Аннотация:
In this paper we consider the generalized shift operator, generated by the Gegenbauer differential operator \[G =(x^2-1)^{\frac{1}{2}-\lambda } \frac{d}{dx} (x^2-1)^{\lambda+\frac{1}{2}}\frac{d}{dx}.\] 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.
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