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Convolution in Rearrangement-Invariant Spaces Defined in Terms of Oscillation and the Maximal Function
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
2014 (English)In: Zeitschrift für Analysis und ihre Anwendungen, ISSN 0232-2064, E-ISSN 1661-4534, Vol. 33, no 4, 369-383 p.Article in journal (Refereed) PublishedText
Abstract [en]

We characterize boundedness of a convolution operator with a fixed kernel between the classes S p ( v), de fined in terms of oscillation, and weighted Lorentz spaces Gamma(q)(w), defined in terms of the maximal function, for 0 < p; q <= infinity. We prove corresponding weighted Young-type inequalities of the form parallel to f * g parallel to Gamma(q)(w) <= C parallel to f parallel to S-p(v)parallel to g parallel to Y and characterize the optimal rearrangement-invariant space Y for which these inequalities hold.

Place, publisher, year, edition, pages
2014. Vol. 33, no 4, 369-383 p.
Keyword [en]
Convolution, Young inequality, weighted Lorentz spaces, oscillation
National Category
Research subject
URN: urn:nbn:se:kau:diva-41570DOI: 10.4171/ZAA/1517ISI: 000347639000001OAI: diva2:922437
Available from: 2016-04-22 Created: 2016-04-11 Last updated: 2016-08-17Bibliographically 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.
Karlstad University Studies, ISSN 1403-8099 ; 2014:21
Convolution, Young inequality, Lorentz spaces, weights, rearrangement-invariant spaces
National Category
Mathematical Analysis
Research subject
urn:nbn:se:kau:diva-31754 (URN)978-91-7063-552-6 (ISBN)
2014-05-09, 3B426, Karlstads universitet, Universitetsgatan 2, Karlstad, 10:15 (English)

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

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Křepela, Martin
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