Partial differential equations Definition

Fourier integral operator

Builds on Symbol class Pseudo-differential operator Fourier transform Schwartz space Lagrangian submanifold

Definition

Let $n \ge 1$ and $m \in \R$ . Let $\varphi \colon \R^n \times (\R^n \setminus \set{0}) \to \R$ be a smooth function such that:

$\varphi(x, t\xi) = t \, \varphi(x, \xi)$ for all $t > 0$ , $x \in \R^n$ and $\xi \ne 0$ .
$\det \big( \partial^2 \varphi / \partial x_j \partial \xi_k \big)_{j,k} \ne 0$ everywhere.

Let $a \in S^m(\R^n \times \R^n)$ be a symbol in the symbol class $S^m = S^m_{1,0}$ with $a(x, \xi) = 0$ for $\abs{\xi} \le 1$ . For $u$ in the Schwartz space $\mathcal{S}(\R^n)$ with Fourier transform $\hat{u}$ , put $T u(x) = \frac{1}{(2\pi)^n} \int_{\R^n} e^{i \varphi(x, \xi)} a(x, \xi) \, \hat{u}(\xi) \, d\xi.$ The operator $T$ is a Fourier integral operator of order $m$ with phase function $\varphi$ and amplitude $a$ .

Remarks

Example: the phase $\varphi(x, \xi) = x \cdot \xi$ satisfies both conditions, with mixed Hessian the identity matrix, and gives the pseudo-differential operator $\mathrm{Op}(a)$ . For fixed $t \in \R$ , the phase $\varphi(x, \xi) = x \cdot \xi + t \abs{\xi}$ also satisfies both conditions, and with a symbol $a$ equal to $1$ for $\abs{\xi} \ge 2$ it gives $e^{i t \sqrt{-\Delta}}$ up to a smoothing operator.

Non-example: $\varphi(x, \xi) = x \cdot \xi + \abs{\xi}^2$ fails the first condition, since it is not homogeneous of degree one in $\xi$ ; it is the phase of a Schrödinger evolution.

The general definition, due to Hörmander, allows phases $\varphi(x, y, \theta)$ with extra variables and works on manifolds. There a Fourier integral operator is associated with a canonical relation, a Lagrangian submanifold of the product of the cotangent bundles, and it moves wave front sets along that relation.

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Tag	`MSMD2BS9C5`
URI	`m2://MSMD2BS9C5`
Kind	definition
Writing	W0 stub
MSC	35S30, 58J40
Defines	Fourier integral operator
Uses	Symbol class, Pseudo-differential operator, Fourier transform, Schwartz space, Lagrangian submanifold

Fourier integral operator

Definition

Remarks

Reading path

Further reading

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