Twisted spherical means in annular regions in $C^n$ and support theorems

Let $Z(Ann(r,R))$ be the class of all continuous functions $f$ on the annulus $Ann(r,R)$ in ${\mathbb{C}}^n$ with twisted spherical mean $f\times {\mu }_s(z)=0,$ whenever $z\in {\mathbb{C}}^n$ and $s>0$ satisfy the condition that the sphere $S_s(z)\subseteq Ann(r,R)$ and ball $B_r(0)\subseteq B_s(z)\mathrm{.}$ In this paper, we give a characterization for functions in $Z(Ann(r,R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in ${\mathbb{C}}^n$ which improve some of the earlier results.