The Radon Transform on the Heisenberg Group and the Transversal Radon Transform

TL;DR

This paper studies the Radon transform on the Heisenberg group and the transversal Radon transform, providing new boundedness results, explicit inversion formulas, and demonstrating their isomorphism properties on specific function spaces.

Contribution

It introduces new boundedness results and explicit inversion formulas for the Radon transforms on $L^p$ spaces and Semyanistyi-Lizorkin spaces, extending previous work.

Findings

New boundedness results for the transforms.

Explicit inversion formulas for $L^p$ functions.

Transforms are isomorphisms on certain smooth function spaces.

Abstract

The notion of the Radon transform on the Heisenberg group was introduced by R. Strichartz and inspired by D. Geller and E.M. Stein's related work. The more general transversal Radon transform integrates functions on the m-dimensional real Euclidean space over hyperplanes meeting the last coordinate axis. We obtain new boundedness results and explicit inversion formulas for both transforms on $L^p$ 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.