No abstract

In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential {I_{G}^{\alpha}} , {0<\alpha<2\lambda+1} , is bounded from the G-Morrey space {L_{p,\lambda,\gamma}} to {L_{q,\lambda,\gamma}} if and only if \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1<p<\frac{2% \lambda+1-\gamma}{\alpha}. Also, we prove that the G-Riesz potential {I_{G}^{\alpha}} is bounded from the G-Morrey space {L_{1,\lambda,\gamma}} to the weak G-Morrey space {WL_{q,\lambda,\gamma}} if and only if 1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

In this paper, the Gegenbauer transformation is constructed and some of its properties similar to the Fourier transformation are proved. An equation of Parseval–Plancherel type is obtained. The inversion theorem for the Gegenbauer transformation is proved.

In this paper, we introduce the generalized shift operator generated by the Gegenbauer differential op-and define a generalized convolution ⊗ on the half-line corresponding to the Gegenbauer differential operator. We investigate the Calderon reproducing formula associated with the convolution ⊗ involving finite Borel measures, leading to results on the L p -norm and pointwise approximation for functions on the half-line.

Using the generalized shift operator (GSO) generated by the Gegenbauer differential operator we introduce the notion of a Lebesgue–Gegenbauer (L-G)-point of a summable function 𝑓 on the interval [1,∞) and prove that almost all points of this interval are (L-G)-points of 𝑓. Furthermore, we give an exact (by order) estimation of the approximation of locally summable functions by singular integrals generated by GSO (Gegenbauer singular integrals).