Radial Wavelet, Radon Transform And Singular Integrals On The Heisenberg Group

Let Hnbe the Heisenberg group. In this paper we construct a type of radialwavelets on L2(Hn), for which the Caldero′n reproducing formula is valid. In addi-tion, we devise a subspace of Schwartz functions on which the Radon transform isa bijection. Furthermore, we also introduce two subspaces of L2(Hn) such that theRadon transform and the inverse Radon transform hold by using the wavelet trans-forms. In our new formulae the inverse Radon transforms are associated with thesub-Laplacian on Hn, and the smoothness on f can be neglected if wavelet functionsare differential.An important part of Harmonic Analysis is singular integrals. So we also dealwith singular integrals on the reduced Heisenberg group. We show that singular inte-grals on the reduced Heisenberg group is of (p, p) type with1≤p≤∞if the kernelK(z, t) is controlled by functions which are only related to z. These conditions aredifferent from those of the Heisenberg group.