A generalization of Borel's theorem and microlocal Gevrey regularity in involutive structures (Q952526)

The first result proved by the authors is the construction, through Borel's procedure, of approximate solutions for involutive systems of rank one [see \textit{S. Berhanu, P. Cordaro} and \textit{J. Hounie}, An introduction to involutive structures. New Mathematical Monographs 6. Cambridge: Cambridge University Press (2008; Zbl 1151.35011)] with coefficients in the Gevrey class \(G^s\) of order \(s>1.\) They prove the existence of approximate solutions of class \(G^{s'}\) for all \(s'>s+1.\) Their next result, which is based upon the first one, is the description of the Gevrey wave-front set of the (ultradistribution) boundary values of \(G^s\)-approximate solutions in wedges \(W\) of Gevrey involutive structures \((M,{\mathcal V})\). They prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone contained in \(\operatorname{Re}{\mathcal V}\cap TX,\) \(X\) being a maximally real edge of \(W\). They also give some corollaries and a partial converse of the latter theorem.