
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


Factors determining the alpine treeline
Oľhava, Rastislav ; Sklenář, Petr (advisor) ; Treml, Václav (referee)
The alpine treeline is a global phenomenon which represents sudden change between two significantly different vegetation types: mountain forest and lowstature alpine vegetation. The primary climatic factor that determines its position seems to be the temperature. We summarize the results that describe the relation between temperature and the altitudinal position of the treeline and search for a general formula that would determine its position. There are two hypotheses which try to explain what is the main limiting factor above tre eline: Climitation and growthlimitation. The first one claims trees are limited by carbon shortage, while the latter one favours lowtemperature limitation of tissue formation al though the carbon income is sufficient. This thesis provides a summary of the arguments for each of them and describes the state of knowledge on this matter.


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

 