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Algervik, Robert
Publications (3 of 3) Show all publications
Algervik, R. & Kolyada, V. (2011). On Fournier-Gagliardo mixed norm spaces. Annales Academiae Scientiarum Fennicae Mathematica, 36, 493-508
Open this publication in new window or tab >>On Fournier-Gagliardo mixed norm spaces
2011 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 36, p. 493-508Article in journal (Refereed) Published
Abstract [en]

We study mixed norm spaces

V (Rn)

that arise in connection with embeddings of

Sobolev spaces

W

1

1

(Rn). We prove embeddings of V (Rn)

into Lorentz type spaces defined in terms

of iterative rearrangements. Basing on these results, we introduce the scale of mixed norm spaces

V

p

(Rn). We prove that V ½ V p

and we discuss some questions related to this embedding.

Place, publisher, year, edition, pages
Helsinki: Finnish Academy of Science and Letters, 2011
Keywords
Mixed norms; rearrangements, embeddings, Sobolev spaces
National Category
Natural Sciences
Identifiers
urn:nbn:se:kau:diva-11683 (URN)10.5186/aasfm.2011.3624 (DOI)000295069400008 ()
Available from: 2012-02-16 Created: 2012-02-16 Last updated: 2016-04-08Bibliographically approved
Algervik, R. (2010). Embedding Theorems for Mixed Norm Spaces and Applications. (Doctoral dissertation). Karlstad: Karlstad University
Open this publication in new window or tab >>Embedding Theorems for Mixed Norm Spaces and Applications
2010 (English)Doctoral thesis, monograph (Other academic)
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. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005).

Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale.

We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev 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 by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces 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.

In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

Place, publisher, year, edition, pages
Karlstad: Karlstad University, 2010. p. 134
Series
Karlstad University Studies, ISSN 1403-8099 ; 2010:16
Keywords
mixed norms, rearrangements, modulus of continuity, embeddings, Sobolev spaces, Besov spaces, Lorentz spaces
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-5646 (URN)978-91-7063-306-5  (ISBN)
Public defence
2010-09-01, 21A 342, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2010-05-28 Created: 2010-05-14 Last updated: 2011-10-04Bibliographically approved
Algervik, R. (2008). Embedding Theorems for Mixed Norm Spaces and Applications. (Licentiate dissertation). Karlstad: Karlstad University
Open this publication in new window or tab >>Embedding Theorems for Mixed Norm Spaces and Applications
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.

Place, publisher, year, edition, pages
Karlstad: Karlstad University, 2008. p. 63
Series
Karlstad University Studies, ISSN 1403-8099 ; 2008:31
Keywords
mixed norms, embeddings, rearrangements, anisotropic fractional Sobolev spaces, anisotropic Besov spaces, mixade normer, inbäddningar, omarrangeringar, anisotropa fractionära Sobolev rum, anisotropa Besov rum
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-2874 (URN)978-91-7063-190-0 (ISBN)
Presentation
2008-10-17, 21A 342, Karlstads universitet, Karlstad, 13:15 (Swedish)
Opponent
Supervisors
Available from: 2008-10-17 Created: 2008-10-17 Last updated: 2011-10-04Bibliographically approved
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