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}
}
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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