Abstract
Let (X, μ, d) be a space of homogeneous type, where d and p are a metric and a measure, respectively, related to each other by the doubling condition with γ > 0. Let W pα (X) be generalized Sobolev classes, let Capα p (where p > 1 and 0 < α ≤ 1) be the corresponding capacity, and let dimH be the Hausdorff dimension. We show that the capacity Capα p is related to the Hausdorff dimension; we also prove that, for each function u ∈ W α/p (X), p > 1, 0 < a < γ/p, there exists a set E ⊂ X such that dim H (E) ≤ γ - αp, the limit
exists for each x ∈ X\E, and moreover
.
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Prokhorovich, M.A. Hausdorff dimension of Lebesgue sets for W p α classes on metric spaces. Math Notes 82, 88–95 (2007). https://doi.org/10.1134/S0001434607070115
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DOI: https://doi.org/10.1134/S0001434607070115