1007/s11425-015-5001-9

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 ∈ LQ(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 ∈ W1,Q(X) where (X, d, µ) is a complete Q-doubling space supporting a Q-Poincaré inequality for Q > 1.

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