
Optimal function spaces in weighted Sobolev embeddings with monomial weight
Drážný, Ladislav ; Mihula, Zdeněk (advisor) ; Vybíral, Jan (referee)
In this thesis we study a weighted Sobolevtype inequality for functions from a certain Sobolevtype space that is built upon a rearrangementinvariant space. Considered rear rangementinvariant spaces are defined on the space Rn endowed with the measure that is given by a monomial weight. We prove a socalled reduction principle for the Sobolev type inequality. The reduction principle represents a method of how to characterize the rearrangementinvariant spaces that satisfy the Sobolevtype inequality by means of one dimensional inequalities. Next, for a fixed domain rearrangementinvariant space, we describe the optimal, i.e. the smallest target rearrangementinvariant space such that the Sobolevtype inequality holds. Finally, we describe some concrete examples. We describe the optimal spaces for LorentzKaramata spaces. 1


Volumes of unit balls of Lorentz spaces
Doležalová, Anna ; Vybíral, Jan (advisor)
This thesis studies the volume of the unit ball of finitedimensional Lorentz sequence spaces p,q n . Lorentz spaces are a generalisation of Lebesgue spaces with a quasinorm described by two parameters 0 < p, q ≤ ∞. The volume of the unit ball Bp,q n of a general finitedimensional Lorentz space was so far an unknown quantity, even though for the Lebesgue spaces it has been wellknown for many years. We present the explicit formula for Vol(Bp,∞ n ) and Vol(Bp,1 n ). We also describe the asymptotic behaviour of the nth root of Vol(Bp,q n ) with respect to the dimension n and show that [Vol(Bp,q n )]1/n ≈ n−1/p for all 0 < p < ∞, 0 < q ≤ ∞. Furthermore, we study the ratio of Vol(Bp,∞ n ) and Vol(Bp n). We conclude by examining the decay of entropy numbers of embeddings of the Lorentz spaces.


Entropy numbers
Kossaczká, Marta ; Vybíral, Jan (advisor) ; Hencl, Stanislav (referee)
In this work we study entropy numbers of linear operators. We focus on entropy numbers of identities between real finitedimensional sequence spaces and present detailed proofs of their estimates. Then we describe relation between entropy numbers of identities between real spaces and between complex spaces, which allows us to establish similar estimates for complex spaces. Powered by TCPDF (www.tcpdf.org)


Laplace equation in fractional Sobolev spaces
Bartoš, Ondřej ; Bárta, Tomáš (advisor) ; Vybíral, Jan (referee)
The goal of this thesis is to study Laplace's equation on a unit disc. The given function values on a unit circle can be interpreted as a 2πperiodic function and the solution can be derived using Fourier method. We introduce general integer Sobolev spaces and their alternatives useful for describing functions on a unit disc and a unit circle. Using elementary methods, we show how they are related to each other. The same results are shown for fractional Sobolev spaces. The main result is that functions from some Sobolev space on a unit disc that solve Laplace's equation correspond to functions from a one half lower Sobolev space on a unit circle. These results can be used to show for a function from some Sobolev space on a unit circle in how strong norm the solution of Laplace's equation converges to the given function. 1

 

Volumes of unit balls of Lorentz spaces
Doležalová, Anna ; Vybíral, Jan (advisor) ; Lang, Jan (referee)
This thesis studies the volume of the unit ball of finitedimensional Lorentz sequence spaces p,q n . Lorentz spaces are a generalisation of Lebesgue spaces with a quasinorm described by two parameters 0 < p, q ≤ ∞. The volume of the unit ball Bp,q n of a general finitedimensional Lorentz space was so far an unknown quantity, even though for the Lebesgue spaces it has been wellknown for many years. We present the explicit formula for Vol(Bp,∞ n ) and Vol(Bp,1 n ). We also describe the asymptotic behaviour of the nth root of Vol(Bp,q n ) with respect to the dimension n and show that [Vol(Bp,q n )]1/n ≈ n−1/p for all 0 < p < ∞, 0 < q ≤ ∞. Furthermore, we study the ratio of Vol(Bp,∞ n ) and Vol(Bp n). We conclude by examining the decay of entropy numbers of embeddings of the Lorentz spaces.


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

 

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

 