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