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

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 ‐Morrey spaces. We consider the generalized shift operator, associated with the Gegenbauer differential operator . The commutator of fractional integral and the commutator of fractional maximal operator associated with the generalized shift operator are investigated. At first, we prove that the commutator is bounded from ‐Morrey space to . We prove that the commutator is bounded from the ‐Morrey space to by the conditions , , , and , if and only if . Also, we prove that commutator is bounded from to under the same conditions.