Boundedness criteria for the commutators of fractional integral and fractional maximal operators on Morrey spaces generated by the Gegenbauer differential operator
MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.47, no.9, pp.7243-7254, 2024 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 47 Issue: 9
- Publication Date: 2024
- Doi Number: 10.1002/mma.9970
- Journal Name: MATHEMATICAL METHODS IN THE APPLIED SCIENCES
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
- Page Numbers: pp.7243-7254
- Open Archive Collection: Article
- Azerbaijan State University of Economics (UNEC) Affiliated: Yes
Abstract
In this paper, we find the necessary and sufficient conditions for the boundedness of commutators of fractional integral and fractional maximal operators generated by Gegenbauer differential operator in G-Morrey spaces. We consider the generalized shift operator, associated with the Gegenbauer differential operator G(lambda) = (x(2) - 1)(1/2) (-) (lambda) d/dx x x(x(2) - 1)(lambda+1/2) d/dx. The commutator J(G)(b,alpha) of fractional integral J(G)(alpha) and the commutator M-G(b,alpha) of fractional maximal operator M-G(alpha) associated with the generalized shift operator are investigated. At first, we prove that the commutator J(G)(b,alpha) is bounded from G-Morrey space M-p,M-lambda,M-v to M-q,M-lambda,M-v. We prove that the commutator J(G)(b,alpha) is bounded from the G-Morrey space M-p,M-lambda,M-v to M-q,M-lambda,M-v by the conditions 0 < alpha < gamma, 1 < p < gamma/alpha, 0 < v < gamma - alpha p, and 1/p - 1/q = alpha/gamma-v, if and only if b is an element of BMOG. Also, we prove that commutator M-G(b,alpha) is bounded from M-p,M-lambda,M-v to M-q,M-lambda,M-v under the same conditions.