Propagation of singularities theorem

In microlocal analysis, the propagation of singularities theorem (also called the Duistermaat–Hörmander theorem) is theorem which characterizes the wavefront set of the distributional solution of the partial (pseudo) differential equation

for a pseudodifferential operator on a smooth manifold. It says that the propagation of singularities follows the bicharacteristic flow of the principal symbol of .

The theorem appeared 1972 in a work on Fourier integral operators by Johannes Jisse Duistermaat and Lars Hörmander and since then there have been many generalizations which are known under the name propagation of singularities.

Propagation of singularities theorem

We use the following notation:

• is a -differentiable manifold, and is the space of smooth functions with a compact set , such that .
• denotes the class of pseudodifferential operators of type with symbol .
• is the Hörmander symbol class.
• .
• is the space of distributions, the Dual space of .
• is the wave front set of
• is the characteristic set of the principal symbol

Statement

Let be a properly supported pseudodifferential operator of class with a real principal symbol , which is homogeneous of degree in . Let be a distribution that satisfies the equation , then it follows that

Furthermore, is invariant under the Hamiltonian flow induced by .

Bibliography

References