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Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  23 June 2016

António Caetano ,
Amiran Gogatishvili  and
Bohumír Opic
António Caetano*
Affiliation:
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal (acaetano@ua.pt)
Amiran Gogatishvili
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, , 25, 11567 Prague 1, Czech Republic (gogatish@math.cas.cz)
Bohumír Opic
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic (opic@karlin.mff.cuni.cz)
*
*Corresponding author.
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Extract

There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp (ℝn ), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

Keywords

quasi-Banach function spacecompactnesscompact embeddingabsolute continuityBesov spaceLorentz space

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46B42: Banach lattices 46B50: Compactness in Banach (or normed) spaces

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 5 , October 2016 , pp. 905 - 927
Copyright
Copyright © Royal Society of Edinburgh 2016 

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