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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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Famous unsolvable problems.
Kesely, Michal ; Pick, Luboš (referee) ; Pražák, Dalibor (advisor)
Title: Famous nnsolvable problems Author: Michal Kesely Department,: Deportment of Mathematical Analysis Supervisor: RNDr. Dalibor Prazak, Ph.D. Supervisor's e-mail address; prazak^karlin.inff.cuni.cz Abst.ra.ct: In the present work we study three famous problems of antiquity (the Delian problem, the trisect,ion of an angle and the squaring of a. cir- cle), which turned to be nnsolvable much later. In the first chapter we will formalize the concept of Euclidean construction, prove few theorems about algebraic numbers and show an interesting connection between con- structible numbers and algebraic numbers. In the next, two chapters we will prove the insolvability of the Delia.ii problem and the trisection of an an- gle using the properties of constructible numbers. Furthermore in (.he third chapter we will mention some incorrect solutions of the trisection problem, In the last, chapter we will prove the existence of transcendental numbers, build an appropriate apparatus and finally we will prove the transcendence of two famous const.nnts - c and TV. The insolvabilityof the squaring problem is a direct, consequence of the transcendence of T\. Keywords: unsolvable problem, constrnctible. transcendental
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Banach Function Spaces
Marko, Ján ; John, Oldřich (referee) ; Pick, Luboš (advisor)
N'a/ev prace: Banarhovy prostory fuiikci Autor: Jan Marko Katcdr;i: KaTedra inaleniaticke analy/y Yedouci bakalarske prace: doc1. UN Dr. Lubus Pick. CSc.. DSc. t'-inail vedoudho: Lubos.Pick'i'inff.cuni.c/ Abst.ra.kt.: V t.ejto ])riici su popisane xakladne vlastnosti Baimchovho priostoru funkcii. jeho podpriestor funkfii s absolutnc spujitou iiorniou a asociovany Banai-liov priestor I'luikcii. Zaobcra sa lie/ problcinal.ikou Lcbcsgucovycli pricslorov ['unkcii. branycli ako Banadiovo priest ory funkcif. \ Icxte su vyprarovaiu'1 priklady lykajuce sa.vlastnosti niicr ineratel'nych priosl.orov a k-h vplyv na ist udovanc'1 podpricstory. Taklio/ su vypracovaiK'1 priklady Baiia- chovyi'li iiorioni. iin ])n'sliisiio limiacliovc priest ory funkfii a. ich /;ikladne. N'la Klfcova slova: Banacliuva noriiia. Banacliov prieslor I'uukcif, asociovany prieslor. spojita noriua Title: Banach funct ion spaces Author: Jan Marko Department: Department of Mathematical Analysis Supervisor: doc. HNDr. Lubus Pick, CSc.. DSc. 'rvisor's e-mail address: Lubos.Pick'imir.cuni.cx Abstract: This thesis describes basic properties of Banach function spaces, its subspace of functions of absolutely continuous norm and its associa.lt; space.. II.also deals with problems of Lebesgue spaces considered to be Banach function spaces. Several problems...
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Weighted inequalities for Hardy-type operators and their application in the Interplation Theory
Pražák, David ; Pick, Luboš (advisor) ; Krbec, Miroslav (referee)
We study real interpolation spaces (Xo, X1) 12,q, where {} is a parameter function, not necessarily a power weight. Using a discretization method we "discretize" the norm in (Xo, X1) 12,q. The resulting norm is given by the corresponding quasiconcave function h and its discretizing sequence, we denote the space endowed with this norm by (Xo, X1)h,q· We give a direct proof of a theorem dueto V. I. Ovchinnikov and A. S. Titenkov, which characterizes the space (Lp0 , Lp1 )h,q in terms of the non- increasing rearrangement. Further, we find a relation between the dilation indices of a quasiconcave function h and its discretizing sequence. In the case when the dilation indices of h are not limiting, the space ( Lp0 , Lp1 ) h,q coincides wi th some classical Lorentz space A q ( r.p). If the dilation indices are limiting, then we characterize the space (Lp0 , Lp1 )h,q as an extrapolation space. Powered by TCPDF (www.tcpdf.org)
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Weighted rearrangement-invariant spaces and their basic properties
Soudský, Filip ; Pick, Luboš (advisor)
In this thesis we shall provide the reader with results in the field of classical Lorentz spaces. These spaces have been studied since the 50's and have many applications in partial differential equations and interpolation theory. This work includes five papers. The first paper studies the properties of Generalized Gamma spaces. The second paper provides an alternative proof of normability characterization of classical Lorentz spaces. The third paper discusses conditions of linearity and quasi-norm property of rearrangement-invariant lattices. The following paper gives a characterization of normability of Gamma spaces. And finally the last paper characterizes the embeddings between GΓ spaces. 1
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