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Gauged Lagrangian distributions. (English) Zbl 0803.58052
Consider an \(n\)-dimensional manifold \(X\) and a closed conic Lagrangian submanifold \(\Lambda\) of \(T^* X - 0\). The author is interested in the holomorphic maps \(u : \mathbb{C} \to I(X;\Lambda)\) with the property \(u(z) \in I^{m + \text{Re }z}\), where \(I^ m(X;\Lambda)\) is the space of Lagrangian distributions of order \(m\) with microsupport on \(\Lambda\). The problem is to extend the map \(z \to \langle u(z), f\rangle\) from \(\text{Re }z \ll 0\) to the whole complex plane as a meromorphic function with simple isolated poles. The answer is affirmative in the cases where \(f\) is itself a Lagrangian distribution and its microsupport intersects \(\Lambda\) cleanly and \(f\) is a Fourier-Hermite distribution and the microsupport of \(f\) intersects \(\Lambda\) cleanly. The main result of the paper is a simple criterion for the vanishing of \(\text{res} \langle u, f\rangle = \text{res}_{z = 0}\langle u(z),f\rangle\).
Theorem 1.1. If \(\text{res}\langle u,f\rangle = 0\), there exist compactly supported distributions \(u_ i \in I(X;\Lambda)\) and pseudodifferential operators \(P_ i \in \text{Ann}(f)^ t\), such that \(u = \sum^ K_{i = 1} P_ i u_ i\).
Reviewer: V.Oproiu (Iaşi)

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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