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Convolution in weighted Lorentz spaces of type Γ
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.ORCID-id: 0000-0003-0234-1645
2016 (Engelska)Ingår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, nr 1, s. 113-132Artikel i tidskrift (Refereegranskat) Published
Ort, förlag, år, upplaga, sidor
Aarhus Universitetsforlag, 2016. Vol. 119, nr 1, s. 113-132
Nyckelord [en]
Convolution, Young inequality, Lorentz spaces, weights
Nationell ämneskategori
Matematisk analys
Forskningsämne
Fysik
Identifikatorer
URN: urn:nbn:se:kau:diva-31752DOI: 10.7146/math.scand.a-24187ISI: 000383815600007OAI: oai:DiVA.org:kau-31752DiVA, id: diva2:706896
Anmärkning

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

Tillgänglig från: 2014-03-23 Skapad: 2014-03-23 Senast uppdaterad: 2017-12-06Bibliografiskt granskad
Ingår i avhandling
1. Forever Young: Convolution Inequalities in Weighted Lorentz-type Spaces
Öppna denna publikation i ny flik eller fönster >>Forever Young: Convolution Inequalities in Weighted Lorentz-type Spaces
2014 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Karlstad: Karlstads universitet, 2014. s. 23
Serie
Karlstad University Studies, ISSN 1403-8099 ; 2014:21
Nyckelord
Convolution, Young inequality, Lorentz spaces, weights, rearrangement-invariant spaces
Nationell ämneskategori
Matematisk analys
Forskningsämne
Matematik
Identifikatorer
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 (Engelska)
Opponent
Handledare
Anmärkning

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

Tillgänglig från: 2014-04-17 Skapad: 2014-03-24 Senast uppdaterad: 2019-07-12Bibliografiskt granskad
2. The Weighted Space Odyssey
Öppna denna publikation i ny flik eller fönster >>The Weighted Space Odyssey
2017 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Karlstad: Karlstads universitet, 2017. s. 57
Serie
Karlstad University Studies, ISSN 1403-8099 ; 2017:1
Nyckelord
integral operators, supremal operators, weights, weighted function spaces, Lorentz spaces, Lebesgue spaces, convolution, Hardy inequality, multilinear operators, nonincreasing rearrangement
Nationell ämneskategori
Matematisk analys
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kau:diva-41944 (URN)978-91-7063-734-6 (ISBN)978-91-7063-735-3 (ISBN)
Disputation
2017-02-10, 9C203, Karlstads universitet, Karlstad, 09:00 (Engelska)
Opponent
Handledare
Anmärkning

Artikel 9 publicerad i avhandlingen som manuskript med samma titel.

Tillgänglig från: 2017-01-18 Skapad: 2016-04-28 Senast uppdaterad: 2019-07-12Bibliografiskt granskad

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