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Optimal function spaces in weighted Sobolev embeddings with monomial weight
Drážný, Ladislav ; Mihula, Zdeněk (advisor) ; Vybíral, Jan (referee)
In this thesis we study a weighted Sobolev-type inequality for functions from a certain Sobolev-type space that is built upon a rearrangement-invariant space. Considered rear- rangement-invariant spaces are defined on the space Rn endowed with the measure that is given by a monomial weight. We prove a so-called reduction principle for the Sobolev- type inequality. The reduction principle represents a method of how to characterize the rearrangement-invariant spaces that satisfy the Sobolev-type inequality by means of one- dimensional inequalities. Next, for a fixed domain rearrangement-invariant space, we describe the optimal, i.e. the smallest target rearrangement-invariant space such that the Sobolev-type inequality holds. Finally, we describe some concrete examples. We describe the optimal spaces for Lorentz-Karamata 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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Compact and weakly compact operators in Banach function spaces
Drážný, Ladislav ; Spurný, Jiří (advisor) ; Mihula, Zdeněk (referee)
In this work, we study properties of compact integral operators in Banach function spaces. At first, there are introduced Banach function spaces and their basic characte- ristics. Then the work deals with some properties of weakly sequentially compact sets in Banach function spaces. The main aim of the work is to characterise compact ope- rators with Lρ-kernel, which is a specific kind of the integral kernel. To this end, there are applied properties of the sets with uniformly absolutely continuous norm. Finally, the work describes some specific characteristics of the space L1 ([0, 1]) and proves that the Volterra operator in this space is compact. 1
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Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor)
We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1
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Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor)
We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1
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Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor) ; Vybíral, Jan (referee)
We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1
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Function Spaces and Algebras
Mihula, Zdeněk ; Pick, Luboš (advisor) ; Hencl, Stanislav (referee)
The primary purpose of this thesis is to determine when a function space is equivalent to an algebra, that is, when it is closed with respect to pointwise multiplication. Firstly, the theory of some function spaces, namely Lebesgue Lp spaces, the class of Banach function spaces, rearrangement-invariant Banach function spaces, Morrey spaces, Campanato spaces, and weak−L∞ , is introduced. Secondly, a general necessary condition, as well as a general sufficient condition, for a function space to be equivalent to an algebra is given. In each of these two conditions, a crucial role is played by the space L∞ . Furthermore, as a corollary, a characterisation when a Banach function space is equivalent to an algebra is obtained. Thereafter, a few examples illustrating possible usage of these results are presented. After that, a special case when a Banach function space is rearrangement invariant is dealt with. Lastly, the matter of equivalence to an algebra is addressed for the function spaces introduced before. 1
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