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Weighted inequalities, limiting real interpolation and function spaces
Grover, Manvi ; Opic, Bohumír (advisor) ; Persson, Lars-Erik (referee) ; Nekvinda, Aleš (referee)
This thesis is focused on studying limiting interpolation spaces with weight func- tions of slowly varying type and properties of operators defined on them. In Paper 1 we establish conditions under which K-spaces in the limiting real interpolation involving slowly varying functions can be described by means of J-spaces and we also solve the reverse problem. Further, we apply our results to obtain density theorems for the corresponding limiting interpolation spaces. In paper 2 we study the properties of compactness of operators defined on lim- iting interpolation spaces and derive the quantitative estimates of measure of non-compactness. In paper 3 we estimate dual spaces of limiting interpolation spaces that involve weight functions of slowly varying type. 1
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Properties of function spaces and operators acting on them
Turčinová, Hana ; Nekvinda, Aleš (advisor) ; Edmunds, David Eric (referee) ; Sickel, Winfried (referee)
The present thesis is focused on the study of properties of function spaces con- taining measurable functions, and operators acting on them. It consists of four papers. In the first paper, we establish a new characterization of the set of Sobolev functions with zero traces via the distance function from the boundary of a do- main. This characterization is innovative in that it is based on the space L1,∞ a of functions having absolutely continuous quasinorms in L1,∞ . In the second paper, we investigate properties of certain new scale of spaces governed by a functional involving the maximal nonincreasing rearrangement and powers. Motivation for studying such structures stems from a recent research of sharp Sobolev embeddings into spaces furnished with Ahlfors measures. In the third paper, we extend discretization techniques for Lorentz norms by eliminating nondegeneracy restrictions on weights. We apply the method to characterize general embeddings between classical Lorentz spaces. In the fourth paper, we characterize triples of weights for which an inequality involving the superposition of two integral operators holds. We apply results from the third paper to avoid duality and to obtain thereby a general result. 1
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Summation Methods
Berkman, Pavel ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In the given thesis, we study limiting (summation) methods. The problems are divided into two main groups, the first one focusing on elementary limiting methods, and the second one dealing with methods which generalise those from the first group, for instance the class of matrix limiting methods. The base of the thesis is the Toeplitz theorem, which characterises regular matrix methods. Furthermore, we invent the term improper regularity, which we subsequently apply to individual methods. By doing that we extend our knowledge of their field of convergence. We especially deal with Hutton's method, where we present some of our own results. All findings are illustrated with examples for better understanding. 1
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Characterization of functions with zero traces via the distance function
Turčinová, Hana ; Nekvinda, Aleš (advisor) ; Edmunds, David Eric (referee)
Consider a domain Ω ⊂ RN with Lipschitz boundary and let d(x) = dist(x, ∂Ω). It is well known for p ∈ (1, ∞) that u ∈ W1,p 0 (Ω) if and only if u/d ∈ Lp (Ω) and ∇u ∈ Lp (Ω). Recently a new characterization appeared: it was proved that u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1 (Ω) and ∇u ∈ Lp (Ω). In the author's bachelor thesis the condition u/d ∈ L1 (Ω) was weakened to the condition u/d ∈ L1,p (Ω), but only in the case N = 1. In this master thesis we prove that for N ≥ 1, p ∈ (1, ∞) and q ∈ [1, ∞) we have u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1,q (Ω) and ∇u ∈ Lp (Ω). Moreover, we present a counterexample to this equivalence in the case q = ∞. 1
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Laplaceova transformace na prostorech funkcí
Buriánková, Eva ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this manuscript we study the action of the Laplace transform on rearrangement-invariant Banach function spaces. Our principal goal is to characterize the optimal range space corresponding to a given domain space within the category of rearrangement-invariant Banach function spaces. We first prove a key pointwise estimate of the non-increasing rearrangement of the image under the Laplace transform of a given function. Then we use this inequality to carry out the construction of the optimal range space. We apply this general result to establish an optimality relation between the Lebesgue and Lorentz spaces under the Laplace transform.
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Weighted rearrangement-invariant function spaces
Soudský, Filip ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis we focus on generalized Gamma spaces GΓ(p, m, v) and classify some of their intrinsic properties. In an article called Relative Re- arrangement Methods for Estimating Dual Norm (for details see references), the authors attempted to characterize their associate norms but obtained only several one-sided estimates. Equipped with these, they further showed reflexivity of gener- alized Gamma spaces for p ≥ 2 and m > 1 under an additional restriction that the underlying measure space is of finite measure. However, the full characterization of the associate norm and of the reflexivity of such spaces for 2 > p > 1 remained an open problem. In this thesis we shall fill this gap. We extend the theory to a σ-finite measure space. We present a complete characterization of the associate norm, and we find necessary and sufficient conditions for the reflexivity of such spaces. 1
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Compactness of higher-order Sobolev embeddings
Slavíková, Lenka ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
The present work deals with m-th order compact Sobolev embeddings on a do- main Ω ⊆ Rn endowed with a probability measure ν and satisfying certain isoperi- metric inequality. We derive a condition on a pair of rearrangement-invariant spaces X(Ω, ν) and Y (Ω, ν) which suffices to guarantee a compact embedding of the Sobolev space V m X(Ω, ν) into Y (Ω, ν). The condition is given in terms of compactness of certain operator on representation spaces. This result is then applied to characterize higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, among them the Gauss space is the most stan- dard example. 1
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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
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