Denjoy-Carleman Microlocal Regularity on Smooth Real Submanifolds of Complex Space

TL;DR

This paper develops a framework for Denjoy-Carleman microlocal regularity on real submanifolds of complex space, introducing approximate solutions and characterizing wave front sets via Fourier transforms, with applications to PDEs.

Contribution

It establishes the existence of approximate solutions in the Denjoy-Carleman setting for complex vector fields and defines a wave front set for distributions on real submanifolds.

Findings

Defined Denjoy-Carleman wave front set via Fourier-Bros-Iagolnitzer transform.

Applied approximate solutions to analyze microlocal regularity of nonlinear PDEs.

Characterized regularity in terms of decay properties of transforms.

Abstract

We prove the existence of approximate solutions in the (regular) Denjoy-Carleman sense for some systems of smooth complex vector fields. Such approximate solutions provide a well defined notion of Denjoy-Carleman wave front set of distributions on maximally real submanifolds in complex space which can be characterized in terms of the decay of the 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.