This course develops the microlocal viewpoint on distributions: not just where singularities live, but how they are distributed in direction as well. The central object is the wave front set, which refines the notion of singular support by recording the cotangent directions in which a distribution fails to be smooth. From there, the course builds a calculus for detecting, tracking, and transforming singularities under pseudodifferential operators, products, pullbacks, and other basic operations on distributions.

The main themes are singularity analysis, symplectic geometry, and the dynamics of linear partial differential equations. After introducing wave front sets and their behavior under [Microlocal Analysis I: Pseudodifferential Operators](/page/Microlocal%20Analysis%20I%3A%20Pseudodifferential%20Operators), the course studies conormal and Lagrangian distributions as model classes of singularities, then turns to real principal type operators and propagation along bicharacteristics. Oscillatory integrals and the [Stationary Phase Lemma](/theorems/645) provide the analytic bridge to Fourier integral operators, whose canonical relations encode how singularities move under generalized changes of variables. Composition, adjoints, and [Egorov's theorem](/theorems/896) then show how this geometric picture controls operator calculus.

The later chapters apply this framework to hyperbolic equations and parametrix construction, where Fourier integral operators capture the structure of solution operators and the propagation of wave fronts. The course is cumulative: the early chapters establish the language of microlocal regularity, the middle chapters develop the geometric and oscillatory machinery, and the final chapters use that machinery to analyze PDEs and concrete singular phenomena in a unified way.

# Introduction

Microlocal analysis refines the study of singularities by recording both their position and their cotangent direction. The prerequisite course [Microlocal Analysis I: Pseudodifferential Operators](/page/Microlocal%20Analysis%20I%3A%20Pseudodifferential%20Operators) supplied the calculus needed to cut a distribution in phase space; this course uses that calculus to describe how singularities are detected, transformed, and propagated. The central objects are the wave front set, canonical relations, and Fourier integral operators, with hyperbolic PDEs as the guiding class of examples.

These notes assume familiarity with [Distribution](/page/Distribution), [Fourier Transform](/page/Fourier%20Transform), [Sobolev Space](/page/Sobolev%20Space), smooth manifolds, cotangent bundles, and the principal symbol calculus from [Microlocal Analysis I: Pseudodifferential Operators](/page/Microlocal%20Analysis%20I%3A%20Pseudodifferential%20Operators). The aim of this introductory chapter is to set the agenda: why ordinary support and singular support are too coarse, why cotangent directions are the right refinement, and how the later chapters fit into a single geometric picture.

## Why Singular Support Is Not Enough

What information is lost if we only ask where a distribution fails to be smooth? A distribution may be singular at a point for many different geometric reasons: a jump across a hypersurface, a point mass, a conormal concentration, or rapid oscillation approaching the point. The singular support records the base point, but it forgets the frequency directions in which the failure of smoothness is visible.

[definition: Singular Support] Let $u \in \mathcal{D}'(U)$, where $U \subset \mathbb R^n$ is open. The singular support of $u$, denoted $\operatorname{sing\,supp} u$, is the complement in $U$ of the set of points $x_0 \in U$ for which there exists $\chi \in C_c^\infty(U)$ with $\chi = 1$ on a neighbourhood of $x_0$ and $\chi u \in C^\infty_c(U)$. [/definition]

The definition localises in $x$, but it does not localise in the dual variable $\xi$. Microlocal analysis keeps the spatial cutoff and adds a conic frequency cutoff, so that one asks whether $\widehat{\chi u}(\xi)$ decays rapidly near a specified non-zero covector direction.

[example: Jump Across A Hypersurface] Let $u=\mathbb 1_{\{x_1>0\}}$ on $\mathbb R^n$, and write $x=(x_1,x')$. Away from the hyperplane $\{x_1=0\}$ the function is locally constant, hence smooth; at any point $(0,y_0)$ on the hyperplane every neighbourhood contains points where $u=0$ and points where $u=1$, so no smooth representative can agree with $u$ there. Thus \begin{align*} \operatorname{sing\,supp}u=\{x_1=0\}. \end{align*} Choose a cutoff of product form $\chi(x_1,x')=\alpha(x_1)\beta(x')$, with $\alpha,\beta\in C_c^\infty$, $\alpha(0)\ne 0$, and $\int \beta(x')\,dx'\ne 0$. Then \begin{align*} \widehat{\chi u}(\xi_1,\xi')=\left(\int_0^\infty \alpha(x_1)e^{-ix_1\xi_1}\,dx_1\right)\left(\int_{\mathbb R^{n-1}}\beta(x')e^{-ix'\cdot \xi'}\,dx'\right). \end{align*} Along the normal ray $\xi' =0$, set $t=\xi_1$. Since $\alpha$ is compactly supported, [integration by parts](/theorems/210) on $[0,\infty)$ gives \begin{align*} \int_0^\infty \alpha(s)e^{-its}\,ds=\frac{\alpha(0)}{it}+\frac{1}{it}\int_0^\infty \alpha'(s)e^{-its}\,ds. \end{align*} Applying the same [integration by parts](/theorems/2098) to the remaining integral gives \begin{align*} \int_0^\infty \alpha(s)e^{-its}\,ds=\frac{\alpha(0)}{it}+O(t^{-2}). \end{align*} Therefore \begin{align*} \widehat{\chi u}(t,0)=\left(\frac{\alpha(0)}{it}+O(t^{-2})\right)\int_{\mathbb R^{n-1}}\beta(x')\,dx', \end{align*} so $\widehat{\chi u}(t,0)$ decays like $|t|^{-1}$, not faster than every power of $|t|$. In tangential directions with $|\xi'|$ large and $\xi_1$ bounded, repeated integration by parts in the $x'$ variables gives rapid decay because derivatives fall only on the smooth compactly supported factor $\beta$. Thus the singularity is detected in the conormal directions, parallel to $dx_1$, rather than in directions tangent to the hyperplane. [/example]

This example already indicates why cotangent variables enter. A hypersurface singularity is not equally singular in every direction; it is tied to the conormal bundle of the hypersurface.

## Local Fourier Decay As A Smoothness Test

How can Fourier analysis distinguish smooth behaviour from singular behaviour after a spatial cutoff? Compactly supported smooth functions have rapidly decaying Fourier transforms, while compactly supported distributions have at most polynomial growth. Thus rapid decay of $\widehat{\chi u}$ is the Fourier-side signature of smoothness near the region where $\chi$ is equal to $1$.

[definition: Rapid Decay In A Cone] Let $\mathbb R^n_0 := \mathbb R^n \setminus \{0\}$. Let $\Gamma \subset \mathbb R^n_0$ be an open conic set and let $v \in \mathcal{D}'(\mathbb R^n)$ have compact support. We say that $\hat v$ is rapidly decreasing in $\Gamma$ if for every $N \in \mathbb N$ there exists $C_N > 0$ such that \begin{align*} |\hat v(\xi)| \le C_N(1+|\xi|)^{-N} \end{align*} for all $\xi \in \Gamma$. [/definition]

The cone condition matters because covectors are directional objects: multiplying $\xi$ by a positive scalar does not change the direction of oscillation. Before using conic decay to define directional regularity, we first need the non-directional benchmark showing that rapid decay in every direction is the same as ordinary smoothness.

[quotetheorem:8161]

[citeproof:8161]

The compact support hypothesis after applying $\chi$ is essential: the constant function $1$ is smooth, but its global Fourier transform is a multiple of $\delta_0$ rather than an ordinary rapidly decreasing function. The requirement that $\chi=1$ near $x_0$ is also essential. If $u=\delta_0$ and $x_0=0$, choosing a cutoff with $\chi(0)=0$ may give $\chi u=0$, falsely certifying smoothness at the singular point. Finally, rapid decay has to hold in every frequency direction for ordinary smoothness: the hypersurface jump $u=\mathbb{1}_{\{x_1>0\}}$ has improved decay in tangential directions after localisation, but it remains singular at $x_1=0$ because the normal covector directions fail rapid decay. The theorem therefore supplies the non-directional benchmark from which the wave front set is obtained by replacing all of $\mathbb R^n_0$ with a single conic neighbourhood.

## The Microlocal Question