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Skorokompaktní vnoření prostorů funkcí
Křepela, Martin ; Pick, Luboš (advisor) ; Spurný, Jiří (referee)
This work is dealing with almost-compact embeddings of function spaces, in particular, the class of classical and weak Lorentz spaces with a norm given by a general weight fuction is studied. These spaces are not Banach function spaces in general, thus the almost-compact em- bedding is defined for more general sturctures of rearrangement-invariant lattices. A general characterization of when an r.i. lattice is almost-compactly embedded into a Lorentz space, involving an optimal constant of a certain continuous embedding, is proved. Based on this the- orem and appropriate known results about continuous embeddings, explicit characterizations of mutual almost-compact embeddings of all subtypes of Lorentz spaces are obtained. 1
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Characterization of functions vanishing at the boundary
Turčinová, Hana ; Nekvinda, Aleš (advisor) ; Edmunds, David Eric (referee)
Let Ω ⊂ Rn be a domain with a moderate boundary regularity, p ∈ (1, ∞) and let d be the distance function defined by d(t) = dist(t, ∂Ω), t ∈ Rn . Assume that u belongs to the Sobolev space W1,p (Ω). A classical result states that u ∈ W1,p 0 (Ω) if and only if u d ∈ Lp (Ω) and ∇u ∈ Lp (Ω). This fact has been several times consecutively refined, and each time the required condition u d ∈ Lp (Ω) was relaxed to a weaker one. The first such improvement shows that the condition u d ∈ Lp,∞ (Ω) is sufficient. In the next such result the condition u d ∈ L1 (Ω) was considered. Moreover, this result was extended to Sobolev spaces of higher order. In this thesis we improve the previous results in the case when n = 1 and Ω is an open interval I. In our principal result we prove that u ∈ W1,p 0 (I) if and only if u d ∈ L1,p (I) and u′ ∈ Lp (I). 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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Skorokompaktní vnoření prostorů funkcí
Křepela, Martin ; Pick, Luboš (advisor) ; Spurný, Jiří (referee)
This work is dealing with almost-compact embeddings of function spaces, in particular, the class of classical and weak Lorentz spaces with a norm given by a general weight fuction is studied. These spaces are not Banach function spaces in general, thus the almost-compact em- bedding is defined for more general sturctures of rearrangement-invariant lattices. A general characterization of when an r.i. lattice is almost-compactly embedded into a Lorentz space, involving an optimal constant of a certain continuous embedding, is proved. Based on this the- orem and appropriate known results about continuous embeddings, explicit characterizations of mutual almost-compact embeddings of all subtypes of Lorentz spaces are obtained. 1
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