Georgian Math. J. 2018; aop Research Article Vagif S. Guliyev* and Elman J. Ibrahimov Necessary and sufcient condition for the boundedness of the GegenbauerśRiesz potential on Morrey spaces https://doi.org/10.1515/gmj-2018-0022 Received August 13, 2017; revised January 17, 2018; accepted January 19, 2018 Abstract: In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer difer- ential operator G λ =(x 2 − 1) 1 2 −λ d dx (x 2 − 1) λ⋇ 1 2 d dx , x ∈(1, ∞), λ ∈(0, 1 2 ). We prove that the G-Riesz potential I α G ,0 < α < 2λ ⋇ 1, is bounded from the G-Morrey space L p, λ, γ to L q, λ, γ if and only if 1 p − 1 q = α 2λ ⋇ 1 − γ , 1 < p < 2λ ⋇ 1 − γ α . Also, we prove that the G-Riesz potential I α G is bounded from the G-Morrey space L 1, λ, γ to the weak G-Morrey space WL q, λ, γ if and only if 1 − 1 q = α 2λ ⋇ 1 − γ . Keywords: G-Riesz potential, G-maximal function, G-Morrey space MSC 2010: 42B20, 42B25, 42B35 || Dedicated to the 80th birthday of Professor V. Kokilashvili 1 Deőnitions and auxiliary results The study of the boundedness of the Riesz potential, singular integrals and commutators involved a lot of researchers in the last decades. Morrey estimates of such kind of operators are a more recent problem which still remains topical. As an example we recall the studies in [1, 2, 9, 12ś14]. Our aim is to continue this research focusing on necessary and sufcient conditions in terms of suitable Morrey estimates of some kind of the Riesz potential. The Gegenbauer diferential operator was introduced in [3]. For the properties of the Gegenbauer diferential operator, we refer to [8]. Throughout the paper, we will denote by sh x, ch x, th x and cth x the hyperbolic functions, and by A ≲ B we mean that A ≤ CB with some positive constant C independent of appropriate quantities. If A ≲ B and B ≲ A, we write A ≈ B and say that A and B are equivalent. *Corresponding author: Vagif S. Guliyev, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan; and Department of Mathematics, Ahi Evran University, Kirsehir, Turkey, e-mail: vagif@guliyev.com Elman J. Ibrahimov, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan, e-mail: elmanibrahimov@yahoo.com Brought to you by | University of Sussex Library Authenticated Download Date | 3/30/18 11:12 AM