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

Abstract 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 $$\mathcal {C}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -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.