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

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 λ = ( x 2 − 1 ) 1 2 − λ d d x ( x 2 − 1 ) λ + 1 2 d d x . The L p , λ boundedness of the G -maximal operator is obtained. Hardy–Littlewood–Sobolev theorem of G -Riesz potential on L p , λ spaces is established.