The Hardy-Littlewood-Sobolev theorem for Riesz potential generated by Gegenbauer operator

Abstract

In this paper we introduced and studied the maximal function (G-maximal function) and the Riesz potential (G-Riesz potential) generated by Gegenbauer differential operator G(gimel) = (x(2) - 1)(1/2) (-gimel) d/dx (x(2) - 1)(gimel) (+) (1/2) d/dx. The L-p,L-gimel boundedness of the G-maximal operator is obtained. Hardy-Littlewood-Sobolev theorem of G-Riesz potential on L-p,L-gimel spaces is established. (C) 2016 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V.