Some aspects of harmonic analysis related to Gegenbauer expansions on the half-line

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.