The Boundedness of High Order Riesz-Bessel Transforms via Atomic-molecular Characterization on Weighted Hp ∆v,w Spaces

Let 0 < p ≤ 1 and w in the Muckenhoupt class Ap,ν . In this paper, we will characterize the weighted Hardy spaces in the setting of the Laplace-Bessel differential operators by means of the high order Riesz-Bessel transforms generated by generalized shift operator. We give the elemantary result that for certain weight functions w. We also obtain the boundedness of high order Riesz-Bessel transform within the bounds of some necessary conditions through the atomic-molecular characterization.