New Inversion Formulas for the Horospherical Transform

TL;DR

This paper develops new inversion formulas for the horospherical transform in hyperbolic space, extending classical methods to broader function classes using harmonic analysis and fractional integrals.

Contribution

It introduces two novel inversion methods for the horospherical transform applicable to $L^p$ functions and smooth compactly supported functions in hyperbolic space.

Findings

Successfully inverts the horospherical transform for $L^p$ functions.

Provides inversion formulas for compactly supported smooth functions.

Utilizes harmonic analysis and fractional integrals as main tools.

Abstract

The following two inversion methods for Radon-like transforms are widely used in integral geometry and related harmonic analysis. The first method invokes mean value operators in accordance with the classical Funk-Radon-Helgason scheme. The second one employs integrals of the potential type and polynomials of the Beltrami-Laplace operator. Applicability of these methods to the horospherical transform in the hyperbolic space was an open problem. In the present paper we solve this problem for $L^p$ functions in the maximal range of the parameter $p$ and for compactly supported smooth functions, respectively. The main tools are harmonic analysis in the hyperbolic space and associated fractional integrals.