
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 The work consists of author's published and unpublished results compiled together with material appearing in the literature.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

 

Integral and supremal operators on weighted function spaces
Křepela, Martin ; Pick, Luboš (advisor) ; Sickel, Winfried (referee) ; Tichonov, Sergey (referee)
Title: Integral and Supremal Operators on Weighted Function Spaces Author: Martin Křepela Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: The common topic of this thesis is boundedness of integral and supre mal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three cate gories according to the particular type of studied operators and function spaces. The first part involves a convolution operator acting on general weighted Lorentz spaces of types Λ, Γ and S defined in terms of the nonincreasing rear rangement, HardyLittlewood maximal function and the difference of these two, respectively. It is characterized when a convolutiontype operator with a fixed kernel is bounded between the aforementioned function spaces. Furthermore, weighted Youngtype convolution inequalities are obtained and a certain optima lity property of involved rearrangementinvariant domain spaces is proved. The additional provided information includes a comparison of the results to the pre viously known ones and an overview of basic properties of some new function spaces...


Integral operators on function spaces
Peša, Dalimil ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis, we consider LorentzKaramata spaces with slowly varying fuc tions and study their properties. We first provide simpler definition of slowly varying functions and derive some of their properties. We then consider LorentzKaramata functionals over an arbi trary sigmafinite measure space equipped with a nonatomic measure and corre sponding LorentzKaramata spaces. We characterise nontriviality of said spaces, then study when they are equivalent to a Banach function space and obtain mul titude of conditions, either sufficient or necessary. We further study embeddings between LorentzKaramata spaces and provide a partial characterisation. At last, we try to describe the associate spaces of LorentzKaramata spaces and succeed even in some of the limiting cases. Our contribution is mainly the characterisation of nontriviality, the partial characterisation of embeddings, the description of associate spaces in some lim iting cases and all the results concerning LorentzKaramata spaces with the sec ondary parameter q smaller than one. Those results are, as far as we are aware, new. 1


Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor) ; Vybíral, Jan (referee)
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

 