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• 1.
Karlstad University, Faculty of Technology and Science, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2010Doctoral thesis, monograph (Other academic)

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.

• 2.
Karlstad University, Faculty of Technology and Science, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2008Licentiate thesis, monograph (Other scientific)

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.

• 3.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On Fournier-Gagliardo mixed norm spaces2011In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 36, p. 493-508Article in journal (Refereed)

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.

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