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Some new Hardy-type inequalities on the cone of monotone functions
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Några nya Hardy-typ olikheter på konen av monotona funktioner (Swedish)
Abstract [en]

This PhD thesis is devoted to the study weighted Hardy-type inequalitieswith quasilinear integral operators on the cone of monotone functions. Thethesis consists of six papers (papers A - F) and an introduction, which givesa brief review of the theory of Hardy-type inequalities and also serves to putthese papers into a more general frame.

In papers A, D and E we characterize some weighted Hardy-type inequal-ities on the cone of non-increasing functions. This problem is related to theboundedness of the Hardy-Littlewood maximal operator in weighted LorentzΓ - spaces. In papers D and E the case with integral operators defined byso called Oinarov’s kernels are treated. In all cases necessary and sufficientconditions are derived.

In paper B we solve the similar problem for the cone of quasi-concavefunctions (i.e. when the function f satisfy two monotonicity conditions,namely that f (t) is non-decreasing and f(t)t is non-increasing). Such functions are of great importance for interpolation theory, approximation theory and related areas in functional analysis. Also here complete characterizations are given in all cases.

Paper C is devoted to characterizing weighted Hardy-type inequalities with supremum operators on the cone of monotone functions. In particular, the study of the case with non-decreasing functions was initiated in this paper.

In paper F we focus only on the much less studied problem, namely to characterize Hardy-type inequalities on the cone of non-decreasing functions. A new reduction method is used in a crucial way. Some complete charac-terizations for all studied cases are discussed and proved. The investigations initiated in paper C are here developed to a more general theory, which cov-ers all studied operators. The obtained results are used to derive some new bilinear Hardy-type inequalities.

Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2018.
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-68294ISBN: 978-91-7790-102-0 (print)ISBN: 978-91-7790-103-7 (electronic)OAI: oai:DiVA.org:ltu-68294DiVA, id: diva2:1196993
Public defence
2018-06-08, E 243, Luleå, 10:00 (English)
Opponent
Supervisors
Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-31Bibliographically approved
List of papers
1. Weighted Hardy type inequalities for supremum operators on the cones of monotone functions
Open this publication in new window or tab >>Weighted Hardy type inequalities for supremum operators on the cones of monotone functions
2016 (English)In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 237Article in journal (Refereed) Published
Abstract [en]

The complete characterization of the weighted Lp− Lr inequalities of supremum operators on the cones of monotone functions for all 0 < p, r≤ ∞ is given.

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-59569 (URN)10.1186/s13660-016-1168-z (DOI)000391726600001 ()2-s2.0-84988910456 (Scopus ID)
Note

Validerad; 2016; Nivå 2; 2016-10-12 (kribac)

Available from: 2016-10-07 Created: 2016-10-07 Last updated: 2018-07-10Bibliographically approved
2. Hardy-type inequalities on the weighted cones of quasi-concave functions
Open this publication in new window or tab >>Hardy-type inequalities on the weighted cones of quasi-concave functions
2015 (English)In: Banach Journal of Mathematical Analysis, ISSN 1735-8787, Vol. 9, no 2, p. 21-34Article in journal (Refereed) Published
Abstract [en]

The complete characterization of the Hardy-type Lp-Lq inequalities on the weighted cones of quasi-concave functions for all 0 < p, q < ∞ is given.

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-15326 (URN)10.15352/bjma/09-2-3 (DOI)2-s2.0-84946114931 (Scopus ID)ed1c416a-eb0a-41dc-9ad9-ee8d87652bed (Local ID)ed1c416a-eb0a-41dc-9ad9-ee8d87652bed (Archive number)ed1c416a-eb0a-41dc-9ad9-ee8d87652bed (OAI)
Note

Validerad; 2015; Nivå 2; 20150311 (johsod)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
3. The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions
Open this publication in new window or tab >>The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions
2014 (English)In: Siberian mathematical journal, ISSN 0037-4466, E-ISSN 1573-9260, Vol. 55, no 4, p. 745-767Article in journal (Refereed) Published
Abstract [en]

We study the problem of characterizing the weighted inequalities on the Lebesgue cones of monotone functions on the semiaxis for a class of quasilinear integral operators.

Place, publisher, year, edition, pages
Pleiades Publishing, Ltd., 2014
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-68283 (URN)10.1134/S0037446614040168 (DOI)000340941400016 ()2-s2.0-84906513547 (Scopus ID)
Available from: 2018-04-10 Created: 2018-04-10 Last updated: 2018-04-12Bibliographically approved
4. Some new Hardy-type inequalities on the cone of non-decreasing functions
Open this publication in new window or tab >>Some new Hardy-type inequalities on the cone of non-decreasing functions
2017 (English)Report (Other academic)
Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2017. p. 18
Series
Gula serien, ISSN 1400-4003
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-68315 (URN)
Available from: 2018-04-12 Created: 2018-04-12 Last updated: 2018-04-12Bibliographically approved
5. On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions
Open this publication in new window or tab >>On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions
2017 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 8, no 2, p. 47-73Article in journal (Refereed) Published
Abstract [en]

We solve the characterization problem of Lp v -Lr p weighted inequalities on Lebesgue cones of monotone functions on the half-axis for quasilinear integral operators of iterated type with Oinarov's kernels.

Place, publisher, year, edition, pages
Eurasian Mathematical Journal, 2017
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-68311 (URN)000412802400005 ()2-s2.0-85029000685 (Scopus ID)
Available from: 2018-04-12 Created: 2018-04-12 Last updated: 2018-04-12Bibliographically approved
6. Boundedness of quasilinear integral operators on the cone of monotone functions
Open this publication in new window or tab >>Boundedness of quasilinear integral operators on the cone of monotone functions
2016 (English)In: Siberian mathematical journal, ISSN 0037-4466, E-ISSN 1573-9260, Vol. 57, no 5, p. 884-904Article in journal (Refereed) Published
Abstract [en]

We study the problem of characterizing weighted inequalities on Lebesgue cones of monotone functions on the half-axis for one class of quasilinear integral operators.

Place, publisher, year, edition, pages
Springer, 2016
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-68310 (URN)10.1134/S0037446616050190 (DOI)000386780100019 ()2-s2.0-84992347136 (Scopus ID)
Available from: 2018-04-12 Created: 2018-04-12 Last updated: 2018-04-12Bibliographically approved

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