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Properties of function spaces and operators acting on them
Turčinová, Hana ; Nekvinda, Aleš (advisor) ; Edmunds, David Eric (referee) ; Sickel, Winfried (referee)
The present thesis is focused on the study of properties of function spaces con- taining measurable functions, and operators acting on them. It consists of four papers. In the first paper, we establish a new characterization of the set of Sobolev functions with zero traces via the distance function from the boundary of a do- main. This characterization is innovative in that it is based on the space L1,∞ a of functions having absolutely continuous quasinorms in L1,∞ . In the second paper, we investigate properties of certain new scale of spaces governed by a functional involving the maximal nonincreasing rearrangement and powers. Motivation for studying such structures stems from a recent research of sharp Sobolev embeddings into spaces furnished with Ahlfors measures. In the third paper, we extend discretization techniques for Lorentz norms by eliminating nondegeneracy restrictions on weights. We apply the method to characterize general embeddings between classical Lorentz spaces. In the fourth paper, we characterize triples of weights for which an inequality involving the superposition of two integral operators holds. We apply results from the third paper to avoid duality and to obtain thereby a general result. 1
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Integral and supremal operators on weighted function spaces
Křepela, Martin ; Pick, Luboš (advisor) ; Sickel, Winfried (referee) ; Tichonov, Sergey (referee)
Title: Integral and Supremal Operators on Weighted Function Spaces Author: Martin Křepela Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: The common topic of this thesis is boundedness of integral and supre- mal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three cate- gories according to the particular type of studied operators and function spaces. The first part involves a convolution operator acting on general weighted Lorentz spaces of types Λ, Γ and S defined in terms of the nonincreasing rear- rangement, Hardy-Littlewood maximal function and the difference of these two, respectively. It is characterized when a convolution-type operator with a fixed kernel is bounded between the aforementioned function spaces. Furthermore, weighted Young-type convolution inequalities are obtained and a certain optima- lity property of involved rearrangement-invariant domain spaces is proved. The additional provided information includes a comparison of the results to the pre- viously known ones and an overview of basic properties of some new function spaces...
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Integral and supremal operators on weighted function spaces
Křepela, Martin ; Pick, Luboš (advisor) ; Sickel, Winfried (referee) ; Tichonov, Sergey (referee)
Title: Integral and Supremal Operators on Weighted Function Spaces Author: Martin Křepela Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: The common topic of this thesis is boundedness of integral and supre- mal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three cate- gories according to the particular type of studied operators and function spaces. The first part involves a convolution operator acting on general weighted Lorentz spaces of types Λ, Γ and S defined in terms of the nonincreasing rear- rangement, Hardy-Littlewood maximal function and the difference of these two, respectively. It is characterized when a convolution-type operator with a fixed kernel is bounded between the aforementioned function spaces. Furthermore, weighted Young-type convolution inequalities are obtained and a certain optima- lity property of involved rearrangement-invariant domain spaces is proved. The additional provided information includes a comparison of the results to the pre- viously known ones and an overview of basic properties of some new function spaces...
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