Mathematische Zeitschrift

, Volume 257, Issue 3, pp 613–630

Pointwise behaviour of M1,1 Sobolev functions

Article

DOI: 10.1007/s00209-007-0139-y

Cite this article as:
Kinnunen, J. & Tuominen, H. Math. Z. (2007) 257: 613. doi:10.1007/s00209-007-0139-y

Abstract

Our main objective is to study Hajłasz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Hajłasz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Hajłasz function coincides with a Hölder continuous Hajłasz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.

Mathematics Subject Classification (2000)

46E35

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of OuluOuluFinland
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland