A generalization of Borel's theorem and microlocal Gevrey regularity in involutive structures

In this paper, we use Borel's procedure to construct Gevrey approximate solutions of an initial value problem for involutive systems of Gevrey complex vector fields. As an application, we describe the Gevrey wave-front set of the boundary values of approximate solutions in wedges W of Gevrey involutive structures (M,V). We prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone ΓT(W) contained in RV∩TX where X is a maximally real edge of W. We also prove a partial converse.