Fourier Integral Operators

General Idea

Generally, these operators naturally extend the set of pseudodifferential operators, realized by a link with the set of the canonical transformations and their graphs viewed as Lagrangian submanifolds of a symplectic manifold. In particular, Fourier integral operators are integral operators exhibiting Lagrangian distribution kernels.

The original local notion of Fourier integral operators was introduced to study the singularities of hyperbolic differential equations.

Global theory

Global Fourier integrals are defined on an immersed conic Lagrangian submanifold in $T^\ast X\setminus 0$, and is a $\frac{1}{2}$-distribution density. The definition of a global Fourier integral operator is more elaborate.

See Lecture Notes on Fourier Integral Operators for the moment.

Local theory

Definition

We follow Shubin for the definition. The definition is heavily analytic and looks messy. Things to really keep in mind are

  1. how the expression for the Fourier integral operator when acting on a compactly supported function looks.
  2. the integral kernal “is” the Fourier integral operator.
Definition(Fourier integral operator) . Let $X,Y$ be open sets in $\mathbb{R}^{n_X}$ and $\mathbb{R}^{n_Y}$. A linear operator $A: C_c^\infty(Y) \rightarrow D(X)$ from the compactly supported smooth functions on $Y$ to the distributions on $X$ is a Fourier integral operator, if $$ (Au)(x) = \int e^{i\Phi(x,y,\theta)} a(x,y,\theta) u(y) dy d\theta $$ where $u\in C^\infty_c(Y), x\in X$, $\Phi(x,y,\theta)$ is a phase function on $X\times Y\times \mathbb{R}^n$, and $a(x,y,\theta)\in S_{\rho,\delta}^m(X\times Y\times \mathbb{R}^N)$ where $\rho > 0$, $\delta <1$.
Remark. The class $S^m_{\rho,\delta}(X\times Y\times \mathbb{R}^n)$ consists of smooth functions in $C^\infty(X\times Y\times \mathbb{R}^n)$ s.t. for any multi-indices $\alpha,\beta$ and any compact set $K\subset X\times Y$ there is a constant $C_{\alpha,\beta,k}$ s.t. $$ |\partial^\alpha_\theta \partial^\beta_{x,y} a(x,y,\theta)| \leq C_{\alpha,\beta,k} \left(\sqrt{1+\sum^n_{i}\theta_{i}^2}\right)^{m-\rho|\alpha|+\delta|\beta|}. $$ We require that $m,\rho,\delta$ are real and $\rho,\delta \in [0,1]$. We will see that for pseudodifferential operators this is the symbol class of the operator.

The integral $$ \langle Au, v\rangle = \iiint e^{i\Phi(x,y,\theta)} a(x,y,\theta) u(y) v(x) dx dy d\theta,\quad v\in C^\infty_0(X) $$ is then defined and is an oscillatory integral. For fixed $u$ we then have a continuous linear functional of $v$ on $X$, i.e. a distribution $Au\in D(X)$. Recall that the (integral) kernel $K_A \in D(X\times Y)$ of $A$ is defined by the oscillatory integral $$ \langle K_A,w\rangle = \iiint e^{i\Phi(x,y,\theta)} a(x,y,\theta) w(x,y) dx dy d\theta, $$ so we have $$ \langle K_A, u(y) v(x) \rangle = \langle Au, v\rangle. $$

Remark. While the operator $A$ is uniquely defined by the distribution $K_A$ and vice versa, two different pairs consisting of a phase function and a function $a(x,y,\theta)$ can give rise to the same operator. Even with the same phase, $a(x,y,\theta)$ is not uniquely defined by $A$. We cannot identify the kernal $K_A$ with $e^{i\Phi(x,y,\theta)}a(x,y,\theta)$.

Operator Phase

Definition(Operator phase function) . A phase function $\Phi(x,y,\theta)$ is an operator phase function if $$ \begin{gather} \Phi'(y,\theta)(x,y,\theta)\neq 0 \quad \text{for}\quad \theta \neq 0\\ \Phi'(x,\theta)(x,y,\theta)\neq 0 \quad \text{for}\quad \theta \neq 0 \end{gather} $$

A Fourier integral operator whose phase is an operator phase function continnuously maps $C_c^\infty(Y)$ into $C^\infty(X)$, therefore $A$ can be continuously extended to a continuous (in the weak topologies on the domain and the codomain) map $$ A: D_c(Y) \rightarrow D(X) $$ where $D_c(Y)$ is the set of compactly supported distributions in $Y$.

Let $S$ be a subset of $X\times Y$, and $K\subset Y$. Denote by $S|K$ the subset of $X$ s.t. there is a $y\in Y$ with $(x,y)\in S$.

Theorem. For $A$ a Fourier integral operator, we have $$ \text{sing supp}\ Au \subset (S_\Phi|\text{sing supp}\ u) $$ where $S_\Phi$ consists of those pairs $(x,y)$ for which there is a $\theta \in \mathbb{R}^N\setminus 0$ with $\Phi'_\theta(x,y,\theta)=0$.

Examples

Linear differential operators

A first example is linear differential operators $A = \sum_{|\alpha|\leq m} a_\alpha (x)D^\alpha$ where $\alpha\in C^\infty(X)$, $X$ a set open in $\mathbb{R}^n$ and $D=-i\partial_x$.

Using (double) Fourier transformation, we have $$ D^\alpha u(x) = \iint \xi^\alpha e^{i(x-y)\cdot\xi}u(y) dyd\xi, $$ hence $$ Au(x) = \iint e^{i(x-y)\cdot \xi} \sigma_A(x,\xi) u(y) dy d\xi $$ where $\sigma_A(x,\xi) = \sum_{|\alpha|\leq m}a_\alpha(x)\xi^\alpha$ is the symbol of $A$.

This is a Fourier integral operator. The phase function is $(x-y)\cdot \xi$, the symbol $\sigma_A(x,\xi)\in S^m(X\times\mathbb{R}^n)$. Linear differential operators have the locality property, $$ \text{sing supp}\ Au \subset \text{sing supp}\ u\quad \text{for}\quad u\in C^\infty_c(X). $$

Pseudodifferential operator

For $X=Y$ and $n_X = n_Y = n$ (recall that $n_X$ is in $\mathbb{R}^{n_X}$ and $X$ is open there), a Fourier integral operator with the phase function $\Phi(x,y,\xi) = (x-y)\cdot \xi$ is a pseudodifferential operator.

We see that $A$ a pseudodifferential operator is pseudo local, namely $\text{sing supp}\ Au \subset \text{sing supp}\ u$ for $u\in D_c(X)$.

The class $S^m$ of the symbol $\sigma_A(x,\xi) \in S^m$ is called the symbol class of the operator $A$.