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Compactness of higher-order Sobolev embeddings
Slavíková, Lenka ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
The present work deals with m-th order compact Sobolev embeddings on a do- main Ω ⊆ Rn endowed with a probability measure ν and satisfying certain isoperi- metric inequality. We derive a condition on a pair of rearrangement-invariant spaces X(Ω, ν) and Y (Ω, ν) which suffices to guarantee a compact embedding of the Sobolev space V m X(Ω, ν) into Y (Ω, ν). The condition is given in terms of compactness of certain operator on representation spaces. This result is then applied to characterize higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, among them the Gauss space is the most stan- dard example. 1
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Spaces of functions with fractional derivatives on interval
Lopata, Jan ; Kaplický, Petr (advisor) ; Hencl, Stanislav (referee)
In literature we can find a variety of ways to introduce Sobolev space W1,1 on bounded and open interval. In this thesis we will put them in context. We will show that completion of set of function with continuous first derivative, the space of functions with weak derivative and space of absolutely continuous functions are isometrically isomorphic. Furthemore, we will demonstrate that the Sobolev space W1,∞ is isometrically isomorphic to space of Lipschitz functions. We will also show several trivial and nontrivial embeddings for Besov spaces. Finnaly, we will examine the question, whether functions from Besov space are, given some parameters, included in set of continuous functions. 1
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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor)
The measure of non-compactness is defined for any continuous mapping T : X Y between two Banach spaces X and Y as β(T) := inf { r > 0: T(BX) can be covered by finitely many open balls with radius r } . It can easily be shown that 0 ≤ β(T) ≤ ∥T∥ and that β(T) = 0, if and only if the mapping T is compact. My supervisor prof. Stanislav Hencl has proved in his paper that the measure of non-compactness of the known embedding W k,p 0 (Ω) → Lp∗ (Ω), where kp is smaller than the dimension, is equal to its norm. In this thesis we prove that the measure of non-compactness of the embedding between function spaces is under certain general assumptions equal to the norm of that embedding. We apply this theorem to the case of Lorentz spaces to obtain that the measure of non-compactness of the embedding Wk 0 Lp,q (Ω) → Lp∗,q (Ω) is for suitable p and q equal to its norm. 1
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Positioning of Orlicz space and optimality
Musil, Vít
Given a rearrangement-invariant Banach function space Y (Ω), we consider the problem of the existence of an optimal (largest) domain Or- licz space LA (Ω) satisfying the Sobolev embedding Wm LA (Ω) !Y (Ω). We present a complete solution of this problem within the class of Marcinkiewicz endpoint spaces which covers several important examples.
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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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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor)
The measure of non-compactness is defined for any continuous mapping T : X Y between two Banach spaces X and Y as β(T) := inf { r > 0: T(BX) can be covered by finitely many open balls with radius r } . It can easily be shown that 0 ≤ β(T) ≤ ∥T∥ and that β(T) = 0, if and only if the mapping T is compact. My supervisor prof. Stanislav Hencl has proved in his paper that the measure of non-compactness of the known embedding W k,p 0 (Ω) → Lp∗ (Ω), where kp is smaller than the dimension, is equal to its norm. In this thesis we prove that the measure of non-compactness of the embedding between function spaces is under certain general assumptions equal to the norm of that embedding. We apply this theorem to the case of Lorentz spaces to obtain that the measure of non-compactness of the embedding Wk 0 Lp,q (Ω) → Lp∗,q (Ω) is for suitable p and q equal to its norm. 1
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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
The measure of non-compactness is defined for any continuous mapping T : X Y between two Banach spaces X and Y as β(T) := inf { r > 0: T(BX) can be covered by finitely many open balls with radius r } . It can easily be shown that 0 ≤ β(T) ≤ ∥T∥ and that β(T) = 0, if and only if the mapping T is compact. My supervisor prof. Stanislav Hencl has proved in his paper that the measure of non-compactness of the known embedding W k,p 0 (Ω) → Lp∗ (Ω), where kp is smaller than the dimension, is equal to its norm. In this thesis we prove that the measure of non-compactness of the embedding between function spaces is under certain general assumptions equal to the norm of that embedding. We apply this theorem to the case of Lorentz spaces to obtain that the measure of non-compactness of the embedding Wk 0 Lp,q (Ω) → Lp∗,q (Ω) is for suitable p and q equal to its norm. 1
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Properties of Sobolev Mappings
Roskovec, Tomáš ; Hencl, Stanislav (advisor) ; Björn, Anders (referee) ; Zürcher, Thomas (referee)
We study the properties of Sobolev functions and mappings, especially we study the violation of some properties. In the first part we study the Sobolev Embedding Theorem that guarantees W1,p (Ω) ⊂ Lp∗ (Ω) for some parameter p∗ (p, n, Ω). We show that for a general domain this relation does not have to be smooth as a function of p and not even continuous and we give the example of the domain in question. In the second part we study the Cesari's counterexample of the continuous mapping in W1,n ([−1, 1]n , Rn ) violating Lusin (N) condition. We show that this example can be constructed as a gradient mapping. In the third part we generalize the Cesari's counterexample and Ponomarev's counte- rexample for the higher derivative Sobolev spaces Wk,p (Ω, Rn ) and characterize the validity of the Lusin (N) condition in dependence on the parameters k and p and dimension. 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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Sobolev mappings and Cantor type counterexamples
Fiala, Martin ; Hencl, Stanislav (advisor) ; Vybíral, Jan (referee)
Sobolev mappings and Cantor type counterexamples Author: Martin Fiala Supervisor: doc. RNDr. Stanislav Hencl, Ph.D. Abstract: The aim of this work is to show one of the general con- structions of the mappings, which can be used to create different coun- terexamples in the theory of Sobolev mappings. The construction is described in detail and then it is used for a number of examples. The last chapter is devoted to a slight generalization of this construction. 1
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