Spherical means in annular regions in the $n$-dimensional real hyperbolic spaces

TL;DR

This paper characterizes functions in hyperbolic space annuli with zero spherical means and provides a new proof of Helgason's support theorem for such functions.

Contribution

It offers a new characterization of functions with zero spherical means in hyperbolic annuli and presents a novel proof of Helgason's support theorem.

Findings

Characterization of functions with zero spherical means in hyperbolic annuli.

New proof of Helgason's support theorem for hyperbolic spaces.

Extension of spherical mean analysis to hyperbolic geometry.

Abstract

Let $Z_{r,R}$ be the class of all continuous functions $f$ on the annulus $\Ann(r,R)$ in the real hyperbolic space $\mathbb B^n$ with spherical means $M_sf(x)=0$, whenever $s>0$ and $x\in \mathbb B^n$ are such that the sphere $S_s(x)\subset \Ann(r, R) $ and $B_r(o)\subseteq B_s(x).$ In this article, we give a characterization for functions in $Z_{r,R}$. In the case $R=\infty$, this result gives a new proof of Helgason's support theorem for spherical means in the real hyperbolic spaces.