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- Burtseva, Evgeniya, 1988-
(författare)
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Boundedness of some linear operators in various function spaces
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis is devoted to boundedness of some classical linear operators in various function spaces. We prove boundedness of weighted Hardy type operators and the weighted Riesz potential in Morrey—Orlicz spaces. Furthermore, we consider central Morrey—Orlicz spaces and prove boundedness of the Riesz potential in these spaces. We also present results concerning boundedness of Hardy type operators in Hölder type spaces. The thesis consists of four papers (Papers A—D), two complementary appendices (A1, B1) and an introduction.The introduction is divided into three parts. In the first part we give main definitions and properties of Morrey spaces, Orlicz spaces and Morrey—Orlicz spaces. In the second part we consider boundedness of the Riesz potential and Hardy type operators in various Banach ideal spaces. These operators have lately been studied in Lebesgue spaces, Morrey spaces and Orlicz spaces by many authors. We briefly describe this development and thereafter we present how these results have been extended to Morrey—Orlicz spaces (Paper A) and central Morrey—Orlicz spaces (Paper B). Finally, in the third part, we introduce Hölder type spaces and present our main results from Paper C and Paper D, which concern boundedness of Hardy type operators in Hölder type spaces. In Paper A we prove boundedness of the Riesz fractional integral operator between distinct Morrey—Orlicz spaces, which is a generalization of the Adams type result. Moreover, we investigate boundedness of some weighted Hardy type operators and weighted Riesz fractional integral operator between distinct Morrey—Orlicz spaces. The Appendix A1 contains detailed calculations of some examples, which illustrate one of our main results presented in Paper A.In Paper B we prove strong and weak boundedness of the Riesz potential in central Morrey—Orlicz spaces. We also give some examples, which illustrate the main theorem. Detailed calculations connected to one of the examples are described in the Appendix B1. In Paper C we consider n-dimensional Hardy type operators and prove that these operators are bounded in Hölder spaces. In Paper D we develop the results from paper C and derive necessary and sufficient conditions for the boundedness of n-dimensional weighted Hardy type operators in Hölder type spaces. We also obtain necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces on compactification of Rn.
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- Kato, Mikio, et al.
(författare)
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On James, Jordan - von Neumann constants and the normal structure coefficient of Banach spaces
- 2000
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant CNJ(X), and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) nad J(X*) are given as an answer to a problem of Gao and Lau[16). Connections between CNJ(X) and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by CNJ(X) constant, which implies that a Banach space with CNJ(X)-constant less than 5/4 has the fixed point property.
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- Kufner, Alois, et al.
(författare)
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The Hardy inequality : about its history and some related results
- 2007
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- The Hardy inequality has a fascinating past and will have (hopefully) also a fascinating future. Here, the authors present some important steps of the development of the classical Hardy inequality , of its early weighted generalizations (i.e. furnished with general weights) and of its various modifications and extensions. Besides the continuous (i.e. integral) case, the (originally) discrete one (i.e. for sequences) is dealt with. Eighteen theorems are formulated, most of them with proof. Although the choice of material is a matter of personal taste and knowledge, the authors intended to include as many results as possible, using papers published up to 2005.
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- Larsson, Leo, et al.
(författare)
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Multiplicative inequalities of Carlson type and interpolation
- 2006
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- Collecting all the results on the particular types of inequalities, the coverage of this book is unique among textbooks in the literature. The book focuses on the historical development of the Carlson inequalities and their many generalizations and variations. As well as almost all known results concerning these inequalities and all known proof techniques, a number of open questions suitable for further research are considered. Two chapters are devoted to clarifying the close connection between interpolation theory and this type of inequality. Other applications are also included, in addition to a historical note on Fritz Carlson himself.
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