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Fine properties of functions and operators
Kubíček, David ; Pick, Luboš (advisor) ; Slavíková, Lenka (referee)
We establish the equivalence between the boundedness of certain supremum operators and optimal spaces in Sobolev embeddings. We do this by exploiting known relations between higher-order Sobolev embeddings and isoperimetric inequalities. We provide an explicit way to compute both the optimal domain norm and the optimal target norm in a Sobolev embedding. Finally, we apply our results to higher-order Sobolev embeddings on John domains and on domains from the Maz'ya classes. Furthermore, our results are partially applicable to embeddings involving product probability spaces. 1
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Properties of integral operators on Orlicz spaces
Beránek, Tomáš ; Pick, Luboš (advisor) ; Mihula, Zdeněk (referee)
Working with function spaces in various branches of mathematical analysis introduces optimality problems, where the question of choosing a function space both accessible and expressive becomes a nontrivial exercise. A good middle ground is provided by Orlicz spaces, parameterized by a single Young function and thus accessible, yet expansive. In this work, we study optimality problems on Sobolev embeddings in the so-called Maz'ya classes of Euclidean domains which are defined through their isoperimetric behavior. In particular, we prove the non-existence of optimal Orlicz spaces in certain Orlicz-Sobolev embeddings in a limiting (critical) state whose pivotal special case is the celebrated embedding of Brezis and Wainger for John domains. 1
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Classical operators of harmonic analysis and Sobolev embeddings on rearrangement-invariant function spaces
Mihula, Zdeněk ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Persson, Lars-Erik (referee)
Boundedness properties of some classical operators of harmonic analysis (namely the Hilbert and Riesz transforms, the Riesz potentials and (fractional and nonfractional) maximal operators) as well as certain Sobolev-type embeddings on the entire space are studied. The compactness of Sobolev trace embeddings is also investigated. The focus is on the optimality of the results within the class of rearrangement-invariant function spaces. The aforementioned questions are reduced to equivalent problems concerning appropriate Hardy-type operators acting on functions of a single variable. The behavior of the Hardy-type operators on rearrangement-invariant function spaces is investigated first. The results concerning the Hardy-type operators are used as the building blocks from which together with known results from the literature the other results are obtained. To illustrate possible applications, general results are accompanied by particular exam- ples. The results presented in this thesis are based on some of the papers authored or coauthored by the author. 1
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Mathematical paradoxes
Wintrová, Lucie ; Pick, Luboš (advisor) ; Zelený, Miroslav (referee)
In the presented bachelor thesis we will focus on mathematical paradoxes, especially the Banach-Tarski paradox. We will show several paradoxes concerning decompositions of sets, such as the Sierpiński-Mazurkiewicz paradox. Next, we perform a constructive proof of the Banach-Tarski theorem in R3 using a special group of rotations. Finally, we generalize the notion of equidecomposability to continuous equidecomposability and prove that the Banach-Tarski pardox holds even under the stricter condition of continuous equidecomposability. This will answer de Groot's question. 1
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Summation Methods
Berkman, Pavel ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In the given thesis, we study limiting (summation) methods. The problems are divided into two main groups, the first one focusing on elementary limiting methods, and the second one dealing with methods which generalise those from the first group, for instance the class of matrix limiting methods. The base of the thesis is the Toeplitz theorem, which characterises regular matrix methods. Furthermore, we invent the term improper regularity, which we subsequently apply to individual methods. By doing that we extend our knowledge of their field of convergence. We especially deal with Hutton's method, where we present some of our own results. All findings are illustrated with examples for better understanding. 1
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Weighted inequalities for Hardy-type operators and their application in the Interplation Theory
Pražák, David ; Pick, Luboš (advisor) ; Krbec, Miroslav (referee)
We study real interpolation spaces (Xo, X1) 12,q, where {} is a parameter function, not necessarily a power weight. Using a discretization method we "discretize" the norm in (Xo, X1) 12,q. The resulting norm is given by the corresponding quasiconcave function h and its discretizing sequence, we denote the space endowed with this norm by (Xo, X1)h,q· We give a direct proof of a theorem dueto V. I. Ovchinnikov and A. S. Titenkov, which characterizes the space (Lp0 , Lp1 )h,q in terms of the non- increasing rearrangement. Further, we find a relation between the dilation indices of a quasiconcave function h and its discretizing sequence. In the case when the dilation indices of h are not limiting, the space ( Lp0 , Lp1 ) h,q coincides wi th some classical Lorentz space A q ( r.p). If the dilation indices are limiting, then we characterize the space (Lp0 , Lp1 )h,q as an extrapolation space. Powered by TCPDF (www.tcpdf.org)
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