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Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. Charles University in Prague, Department of Mathematical Analysis.ORCID iD: 0000-0003-0234-1645
(English)Article in journal (Refereed) Submitted
Abstract [en]

Boundedness of a fundamental Hardy-type operator with a kernel is characterized between weighted Lebesgue spaces $L^p(v)$ and $L^q(w)$ for $0<q<1\le p<\infty$. The conditions are explicit and have a standard integral form.

Keyword [en]
Hardy operators, Oinarov kernel, weighted Lebesgue spaces, weighted inequalities, integral operators
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-41241OAI: oai:DiVA.org:kau-41241DiVA: diva2:917049
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2016-11-03Bibliographically approved
In thesis
1. The Weighted Space Odyssey
Open this publication in new window or tab >>The Weighted Space Odyssey
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The common topic of this thesis is boundedness of integral and supremal operators between weighted function spaces.

The first type of results are characterizations of boundedness of a convolution-type operator between general weighted Lorentz spaces. Weighted Young-type convolution inequalities are obtained and an optimality property of involved domain spaces is proved. Additional provided information includes an overview of basic properties of some new function spaces appearing in the proven inequalities.

In the next part, product-based bilinear and multilinear Hardy-type operators are investigated. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.

Further objects of study are iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, thus completing the theory of the involved fundamental Hardy-type operators.

Abstract [en]

Operators acting on function spaces are classical subjects of study in functional analysis. This thesis contributes to the research on this topic, focusing particularly on integral and supremal operators and weighted function spaces.

Proving boundedness conditions of a convolution-type operator between weighted Lorentz spaces is the first type of a problem investigated here. The results have a form of weighted Young-type convolution inequalities, addressing also optimality properties of involved domain spaces. In addition to that, the outcome includes an overview of basic properties of some new function spaces appearing in the proven inequalities.

 Product-based bilinear and multilinear Hardy-type operators are another matter of focus. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.

 The last part of the presented work concerns iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, completing the theory of the involved fundamental Hardy-type operators.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2017. 57 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2017:1
Keyword
integral operators, supremal operators, weights, weighted function spaces, Lorentz spaces, Lebesgue spaces, convolution, Hardy inequality, multilinear operators, nonincreasing rearrangement
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41944 (URN)978-91-7063-734-6 (ISBN)978-91-7063-735-3 (ISBN)
Public defence
2017-02-10, 9C203, Karlstads universitet, Karlstad, 09:00 (English)
Opponent
Supervisors
Available from: 2017-01-18 Created: 2016-04-28 Last updated: 2017-01-18Bibliographically approved

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