Propagation of singularities in a locally integrable structure (Q1313098)

This paper investigates the propagation of singularities for a solution of general first-order system of PDEs, which generalizes a result of Trepreau for CR functions on CR submanifold of \(\mathbb{C}^ n\). Let \(\Omega \subset \mathbb{R}^{m+n}\). The structure bundle \(T'\) is obtained via mapping \(\Omega \to \mathbb{C}^ m\), and \(\mathcal L\) is the orthogonal complement of \(T'\). The orbit of families of vector fields is defined through equivalence relation for points on a manifold via flows of vector fields. The main result of this paper is the following theorem (like Trepreau's): If \(N\) is a \(\text{Re}{\mathcal L}\)-orbit of \(\Omega\) which is a submanifold, distribution \(u\) is a solution for \(\mathcal L\) and \(\gamma\) is a conormal of \(N\), then \[ \gamma \in WF_{ha} u \Leftrightarrow {\mathcal O}(H_{\text{Re }{\mathcal L}},\gamma) \subseteq WF_{ha}u \] where \(WF_{ha}u\) is the hypoanalytic waveform set of \(u\), \(H_{\text{Re }{\mathcal L}}\) is the family of Hamilton fields, and \({\mathcal O}(H_{\text{Re }{\mathcal L}},\gamma)\) is the \(H_{\text{Re }{\mathcal L}}\)-orbit containing \(\gamma\).