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

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 $M_{s}f(x)=0$, 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.

published: Proc. Indian. Acad. Sci., 121 (2011), No.3, 311-325