, Volume 257, Issue 3, pp 613-630
Date: 20 Mar 2007

Pointwise behaviour of M 1,1 Sobolev functions

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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.