Inequality of O'Neil-type for convolutions associated with the Laplace-Bessel differential operator and applications

Abstract

In this paper we prove an O'Neil-type inequality for the convolution operator (B- convolution) associated with the Laplace-Bessel differential operator. By using an O'Neil- type inequality for rearrangements we obtain a pointwise rearrangement estimate of the B- convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional B-maximal operator and B-fractional integral operator with rough kernels from the spaces Lp,γ to Lq,γ and from the spaces L1,γ to the weak spaces WLq,γ .