A normal form around a Lagrangian submanifold of radial points (Q2925296)

For a smooth \(n\)-manifold, \(M\), denote by \(\Psi^m(M)\) the space of classical pseudodifferential operators on \(M\) of order \(m\). \textit{J. J. Duistermaat} and \textit{L. Hörmander} [Acta Math. 128, 183--269 (1972; Zbl 0232.47055)] have shown that, if a classical pseudodifferential operator with real principal symbol \(P\in \Psi^m(M)\) is not radial at \(q\), then the microlocal germ of \(P\) at \(q\) is microlocally equivalent near its characteristic variety to the operator \(\partial_{x_1}\) on \(\mathbb{R}^n\).NEWLINENEWLINEOn the other hand, the study of radial points was initiated by \textit{V. Guillemin} and \textit{D. Schaeffer} [Duke Math. J. 44, 157--199 (1977; Zbl 0356.35080)]. In this work, they were concerned with the microlocal structure of operators around isolated radial points. However, in general, radial points are not isolated and the set of radial points can form a Lagrangian submanifold of \(T^*M\).NEWLINENEWLINEIn the paper under review, the author shows that, if the set of radial points of \(P\in \Psi^m(M)\) with real valued homogeneous principal symbol is a Lagrangian submanifold of \(T^*M\), then there is a function \(p_0(y)\) so that \(P\) is microlocally equivalent to the operator NEWLINE\[NEWLINEz D_z + p_0(y)\, ,NEWLINE\]NEWLINE on \(\mathbb{R}_z \times \mathbb{R}^{n-1}_y\), where \(p_0\) is a smooth function around \(z =0, y = 0, \eta = 0, \zeta = 1\); \(\zeta\) is dual to \(z\); \(\eta\) is dual to \(y\); and \(D_z = \frac{1}{i} \partial_z\).