National Repository of Grey Literature 51 records found  1 - 10nextend  jump to record: Search took 0.00 seconds. 
Amalgam Spaces
Peša, Dalimil ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis we introduce the concept of Wiener-Luxemburg 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 Wiener-Luxemburg 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 Hardy-type 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 three-weight inequality for an iterated discrete Hardy-type 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 The work consists of author's published and unpublished results along with material appearing in the literature.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- ment-invariant (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 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
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 Hardy-Littlewood maximal operator, the Hardy-type integral operators, the maximal operator of fractional order, the Riesz potential, the Laplace transform, and also with Sobolev-type 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 rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned 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
Convexity in normed linear spaces and more general spaces
Zaplatílek, Adam ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
We study questions concerning convexity and the existence of the nearest point for a given set in spaces equipped with either a norm, or with a more gen- eral functional, namely a quasinorm or an α−norm. We characterize convexity in a Hilbert space. We investigate relations between convexity and properties of the distance function. 1
Optimality of function spaces for classical integral operators
Mihula, Zdeněk ; Pick, Luboš (advisor)
We investigate optimal partnership of rearrangement-invariant 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 non-trivial examples involving Generalized Lorentz-Zygmund 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
Boundedness of the average operator on Orlicz sequence spaces
Krejčí, Jan ; Pick, Luboš (advisor) ; Hencl, Stanislav (referee)
The goal of this thesis is to characterize the Average operator on Orlicz sequence spaces and to give a condition equivalent to ∆0 2. 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, Hardy-Littlewood maximal function and the difference of these two, respectively. It is characterized when a convolution-type operator with a fixed kernel is bounded between the aforementioned function spaces. Furthermore, weighted Young-type convolution inequalities are obtained and a certain optima- lity property of involved rearrangement-invariant 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...

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