Abstract
We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r α for r∈(0,1] and any point x from metric space.
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Górka, P. In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure. Potential Anal 47, 13–19 (2017). https://doi.org/10.1007/s11118-016-9605-7
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DOI: https://doi.org/10.1007/s11118-016-9605-7