nLab symbol order

Definition

Definition

(symbol order)

A smooth function $q$ on a cotangent bundle (e.g. the symbol of a differential operator) is of order $m$ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each coordinate chart $((x^i), (k_i))$ we have that for every compact subset $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a real number $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the absolute value of the partial derivatives of $q$ is bounded by

$$\left\vert \frac{\partial^\alpha}{\partial k_\alpha} \frac{\partial^\beta}{\partial x^\beta} q(x,k) \right\vert \;\leq\; C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}$$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

A Fourier integral operator $Q$ is of symbol class $L^m = L^m_{1,0}$ if

1. it is of the form

   $$Q f (x) \;=\; \int \int e^{i k \cdot (x - y)} \hat f(x,y,k) f(y) \, d y d k$$

2. its principal symbol $q$ is of order $m$, in the above sense.

(Hörmander 71, def. 1.1.1 and first sentence of section 2.1)

Examples

Example

The wave operator/Klein-Gordon operator on Minkowski spacetime is of class $L^2$, according to def. .

Related concepts

• propagation of singularities theorem

References

• Lars Hörmander, Fourier integral operators I. Acta Mathematica 127, 79-183 (1971) (Euclid)