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

TL;DR

This paper characterizes functions with vanishing twisted spherical means in annular regions of complex space and establishes support theorems that extend previous results in the field.

Contribution

It provides a new characterization of functions in the class $Z(Ann(r,R))$ via spherical harmonic coefficients and improves existing support theorems for twisted spherical means in $\, ext{C}^n$.

Findings

Characterization of functions in $Z(Ann(r,R))$ using spherical harmonic coefficients.

Support theorems for twisted spherical means in $\, ext{C}^n$ that extend earlier results.

Enhanced understanding of the behavior of functions with vanishing twisted spherical means in annular regions.

Abstract

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 $B_r(0)\subseteq B_s(z).$ 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.