Pseudo-differential operator

Summary

A pseudo-differential operator is an operator of the form $u \mapsto \int e^{i x \cdot \xi} a(x, \xi) \hat{u}(\xi) \, d\xi / (2\pi)^n$, where the symbol $a$ belongs to a symbol class. Linear differential operators with smooth bounded coefficients are the case where $a$ is a polynomial in $\xi$. Pseudo-differential operators form an algebra that also contains approximate inverses of elliptic operators. They are a basic tool of microlocal analysis and of the theory of linear partial differential equations.