JournalsrmiVol. 18, No. 3pp. 685–700

Lebesgue points for Sobolev functions on metric spaces

  • Juha Kinnunen

    Aalto University, Finland
  • Visa Latvala

    University of Joensuu, Finland
Lebesgue points for Sobolev functions on metric spaces cover
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Abstract

Our main objective is to study the pointwise behaviour of Sobolev functions on a metric measure space. We prove that a Sobolev function has Lebesgue points outside a set of capacity zero if the measure is doubling. This result seems to be new even for the weighted Sobolev spaces on Euclidean spaces. The crucial ingredient of our argument is a maximal function related to discrete convolution approximations. In particular, we do not use the Besicovitch covering theorem, extension theorems or representation formulas for Sobolev functions.

Cite this article

Juha Kinnunen, Visa Latvala, Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoam. 18 (2002), no. 3, pp. 685–700

DOI 10.4171/RMI/332