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Compactness of Sobolev-type embeddings with measures

Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy

corresponding author.

and

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic

E-mail Address: mihulaz@karlin.mff.cuni.cz

E-mail Address: mihulzde@fel.cvut.cz

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https://doi.org/10.1142/S021919972150036XCited by:5 (Source: Crossref)

Abstract

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in $ℝ^{n}$ with respect to upper Ahlfors regular measures $ν$ , that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $ν$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

Keywords:

AMSC: 46B50, 46E30, 46E35

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