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#### Open Access in DiVA

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Křepela, Martin
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Křepela, Martin
##### By organisation

Department of Mathematics and Computer Science
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Mathematical Analysis
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The Weighted Space OdysseyPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Abstract [en]

##### Place, publisher, year, edition, pages

Karlstad: Karlstads universitet, 2017. , p. 57
##### Series

Karlstad University Studies, ISSN 1403-8099 ; 2017:1
##### Keywords [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)
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##### Note

##### List of papers

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.

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.

Artikel 9 publicerad i avhandlingen som manuskript med samma titel.

Available from: 2017-01-18 Created: 2016-04-28 Last updated: 2019-07-12Bibliographically approved1. Convolution inequalities in weighted Lorentz spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay706894",{id:"formSmash:j_idt516:0:j_idt520",widgetVar:"overlay706894",target:"formSmash:j_idt516:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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3. Convolution in weighted Lorentz spaces of type Γ$(function(){PrimeFaces.cw("OverlayPanel","overlay706896",{id:"formSmash:j_idt516:2:j_idt520",widgetVar:"overlay706896",target:"formSmash:j_idt516:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Bilinear weighted Hardy inequality for nonincreasing functions$(function(){PrimeFaces.cw("OverlayPanel","overlay788181",{id:"formSmash:j_idt516:3:j_idt520",widgetVar:"overlay788181",target:"formSmash:j_idt516:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Iterating bilinear Hardy inequalities$(function(){PrimeFaces.cw("OverlayPanel","overlay917025",{id:"formSmash:j_idt516:4:j_idt520",widgetVar:"overlay917025",target:"formSmash:j_idt516:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Integral conditions for Hardy-type operators involving suprema$(function(){PrimeFaces.cw("OverlayPanel","overlay917035",{id:"formSmash:j_idt516:5:j_idt520",widgetVar:"overlay917035",target:"formSmash:j_idt516:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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8. Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$$(function(){PrimeFaces.cw("OverlayPanel","overlay917049",{id:"formSmash:j_idt516:7:j_idt520",widgetVar:"overlay917049",target:"formSmash:j_idt516:7:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

9. Convolution inequalities in weighted Lorentz spaces: case 0<q<1$(function(){PrimeFaces.cw("OverlayPanel","overlay924661",{id:"formSmash:j_idt516:8:j_idt520",widgetVar:"overlay924661",target:"formSmash:j_idt516:8:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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