Microlocal analysis provides a framework for understanding differential and pseudo-differential operators through the joint lens of position and frequency, enabling precise control over solutions to partial differential equations. This course develops the theory of pseudodifferential operators from first principles, beginning with Fourier multipliers as a model case and progressing through the symbolic calculus that underpins modern PDE theory. Pseudodifferential operators generalize classical differential operators by replacing polynomial symbol functions with more general smooth symbols, allowing for a vastly richer class of operators that nonetheless retain good analytic properties and a coherent composition algebra.

The course is organized around the construction and manipulation of this symbolic calculus. Chapters 1–7 build the foundational machinery: Fourier multipliers establish the basic intuition; symbol classes formalize the decay and smoothness conditions that make the calculus work; quantisation rules translate symbols into operators via integral kernels; and composition, adjoints, and commutator formulas show how the algebra behaves under operations. Asymptotic expansions emerge as a key theme, allowing us to track operator behavior not just qualitatively but through precise asymptotic formulas in the symbol.

The final chapters apply this machinery to fundamental problems in analysis and geometry. Chapters 8–10 exploit Sobolev regularity, ellipticity, and Fredholm theory to establish solvability and spectral properties of classical differential operators, while Chapter 11 extends the calculus to manifolds through coordinate localisation. Throughout, the microlocal perspective—understanding operators simultaneously in position and momentum—reveals structure hidden by traditional approaches: principal symbols encode the leading-order behavior, commutators vanish asymptotically, and the interplay between regularity and decay becomes transparent. Chapter 12 consolidates these threads through the lens of elliptic theory, showing how pseudodifferential calculus unifies the solution of boundary-value problems and the construction of parametrices.

# Introduction

This introductory chapter fixes the scope, prerequisites, and guiding examples for the course. The central question is how far the Fourier multiplier description of constant-coefficient differential operators can be extended to variable-coefficient operators while retaining an algebraic calculus. The answer developed in the course is the local pseudodifferential calculus on open subsets of $\mathbb R^n$, built from symbols, quantisation, composition, parametrices, and Sobolev estimates.

The course sits between Fourier analysis and elliptic PDE. Fourier analysis supplies the model operators, distributions supply the correct domain of action, and Sobolev spaces supply the scale on which estimates are measured. A later course on wave front sets and Fourier integral operators will attach singularities to directions in phase space; in this course the emphasis stays on the operator algebra needed before that refinement enters.

## What the Course Is Trying to Build

The first problem is that differential operators with variable coefficients are local in $x$ but not diagonal under the Fourier transform. Constant-coefficient operators become multiplication by a polynomial in the frequency variable $\xi$, so their mapping properties are visible from the size and zero set of that polynomial. Variable coefficients destroy exact diagonalisation, but their coefficients vary in $x$ while differentiation still corresponds to powers of $\xi$.

[motivation] ### From Multipliers to Variable Coefficients For a constant-coefficient operator \begin{align*} P(D)u = \sum_{|\alpha| \le m} c_\alpha D^\alpha u, \end{align*} the Fourier transform gives \begin{align*} \widehat{P(D)u}(\xi) = p(\xi)\hat{u}(\xi), \qquad p(\xi)=\sum_{|\alpha|\le m}c_\alpha \xi^\alpha, \end{align*} up to the chosen convention for powers of $D$. The operator is therefore controlled by a function $p$ on frequency space. This is the model for a pseudodifferential operator: replace $p(\xi)$ by a function $a(x,\xi)$ whose $x$-dependence records variable coefficients and whose $\xi$-growth records order. ### Why Estimates Replace Formulas The calculus cannot rely only on closed-form expressions for $a(x,\xi)$. Composition, adjoints, coordinate cutoffs, and parametrices produce expansions rather than finite formulas. The course therefore organises operators by derivative estimates on symbols, since these estimates survive the operations needed in elliptic theory. [/motivation]

This motivation leads to the course's main object. The formal definition will come later after Fourier conventions and distribution spaces have been reviewed, but the guiding shape is already visible: an operator is assembled by taking the Fourier transform, multiplying by an amplitude depending on both $x$ and $\xi$, and inverting the transform.

[definition: Local Pseudodifferential Calculus] Let $U \subset \mathbb R^n$ be open. The local pseudodifferential calculus on $U$ is the framework that starts with smooth functions $a(x,\xi)$ on position-frequency space and turns them into operators, usually written $\operatorname{Op}_U(a)$, acting first on compactly supported smooth test functions. The growth of $a$ as $|\xi|\to\infty$ records the order of the operator, while derivatives in $x$ and $\xi$ measure how stable the operator is under localization, composition, and adjoints. The word local means that the construction is tested after inserting compactly supported cutoffs in $U$. Properly supported operators are the ones whose kernels have support controlled enough that they act on distributions without sending information infinitely far away. Smoothing operators are the error terms with smooth kernels; they are negligible for finite-order local Sobolev regularity. [/definition]

This is not a single operator class in isolation. It is a package: symbol spaces, a way of turning symbols into operators, an [equivalence relation](/page/Equivalence%20Relation) modulo smoothing terms, and estimates strong enough to control Sobolev norms. The later chapters prove the structural properties deliberately kept out of the definition: composition and adjoint formulae modulo smoothing operators, elliptic parametrices, and continuous maps between local Sobolev spaces.

One notation from the later symbol theory is useful already. The class $S^{-\infty}$ denotes the intersection of all symbol classes of finite order: a symbol is in $S^{-\infty}$ when it satisfies symbol estimates of order $N$ for every $N \in \mathbb Z$. Operators with symbols in $S^{-\infty}$ are smoothing, so this notation is a symbolic way of recording that an error term has become negligible for the calculus.

[example: Differential Operators As Symbols] Let $U \subset \mathbb R^n$ be open, and use the course convention $D_j=-i\partial_{x_j}$. For \begin{align*} L u(x)=\sum_{|\alpha|\le m} a_\alpha(x)D^\alpha u(x), \end{align*} with $a_\alpha\in C^\infty(U)$, the coefficient $a_\alpha(x)$ records the $x$-dependence and the factor $D^\alpha$ records the frequency dependence. Indeed, for the oscillation $e^{ix\cdot \xi}$ one has \begin{align*} D_j e^{ix\cdot \xi}=-i\partial_{x_j}e^{ix\cdot \xi}=-i(i\xi_j)e^{ix\cdot \xi}=\xi_j e^{ix\cdot \xi}. \end{align*} Applying this identity once for each derivative in the multi-index $\alpha$ gives \begin{align*} D^\alpha e^{ix\cdot \xi}=\xi^\alpha e^{ix\cdot \xi}. \end{align*} Therefore \begin{align*} L(e^{ix\cdot \xi})=\sum_{|\alpha|\le m}a_\alpha(x)\xi^\alpha e^{ix\cdot \xi}=p(x,\xi)e^{ix\cdot \xi}, \end{align*} where \begin{align*} p(x,\xi)=\sum_{|\alpha|\le m}a_\alpha(x)\xi^\alpha. \end{align*} The same differential expression extends to distributions by defining $\langle Lu,\varphi\rangle=\langle u,L^t\varphi\rangle$ for $\varphi\in C_c^\infty(U)$, where $L^t$ is the transpose differential operator on test functions. For example, if $L=-\Delta+V(x)$ and $D_j=-i\partial_{x_j}$, then \begin{align*} D_j^2=(-i\partial_{x_j})(-i\partial_{x_j})=-\partial_{x_j}^2. \end{align*} Hence \begin{align*} -\Delta=-\sum_{j=1}^n\partial_{x_j}^2=\sum_{j=1}^nD_j^2. \end{align*} On $e^{ix\cdot \xi}$ this gives \begin{align*} (-\Delta)e^{ix\cdot \xi}=\sum_{j=1}^nD_j^2e^{ix\cdot \xi}=\sum_{j=1}^n\xi_j^2e^{ix\cdot \xi}=|\xi|^2e^{ix\cdot \xi}. \end{align*} The multiplication term contributes \begin{align*} V(x)e^{ix\cdot \xi}=V(x)e^{ix\cdot \xi}. \end{align*} Thus \begin{align*} (-\Delta+V(x))e^{ix\cdot \xi}=(|\xi|^2+V(x))e^{ix\cdot \xi}, \end{align*} so the symbol is $p(x,\xi)=|\xi|^2+V(x)$ and the principal symbol is the degree-$2$ part $|\xi|^2$. This is the basic separation used throughout the course: the highest powers of $\xi$ determine order and ellipticity, while lower powers and zeroth-order coefficients enter as lower-order corrections. [/example]

Differential operators give the entry point, but the calculus is larger because inverses of elliptic differential operators are seldom differential operators. The search for approximate inverses is the reason pseudodifferential operators appear naturally in PDE.

## The Guiding Elliptic Problem

The next question is how to solve an elliptic equation microlocally before introducing the full language of wave front sets. If $L$ is elliptic, its principal symbol does not vanish at high frequency, so the first approximation to an inverse should divide by that principal symbol. This is the parametrix idea.

[definition: Parametrix] Let $U \subset \mathbb R^n$ be open and let $L: \mathcal D'(U) \to \mathcal D'(U)$ be a continuous linear operator. A parametrix for $L$ is a continuous linear operator $Q: \mathcal D'(U) \to \mathcal D'(U)$ such that \begin{align*} QL = I - R_1, \qquad LQ = I - R_2, \end{align*} where $R_1,R_2: \mathcal D'(U) \to C^\infty(U)$ are smoothing operators, regarded as maps into $\mathcal D'(U)$ by the canonical inclusion $C^\infty(U) \subset \mathcal D'(U)$. [/definition]

A parametrix gives the right substitute for an inverse when exact inversion is obstructed by lower-order and compactly supported effects. The next result is the course-level form of the definition: ellipticity is the condition that permits inversion modulo smoothing errors, provided the operator is interpreted inside the properly supported local calculus.

[quotetheorem:7662]