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 $$G={(x^2-1)}^{\frac{1}{2}-\lambda }\frac{d}{dx}{(x^2-1)}^{\lambda +\frac{1}{2}}\frac{d}{dx}\mathrm{.}$$ 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.