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Functional Properties of Invariant Spaces Defined in Terms of Oscillations

Show simple item record Salih, Salih Yousuf Mohamed 2013-12-08T07:29:38Z 2013-12-08T07:29:38Z 2010-08-01
dc.identifier.citation Salih,Salih Yousuf Mohamed.Functional Properties of Invariant Spaces Defined in Terms of Oscillations/Salih Yousuf Mohamed Salih;Shawgy Hussein Abdalla.-Khartoum:Sudan University of Science and Technology,College of Science,2010.-311p.: ill. ; 28cm.-PhD. en_US
dc.description Thesis en_US
dc.description.abstract we study the behavior of the weak- type weights and the normability of the Lorentz spaces. The embeddings between classical Lorentz spaces are considered. We show some generalizations of Orlicz- Lorentz spaces and the sharpness of the Orlicz- Sobolev embeddings. We establish the symmetrization and sharp Sobolev inequalities in metric spaces, with isoperimetry and symmetrization for Sobolev spaces on metric spaces and logarithmic Sobolev inequalities. We give a fractional Hardy inequality and optimal Sobolev embedding involving rearrangement- invariant quasinorm and in borderline cases. We have discussed a sharp higher order Sobolev embeddings and inequalities of symmetrization with self improvement. We introduce a rearrangement- invariant function set that appears in optimal Sobolev embeddings. We also show the functional properties of rearrangement- invariant spaces defined interms of oscillations. Isomorphic copies in the lattice and its symmetrization with applications to Orlics – Lorentz are considered. of Talenti. In both cases we are especially interested in when the quasinorms are optimal. The results yield best possible refinements of such limiting Sobolev inequalities as those of Trudinger, Strichartz, Hansson and Brézis-Wainger. In chapter 4, an embedding theorem is given for functions whose gradient belongs to a class slightly larger than the Lebesgue spaces. We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting sobolev embedding theorem due to Brézis and Wainger. We prove a sharp version of the Sobolev embedding theorem, and we compare the result with embeddings due to Hansson, Brézis-Wainger and Malý-Pick. In chapter 5, by using a natural extension of the Lebesgue spaces and a new form of the Pólya Szegö symmetrization principle, we extend the sharp version of the Sobolev embedding theorem. We develop a new method to obtain symmetrization inequalities of Sobolev type. The approach leeds to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement of invariant spaces. We give necessary and sufficient conditions for the set of measurable functions to be normable linear space. We also give a complete characterization of all spaces that can be represented for some space and of all spaces that appear in such representations. In chapter 6, we show some of the main questions concerning the function spaces which measure the oscillation of the function defined as the set of Lebesgue- measurable functions on the real line, relate to their functional properties such as their lattice property, normability and linearity. We devoted to the isomorphic structure of symmetrizations of quasi- Banach ideal function or sequence lattices. The symmeterization of a quasi-Banach ideal lattice of measurable functions, consists of all functions with decreasing rearrangement. We show that every subsymmetric basic sequence which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements. We apply these results to the family of two- weighted Orlicz-Lorentz spaces and Orlicz Marcinkiewicz spaces. We provide a complete characterization of order isomorphic copies in these spaces. en_US
dc.description.sponsorship Sudan University of Science and Technology en_US
dc.language.iso en en_US
dc.publisher Sudan University of Science and Technology en_US
dc.subject Invariant Spaces en_US
dc.subject Oscillations en_US
dc.title Functional Properties of Invariant Spaces Defined in Terms of Oscillations en_US
dc.type Thesis en_US

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