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Kompaktní a slabě kompaktní operátory v Banachových prostorech funkcí
Musil, Vít ; Pick, Luboš (advisor) ; Gurka, Petr (referee)
We study properties of weak topologies induced on Ba- nach function spaces by certain subsets of their associate spaces. We characterise relative sequential compactness in the weak topology and prove that the notions of relative weak compactness and relative weak sequential compactness coincide. Finally we apply the results attained to linear operators and their adjoints acting on Banach function spaces.
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Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor)
In this thesis we introduce the concept of Wiener-Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid- ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the Wiener-Luxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em- beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1
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Weighted rearrangement-invariant spaces and their basic properties
Soudský, Filip ; Pick, Luboš (advisor)
In this thesis we shall provide the reader with results in the field of classical Lorentz spaces. These spaces have been studied since the 50's and have many applications in partial differential equations and interpolation theory. This work includes five papers. The first paper studies the properties of Generalized Gamma spaces. The second paper provides an alternative proof of normability characterization of classical Lorentz spaces. The third paper discusses conditions of linearity and quasi-norm property of rearrangement-invariant lattices. The following paper gives a characterization of normability of Gamma spaces. And finally the last paper characterizes the embeddings between GΓ spaces. 1
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Weighted inequalities and properties of operators and embeddings on function spaces
Slavíková, Lenka ; Pick, Luboš (advisor)
The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...
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Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor)
In this thesis we introduce the concept of Wiener-Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid- ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the Wiener-Luxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em- beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1
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Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis we introduce the concept of Wiener-Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid- ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the Wiener-Luxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em- beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1
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Weighted inequalities and properties of operators and embeddings on function spaces
Slavíková, Lenka ; Pick, Luboš (advisor) ; Pérez, Carlos (referee) ; Malý, Jan (referee)
The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...
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Weighted rearrangement-invariant spaces and their basic properties
Soudský, Filip ; Pick, Luboš (advisor) ; Soria, Javier (referee) ; Barza, Sorina (referee)
In this thesis we shall provide the reader with results in the field of classical Lorentz spaces. These spaces have been studied since the 50's and have many applications in partial differential equations and interpolation theory. This work includes five papers. First paper studies the properties of Generalized Gamma spaces. Second paper provides an alternative proof of normability characterization of classical Lorentz spaces. The third paper discus conditions of linearity and quasi-norm property of rearrangement-invariant lattices. The following paper gives a characterization of normability of Gamma spaces. And finally the last paper characterizes the embeddings between Generalized Gamma spaces. Powered by TCPDF (www.tcpdf.org)
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Weighted inequalities and properties of operators and embeddings on function spaces
Slavíková, Lenka ; Pick, Luboš (advisor)
The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...
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Kompaktní a slabě kompaktní operátory v Banachových prostorech funkcí
Musil, Vít ; Pick, Luboš (advisor) ; Gurka, Petr (referee)
We study properties of weak topologies induced on Ba- nach function spaces by certain subsets of their associate spaces. We characterise relative sequential compactness in the weak topology and prove that the notions of relative weak compactness and relative weak sequential compactness coincide. Finally we apply the results attained to linear operators and their adjoints acting on Banach function spaces.
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