O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and Some Applications

In this paper we prove an O'Neil inequality for the convolution operator ($G$-convolution) associated with the Gegenbauer differential operator $G_{\lambda}$. By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the $G$-convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the $G$-fractional maximal and $G$-fractional integral operators from the spaces $L_{p,\lambda}$ to $L_{q,\lambda }$ and from the spaces $L_{1,\lambda }$ to the weak spaces $WL_{p,\lambda}$.