Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces

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}.