Abstract
In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided that
where Θ−1 is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rn. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.
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Björn, J., Onninen, J. Orlicz capacities and Hausdorff measures on metric spaces. Math. Z. 251, 131–146 (2005). https://doi.org/10.1007/s00209-005-0792-y
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DOI: https://doi.org/10.1007/s00209-005-0792-y