
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


Symmetric approximation numbers
Kossaczká, Marta ; Vybíral, Jan (advisor) ; Gurka, Petr (referee)
This paper deals with the symmetric approximation numbers as well as the other types of snumbers. Concerning the snumbers in the Banach spaces, namely the app roximation numbers the Kolmogorov numbers and the Gelfand numbers, we present a few of possible definitions and some of their properties. We present the symmetric approximation numbers and their relation to the other snumbers. We also focus on the snumbers in the quasiBanach spaces. The situation is a bit different, as we can not use the HahnBanach Theorem. Therefore some of the previous definitions and properties can not be retained. Moreover we define the symmetric approximation num bers in the quasiBanach spaces and discuss the problematics of this definition. Finally, we deal with the Carl's inequality regarding the entropy numbers and the snumbers. We derive the proof for the symmetric approximation numbers in both Banach and quasiBanach case. 1

 

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 CalderónovoZygmundovo lemma, v jejichž d·kazech hrají pokrývací věty klíčovou roli. Dále se zabýváme nerovnostmi mezi operátory, po mocí pokrývacích vět dokazujeme vztahy mezi HardyovýmLittlewoodovým max imálním operátorem, maximálním singulárním integrálním operátorem a ostrým maximálním operátorem. 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 Abstrakt: 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


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.


Compactness of operators on function spaces
Pernecká, Eva ; Gurka, Petr (referee) ; Pick, Luboš (advisor)
Hardytype operators involving suprema have turned out to be a useful tool in the theory of interpolation, for deriving Sobolevtype inequalities, for estimates of the nonincreasing rearrangements of fractional maximal functions or for the description of norms appearing in optimal Sobolev embeddings. This thesis deals with the compactness of these operators on weighted Banach function spaces. We de ne a category of pairs of weighted Banach function spaces and formulate and prove a criterion for the compactness of a Hardytype operator involving supremum which acts between a couple of spaces belonging to this category. Further, we show that the category contains speci c pairs of weighted Lebesgue spaces determined by a relation between the exponents. Besides, we bring an extension of the criterion to all weighted Lebesgue spaces, in proof of which we use characterization of the compactness of operators having the range in the cone of nonnegative nonincreasing functions, introduced as a separate result.
