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

${\displaystyle Pu=f}$

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

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.^[1][2]

We use the following notation:

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

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

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

Furthermore, ${\displaystyle WF(u)\setminus WF(f)}$ is invariant under the Hamiltonian flow induced by ${\displaystyle p_{m}}$.^[3]

• 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.