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Boundedness of quasilinear integral operators on the cone of monotone functions
Peoples’ Friendship University of Russia Steklov Institute of Mathematics, Moscow, Russia.
Financial University Under the Government of the Russian Federation, Moscow, Russia.
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. Vol. 57, no 5, p. 884-904
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-68310DOI: 10.1134/S0037446616050190ISI: 000386780100019Scopus ID: 2-s2.0-84992347136OAI: oai:DiVA.org:ltu-68310DiVA, id: diva2:1197186
Available from: 2018-04-12 Created: 2018-04-12 Last updated: 2018-04-12Bibliographically approved
In thesis
1. Some new Hardy-type inequalities on the cone of monotone functions
Open this publication in new window or tab >>Some new Hardy-type inequalities on the cone of monotone functions
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Några nya Hardy-typ olikheter på konen av monotona funktioner
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:nbn:se:ltu:diva-68294 (URN)978-91-7790-102-0 (ISBN)978-91-7790-103-7 (ISBN)
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

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