Pointwise behaviour of M1,1 Sobolev functions
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- Kinnunen, J. & Tuominen, H. Math. Z. (2007) 257: 613. doi:10.1007/s00209-007-0139-y
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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.