Asymptotics for the Radon transform on hyperbolic spaces

TL;DR

This paper investigates the asymptotic behavior of the Radon transform on hyperbolic spaces, demonstrating that the Abel transform of certain invariant differential operator applications results in Schwartz functions, extending previous results.

Contribution

It extends prior work by proving that the Abel transform of Df is Schwartz for a broader class of functions on hyperbolic spaces.

Findings

The Abel transform of Df is a Schwartz function for L^2-Schwartz functions.

Extension of previous results to non-K-finite functions.

Demonstrates the behavior of the Radon transform on hyperbolic spaces.

Abstract

Let G/H be a hyperbolic space over R C or H, and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H, we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K\cap H-invariant functions.