O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and Some Applications (Journal of Mathematical Study, 2020)
In this paper we prove an O'Neil inequality for the convolution operator ($G$-convolution) associated with the Gegenbauer differential operator $G_{λ}$. 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 $L_{p,λ}$ to $L_{q,λ }$ and from the spaces $L_{1,λ }$ to the weak spaces $WL_{p,λ}$.
Keywords: Gegenbauer differential operator; $G$-convolution; O'Neil inequality; $G$-fractional integral; $G$-fractional maximal function.