Boundedness criteria for the commutators of fractional integral and fractional maximal operators on Morrey spaces generated by the Gegenbauer differential operator

Ibrahimov E. J., Jafarova S. A.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.47, no.9, pp.7243-7254, 2024 (SCI-Expanded) identifier identifier

  • 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
  • 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.