Denjoy–Carleman Microlocal Regularity on Smooth Real Submanifolds of Complex Spaces

We prove the existence of approximate solutions in the regular Denjoy-Carleman sense for some systems of smooth pairwise commuting complex vector fields. Such approximate solutions provide a well-defined notion of Denjoy-Carleman wave front set of distributions on C infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}<^>\infty $$\end{document}-smooth maximally real submanifolds in complex space which can be characterized in terms of the decay of a Fourier-Bros-Iagolnitzer transform. We also apply the approximate solutions to analyze the Denjoy-Carleman microlocal regularity of solutions of certain systems of first-order nonlinear partial differential equations.