kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-4-153-167kaz29-1548Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESНЕРАВЕНСТВА И ТОЖДЕСТВА ХАРДИ, СВЯЗАННЫЕ С ВЕКТОРНЫМИ ПОЛЯМИ БАОУЭНДИ-ГРУШИНА И ЛАНДАУ-ГАМИЛЬТОНИАНОМHARDY INEQUALITIES AND IDENTITIES RELATED TO THE BAOUENDI-GRUSHIN VECTOR FIELDS AND LANDAU-HAMILTONIANhttps://orcid.org/0009-0000-4062-6102ЖангирбаевА.ZhangirbayevA.

бакалавр

г. Алматы;

г. Каскелен

BSc

040900, Kaskelen

amir.zhangirbayev@gmail.comИнститут математики и математического моделирования; Университет СДУКазахстанInstitute of Mathematics and Mathematical Modeling; SDU UniversityKazakhstan202424122024214153167Copyright © Жангирбаев А., 20242024Жангирбаев А.Zhangirbayev A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/1548

В этой статье мы представляем взвешенное тождество Харди, связанное с векторными полями Баоуэнди-Грушина, и его приложения с применениями в контексте дифференциальных неравенств. С помощью выбора соответствующих параметров наше полученное тождество Харди, связанное с оператором Баоуэнди-Грушина, влечет за собой многочисленные формулы точного остатка для неравенств типа Харди. В коммутативном случае мы получаем улучшенные взвешенные неравенства Харди в постановке евклидова пространства. Например, в частном случае, отбрасывая неотрицательные остаточные члены, связанные с оператором Баоуэнди-Грушина, и выбирая подходящие параметры, наше тождество позволяет нам вывести улучшенное критическое неравенство Харди для радиального производного оператора с точной константой, которая, в свою очередь, не зависит от топологической размерности. Мы используем метод факторизации дифференциальных выражений, использованный Гештези и Литтлджоном в [1]. В данной статье мы демон- стрируем применение метода факторизации в некоммутативной постановке Баоуэнди-Грушина. В качестве применения полученного тождества Харди, связанного с векторными полями Баоуэнди-Грушина, мы устанавливаем неравенство Харди для обобщенного гамильтониана Ландау (или искривленного лапласиана) с определенными остаточными членами.

In this paper, we present a weighted Hardy identity related to the Baouendi-Grushin vector fields and its applications in the context of differential inequalities. By selecting appropriate parameters, the Hardy identity related to the Baouendi-Grushin operator implies numerous sharp remainder formulae for Hardy type inequalities. In the commutative case, we obtain improved weighted Hardy inequalities in the setting of the Euclidean space. For example, in a special case, by dropping non-negative remainder terms, related to the Baouendi-Grushin operator, and choosing suitable parameters our identity allows us to derive an improved critical Hardy inequality for the radial derivative operator with a sharp constant that does not depend on the topological dimension. We employ the method of factorizing differential expressions, as used by Gesztesy and Littlejohn in [1]. In this paper, we demonstrate the application of the factorization method in the noncommutative Baouendi-Grushin setting. As an application of the Hardy identity related to the Baouendi-Grushin vector fields, we establish a Hardy inequality for the generalized Landau Hamiltonian (or the twisted Laplacian) with remainder terms.

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The authors declare that there are no conflicts of interest present.