Characterization of Holomorphic Functions by Zero SphericalMeans

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.