Characterization of Holomorphic Functions by Zero Spherical Means

We continue to study the holomorphy problem for functions whose contour integrals overcircles 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 theseconditions cannot be weakened in the general case.