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Positioning of Orlicz space and optimality
Musil, Vít ; Pick, Luboš (advisor) ; Hencl, Stanislav (referee)
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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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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Covering theorems
Jirůtková, Petra ; Pick, Luboš (advisor) ; Gurka, Petr (referee)
V této práci se zabýváme r·znými pokrývacími větami a jejich ap- likacemi. Kromě klasických pokrývacích vět (Vitaliova, Besicovitchova a Whitney- ova věta) zde uvádíme i některá jejich zobecnění a další pokrývací věty. Tyto věty pak používáme v d·kazech dalších vět, některé jsou typickými aplikacemi pokrý- vacích vět jako například Lebesgueova věta o derivování, slabý typ (1,1) maximál- ního operátoru nebo Calderónovo-Zygmundovo lemma, v jejichž d·kazech hrají pokrývací věty klíčovou roli. Dále se zabýváme nerovnostmi mezi operátory, po- mocí pokrývacích vět dokazujeme vztahy mezi Hardyovým-Littlewoodovým max- imálním operátorem, maximálním singulárním integrálním operátorem a ostrým maximálním operátorem. 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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Weighted rearrangement-invariant function spaces
Soudský, Filip ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis we focus on generalized Gamma spaces GΓ(p, m, v) and classify some of their intrinsic properties. In an article called Relative Re- arrangement Methods for Estimating Dual Norm (for details see references), the authors attempted to characterize their associate norms but obtained only several one-sided estimates. Equipped with these, they further showed reflexivity of gener- alized Gamma spaces for p ≥ 2 and m > 1 under an additional restriction that the underlying measure space is of finite measure. However, the full characterization of the associate norm and of the reflexivity of such spaces for 2 > p > 1 remained an open problem. In this thesis we shall fill this gap. We extend the theory to a σ-finite measure space. We present a complete characterization of the associate norm, and we find necessary and sufficient conditions for the reflexivity of such spaces. 1
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Optimal pairs of function spaces for weighted Hardy operators
Oľhava, Rastislav ; Pick, Luboš (advisor) ; Gurka, Petr (referee)
Title: Optimal pairs of function spaces for weighted Hardy operators Author: Rastislav Ol'hava Department: Department of Mathematical Analysis Supervisor of the master thesis: Prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic Abstrakt: We focus on a certain weighted Hardy operator, with a continuous, quasi- concave weight, defined on a rearrangement-invariant Banach function spaces. The op- erators of Hardy type are of great use to the theory of function spaces. The mentioned operator is a more general version of the Hardy operator, whose boundedness was shown to be equivalent to a Sobolev-type embedding inequality. This thesis is con- cerned with the proof of existence of domain and range spaces of our Hardy operator that are optimal. This optimality should lead to the optimality in the Sobolev-type embedding equalities. Our another aim is to study supremum operators, which are also closely related to this issue, and establish some of their basic properties. Keywords: optimality, weighted Hardy operator, supremum operator
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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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Nerovnosti pro integrální operátory
Holík, Miloslav ; Pick, Luboš (advisor) ; Hencl, Stanislav (referee)
The presented work contains a survey of the so far known results about the operator inequalities of the type "good λ", "better good λ" and "rearranged good λ" on the function spaces over the Euclidean space with the Lebesgue measure and their corollaries in the form of more complex operator inequal- ities and norm estimates. However, the main aim is to build similar theory for the Riesz potential operator on the function spaces over the quasi-metric space with the so-called "doubling" measure. Combining the corollaries of this theory with the known norm estimates we obtain the boundedness for the Riesz potential operator on the Lebesgue and Lorentz spaces.
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Kompaktní a slabě kompaktní operátory v Banachových prostorech funkcí
Musil, Vít ; Pick, Luboš (advisor) ; Gurka, Petr (referee)
We study properties of weak topologies induced on Ba- nach function spaces by certain subsets of their associate spaces. We characterise relative sequential compactness in the weak topology and prove that the notions of relative weak compactness and relative weak sequential compactness coincide. Finally we apply the results attained to linear operators and their adjoints acting on Banach function spaces.
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Approximation of a non-increasing rearrangement of a function
Franců, Martin ; Pick, Luboš (advisor) ; Felcman, Jiří (referee)
The non-increasing rearrangement of a measurable real function defined on an appropriate measure space is of the enormous significance in disciplines such as theory of function spaces or interpolation theory and their applications in PDEs. Unfortunately, while it has good and widely applicable mapping properties, it is virtually impossible to calculate the non-increasing rearrangement of a concrete given function precisely. Numerical algorithms for approximation are desirable for this reason. Such method of approximation, based on interpolation by a linear spline, is presented in this thesis. In the first half of this thesis, the developed method is described, while the error estimates of the method are subject to the second part.
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