Abstract
Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)−α, where E is a closed set in X and \(\alpha \in \mathbb {R}\). We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt’s Ap classes of weights, 1 ≤ p < ∞. With the help of general Ap-weighted embedding results, we then prove (global) Hardy–Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.
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B.D. was partially supported by the National Science Centre, Poland, grant no. 2015/18/E/ST1/00239
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Dyda, B., Ihnatsyeva, L., Lehrbäck, J. et al. Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities. Potential Anal 50, 83–105 (2019). https://doi.org/10.1007/s11118-017-9674-2
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DOI: https://doi.org/10.1007/s11118-017-9674-2