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

Abstract

In this paper, we establish necessary and sufficient conditions for the boundedness of the fractional integral on G‐Morrey space generated by Gegenbauer differential operator. A similar problem is studied for the fractional maximal operator.

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

“…G from  1,𝜆,𝜈 (R + ) to  q,𝜆,𝜈 (R + ). Note that this work is closely related to the results to the results of [18] and considered to be its logical conclusion. In fact, in this paper, we use results of [18] to prove the boundedness of commutators of G-fractional maximal and G-fractional integral operators in G-Morrey spaces.…”

“…Note that this work is closely related to the results to the results of [18] and considered to be its logical conclusion. In fact, in this paper, we use results of [18] to prove the boundedness of commutators of G-fractional maximal and G-fractional integral operators in G-Morrey spaces. The proof of the theorem for commutator J b,𝛼 G which is an analogue of Theorem B [3] is the aim of this paper.…”

“…Therefore, we only provide a schematic proof of the necessity. In order to do this, we use the stretch operator 𝑓 t which was introduced in [18]. Let 𝑓 be a positive and increasing function.…”