A sharp sufficient geometric condition for the existence of global real analytic solutions on a bounded domain (Q1089499)

Concerning the surjectivity of a locally hyperbolic linear partial differential operator P(D) with constant coefficients on the space \({\mathcal A}(\Omega)\) of real analytic functions on a bounded open set \(\Omega\), T. Kawai gave in 1972 a new micro-local method to give sufficient conditions expressed in geometric terms of \(\Omega\) in relation to the local propagation cones of P(D). Later, G. Zampieri remarked that as a matter of fact Kawai's condition is not much wider than simply characteristic case, and also gave an example of locally hyperbolic operator for which Kawai's method does not give sharp condition. In this article we introduce a refinement to Kawai's method by means of stratification of the cosphere bundle, and prove that the surjectivity holds if at least one of the local propagation cones at every boundary point of \(\Omega\) and at every characteristic direction \(\xi\) does not intersect \(\Omega\).