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Embedding Theorems for Mixed Norm Spaces and Applications
Karlstad University, Faculty of Technology and Science, Avdelningen för matematik.
2008 (English)Licentiate thesis, monograph (Other scientific)
##### Abstract [en]

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. The work in this direction originates in a paper due to Gagliardo (1958), and was continued by Fournier (1988) and by Kolyada (2005).

We consider fully anisotropic mixed norm spaces. Our main theorem states an embedding of these spaces into Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic fractional Sobolev spaces and anisotropic Sobolev-Besov spaces. The methods used are based on non-increasing rearrangements and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces $B_p^\alpha$ by Bourgain, Brezis, and Mironescu, and for Besov spaces $B^{\alpha_1,\dots,\alpha_n}_p$ by Kolyada.

We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces and in the description of Sobolev type embeddings.

##### Series
Karlstad University Studies, ISSN 1403-8099 ; 2008:31
##### Keywords [en]
mixed norms, embeddings, rearrangements, anisotropic fractional Sobolev spaces, anisotropic Besov spaces
##### Keywords [sv]
mixade normer, inbäddningar, omarrangeringar, anisotropa fractionära Sobolev rum, anisotropa Besov rum
Mathematics
Mathematics
##### Identifiers
ISBN: 978-91-7063-190-0 (print)OAI: oai:DiVA.org:kau-2874DiVA, id: diva2:54555
##### Supervisors
Available from: 2008-10-17 Created: 2008-10-17 Last updated: 2011-10-04Bibliographically approved

#### Open Access in DiVA

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File name FULLTEXT01.pdfFile size 461 kBChecksum SHA-512
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Algervik, Robert

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Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
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Output format
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• text
• asciidoc
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