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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
##### Keywords:
Lagrangian distribution
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