  In this paper we consider the generalized shift operator, generated by the
Gegenbauer differential operator $$ G =\left(x^2-1\right)^{\frac{1}{2}-\lambda}
\frac{d}{dx} \left(x^2-1\right)^{\lambda+\frac{1}{2}}\frac{d}{dx}. $$ Maximal
function ($ G- $ maximal function), generated by the Gegenbauer differential
operator $ G $ is investigated. The $ L_{p,\lambda} $ -boundedness for the $ G-
$ maximal function is obtained. The concept of potential of Riesz-Gegenbauer is
introduced and for it the theorem of Sobolev type is proved.
