
Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor)
In this thesis we introduce the concept of WienerLuxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the WienerLuxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1


Inequalities for discrete and continuous supremum operators
Oľhava, Rastislav ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
Inequalities for discrete and continuous supremum operators Rastislav O , lhava In this thesis we study continuous and discrete supremum operators. In the first part we investigate general properties of Hardytype operators involving suprema. The boundedness of supremum operators is used for characterization of interpo lation spaces between two Marcinkiewicz spaces. In the second part we provide equivalent conditions for boundedness of supremum operators in the situation when the domain space in one of the classical Lorentz spaces Λp w1 or Γp w1 and the target space Λq w2 or Γq w2 . In the case p ≤ q we use inserting technique obtaining continuous conditions. In the setting of coefficients p > q we provide only partial results obtaining discrete conditions using discretization method. In the third part we deal with a threeweight inequality for an iterated discrete Hardytype operator. We find its characterization which enables us to reduce the problematic case when the domain space is a weighted ℓp with p ∈ (0, 1) into the one with p = 1. This leads to a continuous analogue of investigated discrete inequality. The work consists of author's published and unpublished results along with material appearing in the literature.


Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis we introduce the concept of WienerLuxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the WienerLuxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1


Inequalities for discrete and continuous supremum operators
Oľhava, Rastislav ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
Inequalities for discrete and continuous supremum operators Rastislav O , lhava In this thesis we study continuous and discrete supremum operators. In the first part we investigate general properties of Hardytype operators involving suprema. The boundedness of supremum operators is used for characterization of interpo lation spaces between two Marcinkiewicz spaces. In the second part we provide equivalent conditions for boundedness of supremum operators in the situation when the domain space in one of the classical Lorentz spaces Λp w1 or Γp w1 and the target space Λq w2 or Γq w2 . In the case p ≤ q we use inserting technique obtaining continuous conditions. In the setting of coefficients p > q we provide only partial results obtaining discrete conditions using discretization method. In the third part we deal with a threeweight inequality for an iterated discrete Hardytype operator. We find its characterization which enables us to reduce the problematic case when the domain space is a weighted ℓp with p ∈ (0, 1) into the one with p = 1. This leads to a continuous analogue of investigated discrete inequality. The work consists of author's published and unpublished results along with material appearing in the literature.


Optimality of function spaces for integral operators
Takáč, Jakub ; Pick, Luboš (advisor) ; Honzík, Petr (referee)
In this work, we study the behaviour of linear kernel operators on rearrange mentinvariant (r.i.) spaces. In particular we focus on the boundedness of such operators between various function spaces. Given an operator and a domain r.i. space Y, our goal is to find an r.i. space Z such that the operator is bounded from Y into Z, and, whenever possible, to show that the target space is optimal (that is, the smallest such space). We concentrate on a particular class of kernel operators denoted by Sa, which have important applications and whose pivotal instance is the Laplace transform. In order to deal properly with these fairly general operators we use advanced techniques from the theory of rearrangement invariant spaces and theory of interpolation. It turns out that the problem of finding the optimal space for Sa can, to a certain degree, be translated into the problem of finding a "sufficiently small" space X such that a, the kernel of Sa, lies in X. 1


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 Abstract: We focus on a certain weighted Hardy operator, with a continuous, quasi concave weight, defined on a rearrangementinvariant 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 Sobolevtype 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 Sobolevtype 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


Classical operators of harmonic analysis in Orlicz spaces
Musil, Vít ; Pick, Luboš (advisor) ; Kalamajska, Agnieszka (referee) ; Haroske, Dorothee (referee)
Classical operators of harmonic analysis in Orlicz spaces V'ıt Musil We deal with classical operators of harmonic analysis in Orlicz spaces such as the HardyLittlewood maximal operator, the Hardytype integral operators, the maximal operator of fractional order, the Riesz potential, the Laplace transform, and also with Sobolevtype embeddings on open subsets of Rn or with respect to Frostman measures and, in particular, trace embeddings on the boundary. For each operator (in case of embeddings we consider the identity operator) we investigate the question of its boundedness from an Orlicz space into another. Particular attention is paid to the sharpness of the results. We further study the question of the existence of optimal Orlicz domain and target spaces and their description. The work consists of author's published and unpublished results compiled together with material appearing in the literature.


Isoperimetric problem, Sobolev spaces and the Heisenberg group
Franců, Martin ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
In this thesis we study embeddings of spaces of functions defined on Carnot Carathéodory spaces. Main results of this work consist of conditions for Sobolev type embeddings of higher order between rearrangementinvariant spaces. In a special case when the underlying measure space is the socalled XPS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the abovementioned em beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1

 

Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor)
We investigate optimal partnership of rearrangementinvariant 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 nontrivial examples involving Generalized LorentzZygmund 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
