The Radon transform on the Heisenberg group and the transversal Radon transform

The Radon transform on the Heisenberg group was introduced by R. Strichartz. We regard it as a particular case of a more general transversal Radon transform that integrates functions on Rm over hyperplanes meeting the last coordinate axis. The paper contains new boundedness results and explicit inversion formulas for both transforms of Lp functions in the full range of the parameter p. We also show that these transforms are isomorphisms of the corresponding Semyanistyi–Lizorkin spaces of smooth functions. In the framework of these spaces we obtain inversion formulas, which are pointwise analogues of the corresponding formulas by R. Strichartz.