published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2009. / Includes bibliographical references (leaf 49) Also available in print.

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 122-123). Also available for download via the World Wide Web; free to University of Oregon users.

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This thesis is devoted to an investigation of boundedness of a general convolution operator between certain weighted Lorentz-type spaces with the aim of proving analogues of the Young convolution inequality for these spaces. Necessary and sufficient conditions on the kernel function are given, for which the convolution operator with the fixed kernel is bounded between a certain domain space and the weighted Lorentz space of type Gamma. The considered domain spaces are the weighted Lorentz-type spaces defined in terms of the nondecreasing rearrangement of a function, the maximal function or the difference of these two quantities. In each case of the domain space, the corresponding Young-type convolution inequality is proved and the optimality of involved rearrangement-invariant spaces in shown. Furthermore, covering of the previously existing results is also discussed and some properties of the new rearrangement-invariant function spaces obtained during the process are studied. / <p>Paper II was a manuscript at the time of the defense.</p>

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Daniel M. Oberlin, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (June 18, 2004). Includes bibliographical references.

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005). Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale. We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The methods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada. We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings. In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

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In dieser Arbeit behandeln wir das Gebiet der Entropiezahlen - wir definieren sie ganz allgemein, stellen Eigenschaften und ihre Bedeutung heraus und arbeiten uns bis zu den aktuell wichtigsten Resultaten vor. Das Konzept ist so angelegt, dass es schon für Studenten im Hauptstudium Mathematik verstä̈ndlich sein sollte, auf jeden Fall werden Kenntnisse in Analysis, Funktionalanalysis, Kombinatorik und - in Ansätzen - lineare Algebra vorausgesetzt. / We deal with the topic of entropy numbers - we define them abstract, point out their and develop the plot up to the actually important results.