Analysis of the action of a pseudodifferential operator over (C[Omega]|X)T*MX

Let M be a real analytic manifold Ω⊂M an open set, X a complexification of M, P a pseudodifferential operator on X. Using the action of P over holomorphic functions on suitable domains of X, by [B.S], and the theory of representation of micro^-functions at the boundary (C01x)r*μx, by {S.Z], [Z], we show that P defines in a natural manner a sheaf morphism of (C01x)r*μx. Let us note that the hypotheses on ∂Ω are here weaker than in [K 2] where ∂Ω is supposed to be analytic. We also easily prove that P is an isomorphism of (C01x)r*μx out of T*MX∩char(P) (both by composition rule and by non-characteristic deformation). We shall apply the method of this paper in our forthcoming work on regularity at the boundary for solutions of P(cf.[D\u27A-Z])