Characterization of Holomorphic Functions by Zero Spherical Means

Abstract

We continue to study the holomorphy problem for functions whose contour integrals over circles vanish. We consider the case in which a function \(f \) is defined on a deleted ball \(\mathcal {D} \) in \(\mathbb {C}^n\) (without its center) and integrate over all spheres of two fixed radii inside \(\mathcal {D} \). For \(f\in C^{\infty }(\mathcal {D}) \), we find conditions on the radii and size of \(\mathcal {D} \) implying that \(f \) is a holomorphic function. We also show that these conditions cannot be weakened in the general case.