Maximal operators associated with Gegenbauer expansions on the half-line. I

Elman J. Ibrahimov<sup>ab

a</sup> Azerbaijan State Oil Academy
^b Azerbaijan State University of Oil and Industry, Baku

Abstract: 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.

MSC: 42B20, 42B25, 42B35

Received: 07.04.2014
Accepted: 05.06.2014

Language: English