Fourier integral operator

Summary

A Fourier integral operator is an operator of the form $u \mapsto \int e^{i \varphi(x, \xi)} a(x, \xi) \hat{u}(\xi) \, d\xi / (2\pi)^n$. Here the phase $\varphi$ is homogeneous of degree one in $\xi$ and $a$ is a symbol. The phase $x \cdot \xi$ gives the pseudo-differential operators, and other phases move singularities along a canonical transformation of the cotangent bundle. Solution operators of strictly hyperbolic equations, such as the wave equation, are Fourier integral operators. The general theory was developed by Lars Hörmander.