# Radon Transforms over Lower-Dimensional Horospheres in Real Hyperbolic Space

## TL;DR

This paper investigates Radon transforms over lower-dimensional horospheres in real hyperbolic space, providing explicit inversion formulas and existence conditions for smooth and L^p functions, extending known transforms like the Gelfand-Graev case.

## Contribution

It introduces new explicit inversion formulas and existence conditions for horospherical Radon transforms of arbitrary dimension in hyperbolic space, generalizing previous results.

## Key findings

- Derived explicit inversion formulas for the transforms.
- Established existence conditions for smooth and L^p functions.
- Extended the Gelfand-Graev transform to lower-dimensional horospheres.

## Abstract

We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known Gelfand-Graev transform.