
Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Slavík, Antonín (referee) ; Tvrdý, Milan (referee)
Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen stockKurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges  additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called poscillation. We show that our indefinite integrals are a.e. approximately differ entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, HenstockKurzweil in tegral, Divergence theorem, Analysis in metric measure spaces 1


Weighted inequalities and properties of operators and embeddings on function spaces
Slavíková, Lenka ; Pick, Luboš (advisor) ; Pérez, Carlos (referee) ; Malý, Jan (referee)
The present thesis is devoted to the study of various properties of Banach func tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higherorder embeddings of Sobolevtype spaces built upon rearrangementinvariant Banach function spaces. In particular, we show that optimal higherorder Sobolev embeddings follow from isoperimetric inequal ities. In the second paper we focus on the question when the abovementioned Sobolevtype space is a Banach algebra with respect to a pointwise multiplica tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...


Eliptické rovnice v nereflexivních prostorech funkcí
Maringová, Erika ; Bulíček, Miroslav (advisor) ; Malý, Jan (referee)
In the work we modify the wellknown minimal surface problem to a very special form, where the exponent two is replaced by a general positive parameter. To the modified problem we define four notions of solution in nonreflexive Sobolev space and in the space of functions of bounded variation. We examine the relationships between these notions to show that some of them are equivalent and some are weaker. After that we look for assumptions needed to prove the existence of solution to the problem in the sense of definitions provided. We outline that in the setting of spaces of functions of bounded variation the solution exists for any positive finite parameter and that if we accept some restrictions on the parameter then the solution exists in the Sobolev space, too. We also provide counterexample indicating that if the domain is nonconvex, the solution in Sobolev space need not exist. Powered by TCPDF (www.tcpdf.org)


Generalized ordinary differential equations in metric spaces
Skovajsa, Břetislav ; Malý, Jan (advisor)
The aim of this thesis is to build the foundations of generalized ordinary differ ential equation theory in metric spaces. While differential equations in metric spaces have been studied before, the chosen approach cannot be extended to in clude more general types of integral equations. We introduce a definition which combines the added generality of metric spaces with the strength of Kurzweil's generalized ordinary differential equations. Additionally, we present existence and uniqueness theorems which offer new results even in the context of Euclidean spaces.


Sobolevtype Spaces on Metric Measure Spaces
Malý, Lukáš ; Pick, Luboš (advisor) ; Malý, Jan (referee) ; Shanmugalingam, Nages (referee)
Title: SobolevType Spaces on Metric Measure Spaces Author: RNDr. Lukáš Malý Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: is thesis focuses on function spaces related to rstorder analysis in abstract metric measure spaces. In metric spaces, we can replace distributional gra dients, whose de nition depends on the linear structure of Rn , by upper gradients that control the functions' behavior along all recti able curves. is gives rise to the socalled Newtonian spaces. e summability condition, considered in the thesis, is expressed using a general Banach function lattice quasinorm and so an extensive framework is built. Sobolevtype spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid s. Standard toolbox for the theory is set up in this general setting and Newto nian spaces are proven complete. Summability of an upper gradient of a function is shown to guarantee the function's absolute continuity on almost all curves. Ex istence of a unique minimal weak upper gradient is established. Regularization of Newtonian functions via Lipschitz truncations is discussed in doubling Poincaré spaces using weak boundedness of maximal...


Zobecněné obyčejné diferenciální rovnice v metrických prostorech
Skovajsa, Břetislav ; Malý, Jan (advisor) ; Pražák, Dalibor (referee)
The aim of this thesis is to build the foundations of generalized ordinary differ ential equation theory in metric spaces. While differential equations in metric spaces have been studied before, the chosen approach cannot be extended to in clude more general types of integral equations. We introduce a definition which combines the added generality of metric spaces with the strength of Kurzweil's generalized ordinary differential equations. Additionally, we present existence and uniqueness theorems which offer new results even in the context of Euclidean spaces.


Arthur I, Duke of Brittany and His Claim on the English Throne
Malý, Jan ; Drška, Václav (advisor) ; Suchánek, Drahomír (referee)
The death of king Richard The Lion Heart in 1199 caused considerable troubles to the Angevin empire, when there again raised for english medieval history very pressing question  who is legitimate successor to the throne? There were two possible pretendents, both had comparable claim to the crown. First of them was Richard's brother John, the second his nephew, at this time twelve years old duke of Brittany Arthur. Legal customs of this period theoretically admitted the succession of both men, because there were no unified successorial usage and every single part of the Angevin empire looked on this problem differently. While John was generally accepted without problems in Normandy and then he was crowned king of England, the toughest fight blazed out in Anjou, Maine and Touraine, where support was given to Arthur. He had also support of king of France Philip Augustus, who understood well, that Arthur is an ideal tool for his schemes to elimination and mastery over the Angevin empire. Whole long struggle between the nephew and his uncle was finsihed by Arthur's capture in the summer of 1202 and his subsequent death in 1203. However king John was not able to stop the dissolution of Plantagenet empire, which was reduced to the duchy of Aquitaine at the beginning of 13th century.


Sobolev mappings and Luzin condition N
Matějka, Milan ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
A mapping f from R^{n} to R^{n} is said to satisfy the Luzin condition N if f maps sets of measure zero to sets of measure zero. It is known to be valid for mappings in the Sobolev space W^{1,p} for p > n and for p <= n there are counterexamples. The aim of this thesis is to summarize known results and study the validity of Luzin condition N for mappings in the Sobolev space W^{2,p}.


First Baron War and Louis VIII as the Kinf of England (12151217). A view of sources.
Malý, Jan ; Drška, Václav (advisor) ; Suchánek, Drahomír (referee)
In 1215, king of England John the Lackland was forced to seal a document known as Magna Carta. It was a simple legal procedure. Nevertheless it was the beginning of conflict between royal power and english nobility, commonly known as first baron's war. It lasted from 1215 to 1217 and it culminated by the invasion of french crown prince Louis to England when english nobility offered him the crown. After the death of king John in autumn 1216, the original revolt, inspired mostly by personal hate against the ruler changed into the effort of prince Louis to achieve the title of English king. Most of original memebers of the oposition quickly switched sides and joined John's little son Henry (king Henry III) and Louis then had only the support of few leaders of rebelion. After a series of defeats in the first half of 1217, capetian prince was forced to abandon his goals and to retreat from the Isles. This work also follows the view of selected contemporary narrative sources, mainly to the activity of french prince in England and his attitude to the uprising.


Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Rataj, Jan (referee)
Title: Nonabsolutely convergent integrals Author: Kristýna Kuncová Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jan Malý, DrSc., Department of Mathematical Analysis Abstract: Our aim is to introduce an integral on a measure metric space, which will be nonabsolutely convergent but including the Lebesgue integral. We start with spaces of continuous and Lipschitz functions, spaces of Radon measures and their dual and predual spaces. We build up the socalled uniformly controlled integral (UCintegral) of a function with respect to a distribution. Then we investigate the relationship between the UCintegral with respect to a measure and the Lebesgue integral. Then we introduce another kind of integral, called UCNintegral, based on neglecting of small sets with respect to a Hausdorff measure. Hereafter, we focus on the concept of ndimensional metric currents. We build the UCintegral with respect to a current and then we proceed to a very general version of GaussGreen Theorem, which includes the Stokes Theorem on manifolds as a special case. Keywords: Nonabsolutely convergent integrals, Multidimensional integrals, GaussGreen Theorem 1
