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arXiv · 1408.5718

Lebesgue points via the Poincar\'e inequality

Nijjwal Karak·Pekka Koskela
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Abstract

In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$ which satisfies a chain condition, if we have a $Q$-Poincar\'e inequality for a pair of functions $(u,g)$ where $g\in L^Q(X),$ then $u$ has Lebesgue points $H^h$-a.e. for $h(t)=\log^{1-Q-\epsilon}(1/t).$ We also discuss how the existence of Lebesgue points follows for $u\in W^{1,Q}(X)$ where $(X,d,\mu)$ is a complete $Q$-doubling space supporting a $Q$-Poincar\'e inequality for $Q>1.$

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BibTeXRIS

Nijjwal Karak, Pekka Koskela. 2014-08-25. Lebesgue points via the Poincar\'e inequality. https://doi.org/10.1007/s11425-015-5001-9

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