Extension of Solutions to Holomorphic Partial Differential Equations

We derive sufficient conditions on a strictly pseudoconvex domain Omega and a linear, holomorphic partial differential operator P for the existence of solutions to Pu = 0 which are holomorphic in Omega near a point p in the boundary of Omega, but cannot be prolonged holomorphically across p. We first show how, given the existence of a supporting everywhere characteristic analytic hypersurface, a theorem of Tsuno can be used to construct the desired solutions. We then derive necessary conditions for the existence of such a hypersurface under the assumption that Omega is strictly pseudoconvex and simply characteristic at p with respect to P.