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.
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Kinnunen, J., Tuominen, H. Pointwise behaviour of M 1,1 Sobolev functions. Math. Z. 257, 613–630 (2007). https://doi.org/10.1007/s00209-007-0139-y
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DOI: https://doi.org/10.1007/s00209-007-0139-y