Approximate solutions and micro-regularity in the Denjoy–Carleman classes

We begin with a sequence M of positive real numbers and we consider the Denjoy–Carleman class CM. We show how to construct M-approximate solutions for complex vector fields with CM coefficients. We then use our construction to study micro-local properties of boundary values of approximate solutions in general M-involutive structures of codimension one, where the approximate solution is defined in a wedge whose edge (where the boundary value exists) is a maximally real submanifold. We also obtain a CM version of the Edge-of-the-Wedge Theorem.