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

Abstract

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 W L p,λ .