THREE-WEIGHTED INEQUALITIES FOR SOME CLASS OF MATRIX OPERATORS
https://doi.org/10.55452/1998-6688-2025-22-2-220-241
Abstract
The criteria for the fulfillment of continuous and discrete inequalities involving Hardy operators are one of the key problems in the theory of weighted inequalities. The study of discrete inequalities for the class of matrix operators can be considered a new direction of research. In general, since the stability criterion in the weighted Lebesgue space for a discrete operator with a matrix kernel is not defined, various conditions are imposed on the matrix, which allows for obtaining broader results compared to the case without a matrix. In this work, we consider discrete quasilinear operators with matrices satisfying certain conditions. The results obtained for quasilinear inequalities can be applied to the description of bilinear Hardy inequalities.
About the Authors
N. S. Zhangabergenova
L.N. Gumilyov Eurasian National University
Kazakhstan
Kazakhstan
PhD
Astana
A. T. Temirhanova
L.N. Gumilyov Eurasian National University
Kazakhstan
Kazakhstan
PhD
Astana
References
1. Kufner A. and Maligranda L., Persson L.-E. The prehistory of the Hardy inequality, Amer. Math. Monthly., 113 (8), 715–732 (2006).
2. Kufner A., Maligranda L. and Persson L.-E. The Hardy Inequality: about its History and Some Related Results. (Pilsen: Vydavatelsky servis, 2007).
3. Oinarov R. and Kalybay A.A., Three – parameter weighted Hardy type inequalities, Banach Journal Math., 2 (2), 85–93 (2008).
4. Burenkov V.I. and Oinarov R., Necessary and Sufficient conditions for boundedness of the Hardy-type operator from a weighted Lebesque space to a Morrey-type space, Math. Inequal. Appl., 16(1), 1–19 (2013) https://doi.org/10.7153/mia-16-01.
5. Gogatishvili A., Mustafayev R. and et al., Some new iterated Hardy-type inequalities: the case, J. Inequal. Appl., 2013(515), 1–29 (2013).
6. Prokhorov D.V. and Stepanov V.D., O vesovykh neravenstvakh Khardi v smeshannykh normakh, Trudy Matematicheskogo instituta imeni V.A. Steklova, 283, 155–170 (2013 ) [in Russian]
7. Bernardis A.L. and Salvador P.O., Some new iterated Hardy-type inequalities and applications, J. Math. Ineq.,11(2), 577–594 (2017).
8. Stepanov V.D. and Shambilova G.E., On weighted iterated Hardy-type operators, Analysis Math., 44(2), 273–283 (2018).
9. Krepela M., Integral conditions for Hardy-type operators involving suprema, Collectanea Mathematica, 68, 21–50 (2017).
10. Kalybay A., On boundedness of the conjugate multidimensional Hardy operator from a Lebesque space to a local Morrey-type space, Int. J. Math. Anal., 8(11), 539–553 (2014).
11. Oinarov R. and Kalybay A., Weighted estimates of a class of integral operators with three parameters, J. Funct. Spaces. Appl., 2016, 1–11 (2016). https://doi.org/10.1155/2016/1045459
12. Kalybay А., Weighted estimates for a class of quasilinear integral operators, Siberian Mathematical Journal, 60 (2), 291–303 (2019).
13. Kalybay A. and Oinarov R., On weighted inequalities for a class of quasilinear integral operators, Banach Journal of Mathematical Analysis, 17 (3), 2–18 (2023).
14. Oinarov R. and Taspaganbetova Z. Criteria of boundedness and compactness of a class of matrix operators, J. Ineq. Appl., 2012 (53) (2012). https://doi.org/10.1186/1029-242X-2012-53.
15. Kalybay A., Temirkhanova A.M. and Zhangabergenova N., On iterated discrete Hardy type inequalities for a class of matrix operators, Analysis Mathematica, 49 (1), 137–150 (2023).
16. Kalybay A. and Zhangabergenova N, On iterated discrete Hardy type operators, Operators and Matrices, 17(1), 79–91 (2023). https://doi.org/10.7153/oam-2023-17-07
17. Zhangabergenova N. and Weighted estimates for a class of matrix operators, Math. Inequalities and Appl., 26 (3), 627–644 (2023). https://doi.org/10.7153/mia-2023-26-38.
18. Omarbayeva B.K., Persson L.-E. and Temirkhanova A.M., Weighted iterated discrete Hardy-type inequalities, Math. Inequalities and Appl., 23(3), 943–959 (2020). https://doi:10.7153/mia-2020-23-73.
19. Temirkhanova A.M. and Omarbayeva B.K., Vesovaia otsenka odnogo klassa kvazilineinykh diskretnykh operatorov: sluchai , Vestnik Kazakhskogo natsionalnogo pedagogicheskogo universiteta imeni Abaia. Seriia Fiziko-matematicheskie nauki, 67 (3), 109–116 (2019). [in Russian]
20. Oinarov R., Omarbayeva B.K. and Temirkhanova A.M., Discrete iterated Hardy-type inequalities with three weights, Journal of Mathematics, Mechanics, Computer Science, 105 (1), 19–29 (2020). https://doi.org/10.26577/JMMCS.2020.v105.i1.03.
21. Temirkhanova А.М. and Omarbayeva В.К., Weighted estimate of a class of quasilinear discrete operators: the case , Vestnik KazNRTU, 140 (4), 588–595 (2020).
22. Zhangabergenova N. and Temirkhanova A.M. Iterated discrete Hardy-typeinequalities, Eurasian Mathematical Journal, 14 (1), 81–95 (2023). https://doi.org/10.32523/2077-9879-2023-14-1-81-95.
23. Bennett G., Some elementary inequalities, Quart. J. Math. Oxford Ser., 38 (2), 401–425 (1987).
24. Bennett G., Some elementary inequalities III, Quart. J. Math. Oxford Ser., 42 (2), 149–174 (1991).
25. Oinarov R. and Shalgynbaeva S.Kh.,Vesovaia additivnaia otsenka odnogo klassa matrichnykh operatorov. Izvestiia Natsionalnoi akademii nauk Respubliki Kazakhstan, Seriia Fiziko-matematicheskaia, 1, 39–49 (2004). [in Russian]
26. Shaimardan S. and Shalgynbaeva S., Hardy-type inequalities for matrix operators, Bulletin of the Karaganda University. Mathematics, 88 (4), 63–72 (2017).
27. Taspaganbetova Zh. and Temirkhanova A., Boundedness and compactness criteria of a certain class of matrix operators, Math. journal, 11, 2(40), 73–85 (2011).
Review
For citations:
Zhangabergenova N.S., Temirhanova A.T. THREE-WEIGHTED INEQUALITIES FOR SOME CLASS OF MATRIX OPERATORS. Herald of the Kazakh-British Technical University. 2025;22(2):220-241. (In Kazakh) https://doi.org/10.55452/1998-6688-2025-22-2-220-241
Views:
17
Email this article 