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

$Pu=f$

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

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:

• $X$ is a $C^{\infty }$-differentiable manifold, and $C_{0}^{\infty }(X)$ is the space of smooth functions $u$ with a compact set $K\subset X$, such that $u\mid {X\setminus K}=0$.
• $L_{\sigma ,\delta }^{m}(X)$ denotes the class of pseudodifferential operators of type $(\sigma ,\delta )$ with symbol $a(x,y,\theta )\in S_{\sigma ,\delta }^{m}(X\times X\times \mathbb {R} ^{n})$.
• $S_{\sigma ,\delta }^{m}$ is the Hörmander symbol class.
• $L_{1}^{m}(X):=L_{1,0}^{m}(X)$.
• $D'(X)=(C_{0}^{\infty }(X))^{*}$ is the space of distributions, the Dual space of $C_{0}^{\infty }(X)$.
• $WF(u)$ is the wave front set of $u$
• $\operatorname {char} p_{m}$ is the characteristic set of the principal symbol $p_{m}$

Statement

Let $P$ be a properly supported pseudodifferential operator of class $L_{1}^{m}(X)$ with a real principal symbol $p_{m}(x,\xi )$, which is homogeneous of degree $m$ in $\xi$. Let $u\in D'(X)$ be a distribution that satisfies the equation $Pu=f$, then it follows that

$WF(u)\setminus WF(f)\subset \operatorname {char} p_{m}.$

Furthermore, $WF(u)\setminus WF(f)$ is invariant under the Hamiltonian flow induced by $p_{m}$.

Bibliography

• Hörmander, Lars (1972). Fourier integral operators. I. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. pp. 79–183. doi:10.1007/BF02392052.
• Duistermaat, Johannes Jisse; Hörmander, Lars (1972). Fourier integral operators. II. Acta Mathematica. Vol. 128. Institut Mittag-Leffler. pp. 183–269. doi:10.1007/BF02392165.
• Shubin, Mikhail A. Pseudodifferential Operators and Spectral Theory. Springer Berlin, Heidelberg. ISBN 978-3-540-41195-6.
• Taylor, Michael E. (1978). "Propagation, reflection, and diffraction of singularities of solutions to wave equations". Bulletin of the American Mathematical Society. 84 (4). American Mathematical Society: 589–611.