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Convolution in weighted Lorentz spaces of type Γ
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
2016 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, 113-132 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Aarhus Universitetsforlag, 2016. Vol. 119, no 1, 113-132 p.
Keyword [en]
Convolution, Young inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Physics
Identifiers
URN: urn:nbn:se:kau:diva-31752DOI: 10.7146/math.scand.a-24187ISI: 000383815600007OAI: oai:DiVA.org:kau-31752DiVA: diva2:706896
Note

This article was published as manuscript in Martin Křepelas licentiate thesis.

Available from: 2014-03-23 Created: 2014-03-23 Last updated: 2016-12-01Bibliographically approved
In thesis
1. Forever Young: Convolution Inequalities in Weighted Lorentz-type Spaces
Open this publication in new window or tab >>Forever Young: Convolution Inequalities in Weighted Lorentz-type Spaces
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2014. 23 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2014:21
Keyword
Convolution, Young inequality, Lorentz spaces, weights, rearrangement-invariant spaces
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-31754 (URN)978-91-7063-552-6 (ISBN)
Presentation
2014-05-09, 3B426, Karlstads universitet, Universitetsgatan 2, Karlstad, 10:15 (English)
Opponent
Supervisors
Note

Paper II was a manuscript at the time of the defense.

Available from: 2014-04-17 Created: 2014-03-24 Last updated: 2016-08-17Bibliographically approved
2. 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
Note

Artikel 9 publicerad i avhandlingen som manuskript med samma titel.

Available from: 2017-01-18 Created: 2016-04-28 Last updated: 2017-10-19Bibliographically approved

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