Morera Theorems for Complex Manifolds

We prove Morera theorems for the Radon transform integrating on geodesic spheres on complex analytic manifolds of arbitrary dimension. To avoid pathologies, we assume that the radius of each sphere of integration is less than the injectivity radius at its center. The proofs of the main results are local, and they involve the microlocal properties of associated Radon transforms and a theorem of Hِrmander, Kawai, and Kashiwara on microlocal singularities. We consider Morera theorems for spheres of fixed radius and spheres of arbitrary radius.