Introduction

Overview#

Microlocal analysis is a part of harmonic analysis involving geometric aspects in the cotangent bundles. It is, in its original sense, the study of the functional analysis of generalized functions/distributions with attention paid to their singular support and the directions ("microlocal") of the propagation of their singularities as covectors. The set of these covectors are called their wave front set. The wave front set is the set of directions (in the dual space) along which the Fourier transform of distribution is not rapidly decreasing locally, hence, microlocal analysis is mainly concerned with constructions related to Fourier transformation, e.g. pseudodifferential operators. And, as a set of covectors the wave front set resides in the cotangent bundle; more precisely, it is a sub-bundle of the cotangent bundle.

In more details#

A Fourier integral operator \(T\) is given by
 

\[(T f)(x) = \int \int e^{2\pi i \Phi(x,y,\xi)} a(x,y,\xi) f(y) dy d\xi,\]

where \(f \in \mathcal{S}'(\mathbb{R}^n)\) where \(\mathcal{S}\) denotes the Schwartz space, \(a(x,\xi)\) is a standard symbol compactly supported in \(x\), and some more requriements for the phase function \(\Phi\). Observe that the \(y\)-integral is doing a Fourier transform of \(f(y)\) if \(\Phi(x,y,\xi) = (x-y)\xi\); the special examples of Fourier integral operators for which \(\Phi(x,y,\xi)=(x-y)\xi\) are pseudodifferential operators.

The motivation for the study of Fourier integral operator can be illustrated by considering the following initial value problem for the wave operator:

\[\partial^2 u(t,x) / \partial t^2 = \nabla^2 u(t,x)\]

where \((t,x)\in \R^+ \times \R^n\) and \(u(0,x) = 0, \partial u(0,x) \partial t = f(x)\) where \(f\in \mathcal{S}(\R^n)\). The solution is given by

\[u(t,x) = \frac{1}{(2\pi)^n} \int \frac{\exp(i(\langle x,\xi\rangle + t|\xi|))}{2i|\xi|}\hat{f}(\xi)d\xi - \frac{1}{(2\pi)^n} \int \frac{\exp(i(\langle x,\xi\rangle - t|\xi|))}{2i|\xi|}\hat{f}(\xi)d\xi\]

The integrals do not in general converge, and formally looks like a sum of two Fourier integral operators, only the coefficient in each of the integrals are not smooth at the origin. Cut this singularity with a cutoff function (after the singularit is cut, we obtain a standard symbol \(a\)), the so obtained operators still provide solutions to the intial value problem, modulo smooth functions. Thus, when only interested in the propagation of singularities of the initial data, it is sufficient to consider such operators.

Recall that the space \(\mathcal{D}(X) = C^\infty_c(X)\) of compactly supported test functions on \(X\) is the set of compactly supported smooth functions \(X\to\mathbb{R}\) equipped with evident real vector space structure (pointwise addition and multiplication) and the topology that is the metric topology induced from the family of semi-norms \(\rho_{K,\alpha} = \text{sup}_{x\in K}|\partial^\alpha f|\) where \(K\subset X\) is compact and \(\alpha\) is a multi-index for which \(\partial^\alpha\) is the corresponding differential operator. A distribution on \(X\) is a linear functional \(C^\infty_c(X)\to\mathbb{R}\) from the locally convex TVS \(\mathcal{D}(X)\) of compactly supported test functions  on \(X\) to the reals; the space of compactly supported distributions on \(X\), which is the topological dual of \(\mathcal{D}(X)\), is denoted \(\mathcal{D}'(X)\).

Distributions are, in the point of view of PDEs, solutions to the PDEs. To solve a PDE, it is often instrumental to define a differential operator \(D\) s.t. the solution \(u\) should satisfy \(Du= g\). By means of Fourier transform - going to the phase space, or equivalently, the covector space/cotangent bundle, the PDE becomes a polynomial equation, and the differential operator becomes a polynomial. A pseudodifferential operator is the inverse to a differential operator. For example, say \(L = \Delta + 1\), by Fourier transform the inverse is given by,
 

\[(L^{-1} u)(x) = (2\pi)^{-n} \int\int e^{i(x-y)\xi}(1+|\xi|^2)^{-1} u(y) dy d\xi.\]

Generalizing pseudodifferential operators, a Fourier integral operator basically does the following to a test function: “Fourier transform” \(f\), but with a more general phase function \(\Phi\) (instead of  the \((x-y)\xi\) of pseudodifferential operators) and get \(\hat{f}\), multiply with some function \(a\), and transform back. The function \(a\) can be, as an example, a function that is the result of applying a cut-off function that eliminates the singularities, thus a Fourier integral operator applies a cutoff to a configuration space function in the phase space and yield the cut-off applied function in the configuration space. The application of cutoff provides a suitable framework for the study of propagation of the singularities of distributions, since locally these correspond to the directions along which the Fourier transform of distributions is not rapidly decreasing (not Schwartz) - the rate of decrease is then measured by the metric topology induced from the family of semi-norms \(\rho_{K,\alpha}\) given above.