Fourier integral operators

In this post I want to summarize the results of Hörmander’s paper “Fourier Integral Operators I”. I read this paper last summer, but at the time I did not appreciate the geometric aspects of the theory. Here I want to summarize the results of the paper for my own future reference, with a greater emphasis on the geometry.

Generalizing pseudodifferential calculus.
We start by recalling the definition of pseudodifferential calculus on \mathbb R^n.

Definition. A pseudodifferential operator is an operator P of the form

\displaystyle Pu(x) = \iint_{T^* \mathbb R^n} e^{i(x - y)\xi} a(x, y, \xi) u(y) ~dyd\xi

acting on Schwartz space, where dyd\xi is the measure induced by the symplectic structure on the cotangent bundle and a is the symbol. We also call P a quantization of a.

Pseudodifferential operators are useful in the study of elliptic PDE, essentially because if P is elliptic of symbol a, then 1/a is only singular on a compact set in each cotangent space, so if we are willing to restrict to Schwartz functions u which are bandlimited to high frequency, and we are willing to ignore the fact that a \mapsto P is not quite a morphism of algebras (essentially since symbols commute but pseudodifferential operators do not), we can “approximately invert” P by quantizing 1/a.

However, this method of “approximate inversion” does not work for hyperbolic operators, essentially because the singular set of the inverse 1/a of a hyperbolic symbol a is asymptotically the cone bundle of null covectors (with respect to the Lorentz structure induced by a). To fix this problem, one defines the notion of a Fourier integral operator

\displaystyle Pu(x) = \iint_{\mathbb R^{n + N}} e^{i\phi(x, y, \xi)} a(x, y, \xi) u(y) ~dyd\xi

where the so-called operator phase \phi is positively homogeneous of degree 1 on each fiber of \mathbb R^{n + N} \to \mathbb R^n, is smooth away from the zero section, for every x there is no critical point of \phi(x, \cdot, \cdot) away from the zero section, and similarly for y.

For example, the solution to the wave equation is

\displaystyle u(t, x) = (2\pi)^{-n} (2i)^{-1} \int_{T^* \mathbb R^n} (e^{i\phi_+(x, y, \xi)} + e^{i\phi_-(x, y, \xi)}) |\xi|^{-1} f(y) ~dyd\xi

where f is the Fourier transform of the initial data and \phi_\pm(x, y, \xi) = (x - y)\xi \pm t|\xi|. Thus the solution map is a sum of Fourier integral operators.

Equivalence of phase.
Given a Fourier integral operator P, of operator phase \phi and symbol a, we can isolate its Schwartz kernel

\displaystyle P(x, y) = \int_{\mathbb R^N} e^{i\phi(x, y, \xi)} a(x, y, \xi) ~d\xi

using Fubini’s theorem. We call P properly supported if the map that sends the support of the Schwartz kernel to each of the factors \mathbb R^n is proper. Once we restrict to Fourier integral operators of proper support, there is no particular reason to keep dividing the domain of the Schwartz kernel into (x, y) and so we might as well study the following class of distributions:

Definition. An oscillatory integral is a distribution of the form

\displaystyle P(x) = \int_{\mathbb R^N} e^{i\phi(x, \xi)} a(x, \xi) ~d\xi

where \phi is a phase, thus is positively homogeneous of degree 1 and smooth away from the zero section, and a is a symbol.

In particular, the Schwartz kernel of a Fourier integral operator is an oscillatory integral.

We put an equivalence relation on phases by saying that two phases are the same if they induce the same set of oscillatory integrals. Let \phi be a phase on X \times \mathbb R^N and define the critical set

\displaystyle C = \{(x, \xi): \partial_\xi \phi(x, \xi) = 0\}.

Then the differential (x, \xi) \mapsto (x, \partial_x \phi(x, \xi)) of \phi(\cdot, \xi) restricts to a map C \to T^* X \setminus 0 by (x, \xi) \mapsto (x, \partial_x \phi(x, \xi)), and the image of C is an immersed conic Lagrangian submanifold of the cotangent bundle. Moreover, if two phases are equivalent in a neighborhood of x, then they induce the same Lagrangian submanifold.

The local theory is as follows.

Theorem. Let X be an open subset of \mathbb R^n, let \phi_i, i \in \{1, 2\}, be phases defined in neighborhoods of (x, \theta_i) \in X \times \mathbb R^N which induce a Lagrangian submanifold \Lambda of T^* X. Then:

  1. Let s_i be the signature of the Hessian tensor \partial_\xi^2 \phi_i(x, \xi). Then s_1 - s_2 is a locally constant, integer-valued function.
  2. If A is an oscillatory integral with phase \phi_1 and symbol a_1, then there exists a symbol a_2 such that A is also an oscillatory integral with phase \phi_2 and symbol a_2.
  3. Let

    \displaystyle d_i = (\partial_\xi \phi_i)^* \delta_0

    where \delta_0 is the Dirac measure on \mathbb R^N. Then modulo lower-order symbols, we have

    \displaystyle i^{s_1/2} a_1(x, \xi) \sqrt{d_1} = i^{s_2/2} a_2(x, \xi) \sqrt{d_2} (1)

    on \Lambda.

  4. We may take a_i to be supported in an arbitrarily small neighborhood of \Lambda without affecting A modulo lower order terms.

The first three claims here are given by Theorem 3.2.1 in the paper, while the last essentially follows fom the first three and an integration by stationary phase.

Cohomology of oscillatory integration.
The above theorem is fine if we have a global coordinate chart, but the formula (1) looks something like the formula relating the sections of a sheaf. Actually, since \sqrt{d_i} is the formal square root of a measure, it can be viewed intrinsically as a half-density — that is, the formal square root of an unsigned volume form. This is very advantageous to us, because ultimately we want to be able to pair the oscillatory integrals we construct with elements of L^2(X) (at least for symbols of order -m where m is large enough), but elements of L^2(X) are not functions if we do not have a canonical volume form on X, but rather half-densities, and therefore we can pair an oscillatory integral with an element of L^2(X) at least locally.

Let \Omega^{1/2} be the half-density sheaf of a Lagrangian submanifold \Lambda of a given symplectic manifold. We want to define a symbol to be a kind of section of \Omega^{1/2}, but the dimension of integration N is not quite intrinsic to an oscillatory integral (even though in practice we will take N to be the dimension of \Lambda) and neither is the signature s of the Hessian tensor of a phase \phi associated to \Lambda. However, what is true is that s – N mod 2 is intrinsic, so given data (a_j, \phi_j) defining an oscillatory integral in an open set U_j \cong \mathbb R^n in \Lambda, we let

\displaystyle \sigma_{jk} = \frac{(s_k - N_k) - (s_j - N_j)}{2}

which defines a continuous function U_j \cap U_k \to \mathbb Z. Chasing the definition of a Čech cochain around, it follows that \sigma drops to an element of the cohomology group \sigma \in H^1(\Lambda, \mathbb Z/4). We recall that since i^4 = 1, i^{\sigma_{jk}} is well-defined (since \sigma_{jk} \in \mathbb Z/4).

Definition. The Maslov line bundle of \Lambda is the line bundle L on \Lambda such that for sections a_j defined on U_j, we have i^{\sigma_{jk}} a_j = a_k.

So now if we absorb a factor of i^s into a, then a is honestly a section of L, and if we absorb a factor of \sqrt{d}, then a is a section of L \otimes \Omega^{1/2}. Moreover, L is defined independently of anything except \Lambda, so we in fact have:

Theorem. Up to lower-order terms, there is an isomorphism between symbols valued in L \otimes \Omega^{1/2} and oscillatory integrals whose Lagrangian submanifold is \Lambda.

Canonical relations.
We now return to the case that the oscillatory integral A is the Schwartz kernel of a Fourier integral operator, which we also denote by A. Actually we will be interested in a certain kind of Fourier integral operator, and we will redefine what we mean by “Fourier integral operator” to make that precise.

Definition. Let X, Y be manifolds such that the natural symplectic forms on T^*X, T^*Y are denoted by \sigma_X, \sigma_Y. A canonical relation C: Y \to X is a closed conic Lagrangian submanifold of T^* Y \times T^* X \setminus 0 with respect to the symplectic form \sigma_X - \sigma_Y.

The intuition here is that if such a set C is (the graph of) a function, then C is a canonical relation iff C is a canonical transformation. We will mainly be interested in the case that C is a symplectomorphism, and thus is a canonical transformation. However, there is no harm in extending everything to the category of manifolds where the morphisms are canonical relations, or more precisely local canonical graphs (which we define below). Thus we come to the main definition of the paper:

Definition. Let C: Y \to X be a canonical relation. A Fourier integral operator with respect to C is an operator A: C^\infty_c(X) \to C^\infty_c(Y)' such that the Schwartz kernel of A is an oscillatory integral whose Lagrangian submanifold is C.

Definition. A local canonical graph C: Y \to X is a canonical relation C such that the projection \Pi_C: C \to T^* Y is an immersion.

In particular, the graph of a canonical transformation is a local canonical graph. “Locality” here means that \Pi_C is an immersion; obviously it is a submersion, so the only reason that \Pi_C is not a diffeomorphism (and hence C is the graph of a canonical transformation) is that \Pi_C is not assumed to be injective. The reason why it is useful to restrict to the category of local canonical graphs is that in that category, we have a natural measure \omega = \Pi_C^* \sigma_Y^n on C, which induces a natural isomorphism a \mapsto a\sqrt \omega between functions and half-densities. Thus the symbol calculus greatly simplifies, as we can define a symbol in this case to just be a section of the Maslov sheaf. What’s annoying is that if C is a local canonical graph, then X and Y have the same dimension, making it hard to study Fourier integral operators between operators of different dimension.

As an application, pseudodifferential operators on manifolds have an intrinsic definition:

Definition. Suppose that X, Y have the same dimension. A pseudodifferential operator A: C^\infty_c(X) \to C^\infty_c(Y)' is a Fourier integral operator whose Lagrangian submanifold is the graph of the identity.

The paper closes by discussing adjoints and products of Fourier integral operators, and showing that they map Sobolev spaces to Sobolev spaces in the usual way.


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