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Laplaceova transformace na prostorech funkcí
Buriánková, Eva ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this manuscript we study the action of the Laplace transform on rearrangement-invariant Banach function spaces. Our principal goal is to characterize the optimal range space corresponding to a given domain space within the category of rearrangement-invariant Banach function spaces. We first prove a key pointwise estimate of the non-increasing rearrangement of the image under the Laplace transform of a given function. Then we use this inequality to carry out the construction of the optimal range space. We apply this general result to establish an optimality relation between the Lebesgue and Lorentz spaces under the Laplace transform.
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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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Behavior of one-dimensional integral operators on function spaces
Buriánková, Eva ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this manuscript we study the action of one-dimensional integral operators on rearrangement-invariant Banach function spaces. Our principal goal is to characterize optimal target and optimal domain spaces corresponding to given spaces within the category of rearrangement-invariant Banach function spaces as well as to establish pointwise estimates of the non-increasing rearrangement of a given operator applied on a given function. We apply these general results to proving optimality relations between special rearrangement-invariant spaces. We pay special attention to the Laplace transform, which is a pivotal example of the operators in question. Powered by TCPDF (www.tcpdf.org)
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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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Laplaceova transformace na prostorech funkcí
Buriánková, Eva ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this manuscript we study the action of the Laplace transform on rearrangement-invariant Banach function spaces. Our principal goal is to characterize the optimal range space corresponding to a given domain space within the category of rearrangement-invariant Banach function spaces. We first prove a key pointwise estimate of the non-increasing rearrangement of the image under the Laplace transform of a given function. Then we use this inequality to carry out the construction of the optimal range space. We apply this general result to establish an optimality relation between the Lebesgue and Lorentz spaces under the Laplace transform.
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