Revista Matemática Iberoamericana


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Volume 18, Issue 3, 2002, pp. 685–700
DOI: 10.4171/RMI/332

Lebesgue points for Sobolev functions on metric spaces

Juha Kinnunen[1] and Visa Latvala[2]

(1) Department of Mathematics, Aalto University, P.O. Box 11100, FIN-00076, AALTO UNIVERSITY, FINLAND
(2) Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101, JOENSUU, FINLAND

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.

Keywords: Sobolev spaces, spaces of homogeneous type, doubling measures, capacity, regularity, maximal functions

Kinnunen J, Latvala V. Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 18 (2002), 685-700. doi: 10.4171/RMI/332