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Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings

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Abstract

We formulate a new definition of Sobolev function spaces on a domain of a metric space in which the doubling condition need not hold. The definition is equivalent to the classical definition in the case that the domain lies in a Euclidean space with the standard Lebesgue measure. The boundedness and compactness are examined of the embeddings of these Sobolev classes into L q and C α . We state and prove a compactness criterion for the family of functions L p (U), where U is a subset of a metric space possibly not satisfying the doubling condition.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    N. N. Romanovskiĭ

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  1. N. N. Romanovskiĭ
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Correspondence to N. N. Romanovskiĭ.

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Original Russian Text Copyright © 2013 Romanovskiĭ N.N.

The author was supported by the Interdisciplinary Project of the Siberian and Far East Divisions of the Russian Academy of Sciences (Grant No. 56).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 2, pp. 450–467, March–April, 2013.

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Romanovskiĭ, N.N. Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings. Sib Math J 54, 353–367 (2013). https://doi.org/10.1134/S0037446613020171

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  • DOI: https://doi.org/10.1134/S0037446613020171

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