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The Weighted Space Odyssey
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.ORCID iD: 0000-0003-0234-1645
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. , p. 57
Series
Karlstad University Studies, ISSN 1403-8099 ; 2017:1
Keyword [en]
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: urn:nbn:se:kau:diva-41944ISBN: 978-91-7063-734-6 (print)ISBN: 978-91-7063-735-3 (electronic)OAI: oai:DiVA.org:kau-41944DiVA, id: diva2:924668
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
List of papers
1. Convolution inequalities in weighted Lorentz spaces
Open this publication in new window or tab >>Convolution inequalities in weighted Lorentz spaces
2014 (English)In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 17, no 4, p. 1201-1223Article in journal (Refereed) Published
Abstract [en]

We characterize boundedness of a convolution operator with a fixed kernel between the weighted Lorentz spaces Lambda(p)(v) and Gamma(q)(w) for 0 < p <= q <= infinity, 1 <= q < p < infinity and 0 < q <= p = infinity. We provide corresponding weighted Young-type inequalities and also study basic properties of some new involved r.i. spaces.

Place, publisher, year, edition, pages
Croatia: Element, 2014
Keyword
Convolution, Young inequality, O’Neil inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-31751 (URN)10.7153/mia-17-90 (DOI)000345462600001 ()
Available from: 2014-03-23 Created: 2014-03-23 Last updated: 2017-12-06Bibliographically approved
2.
The record could not be found. The reason may be that the record is no longer available or you may have typed in a wrong id in the address field.
3. Convolution in weighted Lorentz spaces of type Γ
Open this publication in new window or tab >>Convolution in weighted Lorentz spaces of type Γ
2016 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, p. 113-132Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Aarhus Universitetsforlag, 2016
Keyword
Convolution, Young inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-31752 (URN)10.7146/math.scand.a-24187 (DOI)000383815600007 ()
Note

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

Available from: 2014-03-23 Created: 2014-03-23 Last updated: 2017-12-06Bibliographically approved
4. Bilinear weighted Hardy inequality for nonincreasing functions
Open this publication in new window or tab >>Bilinear weighted Hardy inequality for nonincreasing functions
2017 (English)In: Publications mathématiques (Bures-sur-Yvette), ISSN 0073-8301, E-ISSN 1618-1913, Vol. 61, no 1, p. 3-50Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Barcelona: Universitat Autonoma de Barcelona, 2017
Keyword
Hardy operators, bilinear operators, weights, inequalities for monotone functions
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-35214 (URN)10.5565/PUBLMAT_61117_01 (DOI)000396538700001 ()
Available from: 2015-02-13 Created: 2015-02-13 Last updated: 2017-10-19Bibliographically approved
5. Iterating bilinear Hardy inequalities
Open this publication in new window or tab >>Iterating bilinear Hardy inequalities
2017 (English)In: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839Article in journal (Refereed) Published
Abstract [en]

An iteration technique to characterize boundedness of certain types of multilinear operators is presented, reducing the problem into a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of a bilinear Hardy inequality in the weighted Lebesgue space setting. More equivalent characterizing conditions are presented. The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

Keyword
Hardy operators; bilinear operators; weights; operator inequalities
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kau:diva-41238 (URN)10.1017/S0013091516000602 (DOI)
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2017-12-06Bibliographically approved
6. Integral conditions for Hardy-type operators involving suprema
Open this publication in new window or tab >>Integral conditions for Hardy-type operators involving suprema
2017 (English)In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 68, no 1, p. 21-50Article in journal (Refereed) Published
Abstract [en]

The article contains characterizations of boundedness of an iterated supremal Hardy-type operator between weighted Lebesgue spaces, and an supremal Hardy operator restricted to positive decreasing functions between the same spaces. The found condtitions have an explicit integral/supremal form and cover all possible cases of positive exponents of the involved spaces.

Place, publisher, year, edition, pages
Springer, 2017
Keyword
supremal operators, Hardy inequalities, weighted function spaces
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41239 (URN)10.1007/s13348-016-0170-6 (DOI)000392229000003 ()
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2017-09-07Bibliographically approved
7.
The record could not be found. The reason may be that the record is no longer available or you may have typed in a wrong id in the address field.
8. Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$
Open this publication in new window or tab >>Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$
2017 (English)In: Revista Matemática Complutense, ISSN 1139-1138, E-ISSN 1988-2807Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer, 2017
Keyword
Hardy operators, Oinarov kernel, weighted Lebesgue spaces, weighted inequalities, integral operators
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41241 (URN)10.1007/s13163-017-0230-9 (DOI)
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2017-11-30Bibliographically approved
9. Convolution inequalities in weighted Lorentz spaces: case 0<q<1
Open this publication in new window or tab >>Convolution inequalities in weighted Lorentz spaces: case 0<q<1
2017 (English)In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 20, no 1, p. 191-201Article in journal (Refereed) Published
Abstract [en]

We characterize boundedness of a convolution operator between weighted Lorentz spaces $\Lambda^p(v)$and $\Gamma^q(w)$ in the case $0<q<1$.

Keyword
Convolution, Young inequality, Lorentz spaces, weights
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-41943 (URN)10.7153/mia-20-13 (DOI)000397414900012 ()
Available from: 2016-04-28 Created: 2016-04-28 Last updated: 2017-10-19

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