PMATH 855: Microlocal Analysis
Estimated study time: 1 hr 53 min
Table of contents
These notes synthesize material from L. Hörmander’s The Analysis of Linear Partial Differential Operators, A. Grigis and J. Sjöstrand’s Microlocal Analysis for Differential Operators, M. Zworski’s Semiclassical Analysis, and M.E. Taylor’s Pseudodifferential Operators, enriched with material from MIT OCW 18.157 (R. Melrose) and S. Dyatlov’s lecture notes.
Chapter 1: Distributions and Sobolev Spaces
The modern theory of partial differential equations rests on a broadened notion of “function” that permits differentiation of objects far more irregular than anything classical analysis can handle. Laurent Schwartz’s theory of distributions, developed in the late 1940s, provided the rigorous framework for this program. In this opening chapter, we set up the functional-analytic foundations: the spaces of test functions, distributions, tempered distributions, and the Sobolev spaces that measure regularity in a quantitative way. These tools are indispensable for everything that follows.
1.1 Test Functions and Distributions
We begin with the space of test functions and the distributions that act on them.
Definition 1.1.1 (Test Function Space). Let \(\Omega \subseteq \mathbb{R}^n\) be open. The space \(C_c^\infty(\Omega)\), also denoted \(\mathcal{D}(\Omega)\), consists of all infinitely differentiable functions \(\varphi : \Omega \to \mathbb{C}\) with compact support in \(\Omega\). We equip \(\mathcal{D}(\Omega)\) with the following topology: a sequence \(\varphi_j \to \varphi\) in \(\mathcal{D}(\Omega)\) if and only if there exists a compact set \(K \subset \Omega\) with \(\operatorname{supp} \varphi_j \subseteq K\) for all \(j\), and \(\partial^\alpha \varphi_j \to \partial^\alpha \varphi\) uniformly on \(K\) for every multi-index \(\alpha\).
This topology makes \(\mathcal{D}(\Omega)\) into a locally convex topological vector space, though not a Fréchet space (it is an LF-space, i.e., a strict inductive limit of Fréchet spaces). The precise topological structure, while important for the general theory, will mostly stay in the background; what matters for us is the notion of convergence described above.
Definition 1.1.2 (Distributions). A distribution on \(\Omega\) is a continuous linear functional \(u : \mathcal{D}(\Omega) \to \mathbb{C}\). The space of all distributions on \(\Omega\) is denoted \(\mathcal{D}'(\Omega)\). Continuity means that if \(\varphi_j \to \varphi\) in \(\mathcal{D}(\Omega)\), then \(\langle u, \varphi_j \rangle \to \langle u, \varphi \rangle\) in \(\mathbb{C}\).
Example 1.1.3. Every locally integrable function \(f \in L^1_{\mathrm{loc}}(\Omega)\) defines a distribution via
\[
\langle u_f, \varphi \rangle = \int_\Omega f(x) \varphi(x) \, dx, \quad \varphi \in \mathcal{D}(\Omega).
\]
The map \(f \mapsto u_f\) is injective (by the du Bois-Reymond lemma), so we identify \(L^1_{\mathrm{loc}}(\Omega)\) with a subspace of \(\mathcal{D}'(\Omega)\). Distributions that arise this way are called regular.
Example 1.1.4 (Dirac Delta). For \(x_0 \in \Omega\), the Dirac delta at \(x_0\) is the distribution defined by
\[
\langle \delta_{x_0}, \varphi \rangle = \varphi(x_0).
\]
This is not a regular distribution: there is no locally integrable function \(f\) such that \(\int f \varphi \, dx = \varphi(x_0)\) for all \(\varphi \in \mathcal{D}(\Omega)\).
Example 1.1.5 (Principal Value). The principal value distribution on \(\mathbb{R}\) is defined by
\[
\langle \mathrm{p.v.}\tfrac{1}{x}, \varphi \rangle = \lim_{\varepsilon \to 0^+} \int_{|x| > \varepsilon} \frac{\varphi(x)}{x} \, dx.
\]
One verifies that this limit exists for every \(\varphi \in \mathcal{D}(\mathbb{R})\) and defines a continuous linear functional.
1.2 Operations on Distributions
The power of distribution theory lies in the ability to extend classical operations — differentiation, multiplication by smooth functions, convolution — to this enlarged setting.
Definition 1.2.1 (Derivative of a Distribution). If \(u \in \mathcal{D}'(\Omega)\) and \(\alpha\) is a multi-index, the distributional derivative \(\partial^\alpha u\) is defined by
\[
\langle \partial^\alpha u, \varphi \rangle = (-1)^{|\alpha|} \langle u, \partial^\alpha \varphi \rangle, \quad \varphi \in \mathcal{D}(\Omega).
\]
This definition is motivated by integration by parts: if \(u\) is smooth with compact support, the formula above holds with equality. The key observation is that every distribution is infinitely differentiable in the distributional sense.
Example 1.2.2. On \(\mathbb{R}\), the Heaviside function \(H(x) = \mathbf{1}_{[0,\infty)}(x)\) has distributional derivative \(H' = \delta_0\). Indeed, for \(\varphi \in \mathcal{D}(\mathbb{R})\),
\[
\langle H', \varphi \rangle = -\langle H, \varphi' \rangle = -\int_0^\infty \varphi'(x) \, dx = \varphi(0) = \langle \delta_0, \varphi \rangle.
\]
Definition 1.2.3 (Multiplication by Smooth Functions). If \(u \in \mathcal{D}'(\Omega)\) and \(a \in C^\infty(\Omega)\), then \(au \in \mathcal{D}'(\Omega)\) is defined by
\[
\langle au, \varphi \rangle = \langle u, a\varphi \rangle, \quad \varphi \in \mathcal{D}(\Omega).
\]
Remark 1.2.4. It is in general not possible to multiply two arbitrary distributions. This fundamental obstruction, formalized by Schwartz's impossibility result, is one of the motivations for microlocal analysis: the wavefront set will tell us precisely when multiplication of distributions is well-defined.
Definition 1.2.5 (Convolution). If \(u \in \mathcal{D}'(\mathbb{R}^n)\) and \(\varphi \in \mathcal{D}(\mathbb{R}^n)\), the convolution \(u * \varphi\) is the smooth function
\[
(u * \varphi)(x) = \langle u, \varphi(x - \cdot) \rangle.
\]
More generally, if \(u \in \mathcal{E}'(\mathbb{R}^n)\) (compactly supported distribution) and \(v \in \mathcal{D}'(\mathbb{R}^n)\), the convolution \(u * v \in \mathcal{D}'(\mathbb{R}^n)\) is defined by
\[
\langle u * v, \varphi \rangle = \langle u_x, \langle v_y, \varphi(x+y) \rangle \rangle.
\]
Proposition 1.2.6 (Regularization). Let \(\rho \in \mathcal{D}(\mathbb{R}^n)\) with \(\int \rho = 1\), and set \(\rho_\varepsilon(x) = \varepsilon^{-n}\rho(x/\varepsilon)\). If \(u \in \mathcal{D}'(\mathbb{R}^n)\), then \(u * \rho_\varepsilon \to u\) in \(\mathcal{D}'(\mathbb{R}^n)\) as \(\varepsilon \to 0\).
1.3 Tempered Distributions and the Fourier Transform
To develop Fourier analysis in the distributional setting, we need a space of distributions adapted to the Fourier transform. This is the space of tempered distributions, introduced by Schwartz precisely for this purpose.
Definition 1.3.1 (Schwartz Space). The Schwartz space \(\mathcal{S}(\mathbb{R}^n)\) consists of all \(\varphi \in C^\infty(\mathbb{R}^n)\) such that
\[
\|\varphi\|_{\alpha,\beta} := \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta \varphi(x)| < \infty
\]
for all multi-indices \(\alpha, \beta\). The topology is generated by the family of seminorms \(\|\cdot\|_{\alpha,\beta}\), making \(\mathcal{S}(\mathbb{R}^n)\) a Fréchet space.
Definition 1.3.2 (Tempered Distributions). A tempered distribution is a continuous linear functional on \(\mathcal{S}(\mathbb{R}^n)\). The space of tempered distributions is denoted \(\mathcal{S}'(\mathbb{R}^n)\). We have the continuous inclusions
\[
\mathcal{D}(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n).
\]
Definition 1.3.3 (Fourier Transform). For \(\varphi \in \mathcal{S}(\mathbb{R}^n)\), the Fourier transform is
\[
\hat{\varphi}(\xi) = \mathcal{F}\varphi(\xi) = \int_{\mathbb{R}^n} e^{-i x \cdot \xi} \varphi(x) \, dx.
\]
The inverse Fourier transform is
\[
\mathcal{F}^{-1}\psi(x) = (2\pi)^{-n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} \psi(\xi) \, d\xi.
\]
Theorem 1.3.4 (Fourier Transform on \(\mathcal{S}\)). The Fourier transform \(\mathcal{F} : \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)\) is a topological isomorphism with inverse \(\mathcal{F}^{-1}\). Moreover:
(i) \(\mathcal{F}(\partial^\alpha \varphi)(\xi) = (i\xi)^\alpha \hat{\varphi}(\xi)\),
(ii) \(\mathcal{F}(x^\alpha \varphi)(\xi) = (i\partial_\xi)^\alpha \hat{\varphi}(\xi)\),
(iii) \(\mathcal{F}(\varphi * \psi) = \hat{\varphi} \cdot \hat{\psi}\),
(iv) (Parseval) \(\int \hat{\varphi} \overline{\hat{\psi}} \, d\xi = (2\pi)^n \int \varphi \overline{\psi} \, dx\).
(i) \(\mathcal{F}(\partial^\alpha \varphi)(\xi) = (i\xi)^\alpha \hat{\varphi}(\xi)\),
(ii) \(\mathcal{F}(x^\alpha \varphi)(\xi) = (i\partial_\xi)^\alpha \hat{\varphi}(\xi)\),
(iii) \(\mathcal{F}(\varphi * \psi) = \hat{\varphi} \cdot \hat{\psi}\),
(iv) (Parseval) \(\int \hat{\varphi} \overline{\hat{\psi}} \, d\xi = (2\pi)^n \int \varphi \overline{\psi} \, dx\).
Definition 1.3.5 (Fourier Transform of Tempered Distributions). For \(u \in \mathcal{S}'(\mathbb{R}^n)\), the Fourier transform \(\hat{u} \in \mathcal{S}'(\mathbb{R}^n)\) is defined by
\[
\langle \hat{u}, \varphi \rangle = \langle u, \hat{\varphi} \rangle, \quad \varphi \in \mathcal{S}(\mathbb{R}^n).
\]
Example 1.3.6. The Fourier transform of \(\delta_0\) is the constant function \(1\):
\[
\langle \hat{\delta}_0, \varphi \rangle = \langle \delta_0, \hat{\varphi} \rangle = \hat{\varphi}(0) = \int \varphi(x) \, dx = \langle 1, \varphi \rangle.
\]
Conversely, \(\hat{1} = (2\pi)^n \delta_0\).
1.4 Sobolev Spaces
Sobolev spaces provide the quantitative framework for measuring regularity of distributions. They are the natural \(L^2\)-based function spaces in which to study elliptic and more general PDE.
Definition 1.4.1 (Sobolev Spaces via Fourier Transform). For \(s \in \mathbb{R}\), the Sobolev space \(H^s(\mathbb{R}^n)\) consists of all \(u \in \mathcal{S}'(\mathbb{R}^n)\) such that
\[
\|u\|_{H^s}^2 := \int_{\mathbb{R}^n} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 \, d\xi < \infty.
\]
We write \(\langle \xi \rangle = (1 + |\xi|^2)^{1/2}\), so the norm becomes \(\|u\|_{H^s} = \|\langle \xi \rangle^s \hat{u}\|_{L^2}\).
Remark 1.4.2. For non-negative integers \(s = k\), \(H^k(\mathbb{R}^n)\) coincides with the space of \(L^2\) functions whose distributional derivatives up to order \(k\) are in \(L^2\), with equivalent norms. For negative \(s\), \(H^s(\mathbb{R}^n)\) contains genuine distributions. For instance, \(\delta_0 \in H^s(\mathbb{R}^n)\) if and only if \(s < -n/2\).
Proposition 1.4.3 (Basic Properties).
(i) \(H^s(\mathbb{R}^n)\) is a Hilbert space with inner product \(\langle u, v \rangle_{H^s} = \int \langle \xi \rangle^{2s} \hat{u}(\xi) \overline{\hat{v}(\xi)} \, d\xi\).
(ii) If \(s > t\), then \(H^s(\mathbb{R}^n) \hookrightarrow H^t(\mathbb{R}^n)\) with continuous inclusion.
(iii) \(\mathcal{S}(\mathbb{R}^n)\) is dense in \(H^s(\mathbb{R}^n)\) for every \(s \in \mathbb{R}\).
(iv) The dual of \(H^s(\mathbb{R}^n)\) is isometrically isomorphic to \(H^{-s}(\mathbb{R}^n)\).
(i) \(H^s(\mathbb{R}^n)\) is a Hilbert space with inner product \(\langle u, v \rangle_{H^s} = \int \langle \xi \rangle^{2s} \hat{u}(\xi) \overline{\hat{v}(\xi)} \, d\xi\).
(ii) If \(s > t\), then \(H^s(\mathbb{R}^n) \hookrightarrow H^t(\mathbb{R}^n)\) with continuous inclusion.
(iii) \(\mathcal{S}(\mathbb{R}^n)\) is dense in \(H^s(\mathbb{R}^n)\) for every \(s \in \mathbb{R}\).
(iv) The dual of \(H^s(\mathbb{R}^n)\) is isometrically isomorphic to \(H^{-s}(\mathbb{R}^n)\).
Definition 1.4.4 (Sobolev Spaces on Domains). For an open set \(\Omega \subseteq \mathbb{R}^n\), we define:
(i) \(H^s(\Omega) = \{ u|_\Omega : u \in H^s(\mathbb{R}^n) \}\) with the quotient norm,
(ii) \(H^s_0(\Omega)\) as the closure of \(C_c^\infty(\Omega)\) in \(H^s(\Omega)\).
(i) \(H^s(\Omega) = \{ u|_\Omega : u \in H^s(\mathbb{R}^n) \}\) with the quotient norm,
(ii) \(H^s_0(\Omega)\) as the closure of \(C_c^\infty(\Omega)\) in \(H^s(\Omega)\).
1.5 Sobolev Embedding Theorems
The embedding theorems relate Sobolev regularity to classical regularity. They are among the most frequently used results in PDE theory.
Theorem 1.5.1 (Sobolev Embedding). If \(s > n/2 + k\) for a non-negative integer \(k\), then \(H^s(\mathbb{R}^n) \hookrightarrow C^k_b(\mathbb{R}^n)\) (bounded continuous functions with bounded derivatives up to order \(k\)), and the embedding is continuous:
\[
\|\partial^\alpha u\|_{L^\infty} \leq C_{s,n} \|u\|_{H^s}, \quad |\alpha| \leq k.
\]
In particular, \(\bigcap_{s \in \mathbb{R}} H^s(\mathbb{R}^n) = \mathcal{S}(\mathbb{R}^n)\) (as sets, though the topologies differ).
Proof. For \(|\alpha| \leq k\) and \(u \in \mathcal{S}(\mathbb{R}^n)\),
\[
\partial^\alpha u(x) = (2\pi)^{-n} \int e^{ix \cdot \xi} (i\xi)^\alpha \hat{u}(\xi) \, d\xi.
\]
By the Cauchy-Schwarz inequality,
\[
|\partial^\alpha u(x)| \leq (2\pi)^{-n} \int |\xi^\alpha| |\hat{u}(\xi)| \, d\xi = (2\pi)^{-n} \int \frac{|\xi^\alpha|}{\langle \xi \rangle^s} \cdot \langle \xi \rangle^s |\hat{u}(\xi)| \, d\xi
\]
\[
\leq (2\pi)^{-n} \left(\int \frac{|\xi|^{2k}}{\langle \xi \rangle^{2s}} \, d\xi\right)^{1/2} \|u\|_{H^s}.
\]
The first integral is finite precisely when \(2s - 2k > n\), i.e., \(s > n/2 + k\). The result extends from \(\mathcal{S}\) to \(H^s\) by density. \(\blacksquare\)
Theorem 1.5.2 (Rellich-Kondrachov Compactness). Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set with Lipschitz boundary. If \(s > t\), then the inclusion \(H^s(\Omega) \hookrightarrow H^t(\Omega)\) is compact.
This compactness result is fundamental for spectral theory of elliptic operators and will appear repeatedly in later chapters.
1.6 Trace Theorems
When studying boundary value problems, one needs to restrict Sobolev functions to lower-dimensional submanifolds. The trace theorems make this precise.
Theorem 1.6.1 (Trace Theorem). Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set with smooth boundary \(\partial\Omega\). For \(s > 1/2\), the restriction map \(\gamma_0 : C^\infty(\overline{\Omega}) \to C^\infty(\partial\Omega)\) defined by \(\gamma_0 u = u|_{\partial\Omega}\) extends to a continuous surjection
\[
\gamma_0 : H^s(\Omega) \to H^{s-1/2}(\partial\Omega),
\]
with a continuous right inverse (extension operator).
Remark 1.6.2. The loss of \(1/2\) derivative is sharp and cannot be improved. This is consistent with the heuristic that restricting to a codimension-\(k\) submanifold costs \(k/2\) derivatives in the \(L^2\)-based Sobolev scale.
1.7 Duality of Sobolev Spaces
Theorem 1.7.1. The dual space of \(H^s_0(\Omega)\) is naturally identified with \(H^{-s}(\Omega)\), and the dual of \(H^s(\Omega)\) is identified with a space of distributions supported in \(\overline{\Omega}\). In the case \(\Omega = \mathbb{R}^n\), we have \((H^s(\mathbb{R}^n))' \cong H^{-s}(\mathbb{R}^n)\) with the duality pairing extending the \(L^2\) inner product.
These duality relations are essential for the weak formulation of PDE and for the theory of pseudodifferential operators on Sobolev spaces that we develop in Chapter 3.
Chapter 2: The Fourier Transform and Symbol Classes
Having set up the distributional framework, we now turn to the algebraic and analytic machinery that underlies the theory of pseudodifferential operators. The key idea, pioneered by Kohn-Nirenberg (1965) and Hörmander (1965), is to associate to a linear operator a function of both position and frequency — its symbol — and to study the operator through the properties of this symbol. This chapter develops the symbol calculus that makes this program precise.
2.1 Oscillatory Integrals
Many of the integrals we encounter will not converge absolutely; they must be interpreted as oscillatory integrals, regularized by the rapid oscillation of the integrand.
Definition 2.1.1 (Oscillatory Integral). Let \(a(x,\xi) \in S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)\) (to be defined below) and \(\Phi(x,\xi)\) be a phase function. The oscillatory integral
\[
I(\varphi) = \int\!\!\!\int e^{i\Phi(x,\xi)} a(x,\xi) \varphi(x) \, dx \, d\xi
\]
is defined as the limit
\[
I(\varphi) = \lim_{\varepsilon \to 0} \int\!\!\!\int e^{i\Phi(x,\xi)} a(x,\xi) \chi(\varepsilon\xi) \varphi(x) \, dx \, d\xi,
\]
where \(\chi \in C_c^\infty(\mathbb{R}^n)\) with \(\chi(0) = 1\). The limit is independent of the choice of \(\chi\).
\[
L = \frac{1 - i\xi \cdot \nabla_x \Phi}{|\nabla_x \Phi|^2 + |\xi|^2}(1 - \Delta_\xi)
\]satisfies \(L^t(e^{i\Phi}) = e^{i\Phi}\) when the phase is non-degenerate, and repeated application of \(L^t\) to the amplitude \(a \cdot \chi(\varepsilon\xi) \cdot \varphi\) produces convergent integrals.
2.2 Symbol Classes
The symbol classes, introduced by Hörmander, provide the natural setting for the symbols of pseudodifferential operators.
Definition 2.2.1 (Symbol Classes \(S^m_{\rho,\delta}\)). Let \(m \in \mathbb{R}\) and \(0 \leq \delta \leq \rho \leq 1\). A smooth function \(a \in C^\infty(\mathbb{R}^n \times \mathbb{R}^n)\) belongs to the symbol class \(S^m_{\rho,\delta}(\mathbb{R}^n \times \mathbb{R}^n)\) if for all multi-indices \(\alpha, \beta\) there exists a constant \(C_{\alpha,\beta}\) such that
\[
|\partial_\xi^\alpha \partial_x^\beta a(x,\xi)| \leq C_{\alpha,\beta} \langle \xi \rangle^{m - \rho|\alpha| + \delta|\beta|}
\]
for all \(x, \xi \in \mathbb{R}^n\). We write \(S^m = S^m_{1,0}\) for the standard class and \(S^{-\infty} = \bigcap_m S^m\).
Remark 2.2.2. The condition \(\delta < \rho\) (or at least \(\delta \leq \rho\) with \(\delta < 1\)) is essential for a well-behaved calculus. The "forbidden" class \(S^m_{1,1}\) fails to give \(L^2\)-bounded operators even at order \(m = 0\), and its pseudodifferential calculus is fundamentally deficient. The standard classes \(S^m_{1,0}\) and the Weyl-Hörmander classes \(S^m_{1/2,1/2}\) are the most important in practice.
Example 2.2.3. The function \(a(x,\xi) = \langle \xi \rangle^m = (1 + |\xi|^2)^{m/2}\) belongs to \(S^m_{1,0}\). More generally, if \(p(x,\xi)\) is a polynomial of degree \(m\) in \(\xi\) with smooth coefficients bounded along with all their derivatives, then \(p \in S^m_{1,0}\).
Example 2.2.4. The symbol \(a(x,\xi) = |\xi|^2 + V(x)\), where \(V \in C^\infty_b(\mathbb{R}^n)\), belongs to \(S^2_{1,0}\). This is the symbol of the Schrödinger operator \(-\Delta + V(x)\).
2.3 Asymptotic Expansions
A central technique in the symbol calculus is the construction of symbols from asymptotic series.
Definition 2.3.1 (Asymptotic Expansion). Let \(a_j \in S^{m_j}_{\rho,\delta}\) with \(m_j \to -\infty\). We say that \(a \in S^{m_0}_{\rho,\delta}\) has the asymptotic expansion \(a \sim \sum_{j=0}^\infty a_j\) if for every \(N\),
\[
a - \sum_{j=0}^{N-1} a_j \in S^{m_N}_{\rho,\delta}.
\]
Theorem 2.3.2 (Borel's Lemma for Symbols). Given \(a_j \in S^{m_j}_{\rho,\delta}\) with \(m_j \to -\infty\), there exists \(a \in S^{m_0}_{\rho,\delta}\) with \(a \sim \sum_j a_j\). The symbol \(a\) is unique modulo \(S^{-\infty}\).
Proof. Choose a cutoff \(\chi \in C^\infty(\mathbb{R}^n)\) with \(\chi(\xi) = 0\) for \(|\xi| \leq 1\) and \(\chi(\xi) = 1\) for \(|\xi| \geq 2\). Set
\[
a(x,\xi) = \sum_{j=0}^\infty \chi(\xi/R_j) a_j(x,\xi),
\]
where \(R_j\) is a sequence increasing sufficiently rapidly to \(\infty\). For \(|\xi| \leq R_j\), the \(j\)-th term vanishes, so the sum is locally finite and defines a smooth function. One verifies that for sufficiently rapidly growing \(R_j\), the tail \(\sum_{j \geq N} \chi(\xi/R_j) a_j\) belongs to \(S^{m_N}_{\rho,\delta}\). The uniqueness modulo \(S^{-\infty}\) is immediate from the definition. \(\blacksquare\)
2.4 Classical Symbols
Many operators arising in geometric and physical applications have symbols with a particularly nice structure.
Definition 2.4.1 (Classical Symbols). A symbol \(a \in S^m_{1,0}\) is classical (or polyhomogeneous) if it admits an asymptotic expansion \(a \sim \sum_{j=0}^\infty a_{m-j}\), where each \(a_{m-j}(x,\xi)\) is positively homogeneous of degree \(m-j\) in \(\xi\) for \(|\xi| \geq 1\):
\[
a_{m-j}(x, t\xi) = t^{m-j} a_{m-j}(x,\xi), \quad t \geq 1, \; |\xi| \geq 1.
\]
The leading term \(a_m\) is called the principal symbol.
Example 2.4.2. The Laplacian \(\Delta = \sum_{j=1}^n \partial_{x_j}^2\) has symbol \(-|\xi|^2\), which is a classical symbol of order 2 with principal symbol \(\sigma_2(\Delta)(x,\xi) = -|\xi|^2\). More generally, any differential operator \(P = \sum_{|\alpha| \leq m} a_\alpha(x) D^\alpha\) (where \(D = -i\partial\)) has symbol \(p(x,\xi) = \sum_{|\alpha| \leq m} a_\alpha(x) \xi^\alpha\), which is a polynomial in \(\xi\) and hence classical.
2.5 The Schwartz Kernel Theorem
The Schwartz kernel theorem provides the bridge between operators and distributions, and is fundamental to the theory of integral operators.
Theorem 2.5.1 (Schwartz Kernel Theorem). Every continuous linear operator \(A : \mathcal{D}(\Omega_2) \to \mathcal{D}'(\Omega_1)\) has a unique distribution kernel \(K_A \in \mathcal{D}'(\Omega_1 \times \Omega_2)\) such that
\[
\langle Au, v \rangle = \langle K_A, v \otimes u \rangle
\]
for all \(u \in \mathcal{D}(\Omega_2)\), \(v \in \mathcal{D}(\Omega_1)\). Conversely, every \(K \in \mathcal{D}'(\Omega_1 \times \Omega_2)\) defines such an operator.
Remark 2.5.2. The Schwartz kernel theorem is a deep result in functional analysis, relying on the nuclear structure of the spaces involved. It tells us that the study of linear operators is, in principle, reducible to the study of distributions on product spaces — a profound unification.
2.6 Fourier Integral Representation of Operators
We can now write down the general form of a pseudodifferential operator as a Fourier integral.
Definition 2.6.1. Given a symbol \(a \in S^m_{\rho,\delta}(\mathbb{R}^n \times \mathbb{R}^n)\), the associated operator \(\mathrm{Op}(a)\) (or \(a(x,D)\)) is defined by
\[
\mathrm{Op}(a)u(x) = (2\pi)^{-n} \int_{\mathbb{R}^n} e^{ix \cdot \xi} a(x,\xi) \hat{u}(\xi) \, d\xi
\]
for \(u \in \mathcal{S}(\mathbb{R}^n)\). Equivalently, using the Fourier inversion formula,
\[
\mathrm{Op}(a)u(x) = (2\pi)^{-n} \int\!\!\!\int e^{i(x-y) \cdot \xi} a(x,\xi) u(y) \, dy \, d\xi.
\]
The latter is an oscillatory integral.
Remark 2.6.2. This is the Kohn-Nirenberg (or standard or left) quantization. Other quantization schemes — notably the Weyl quantization \(\mathrm{Op}^w(a)\), where the symbol is evaluated at the midpoint \((x+y)/2\) rather than at \(x\) — will become important in Chapter 7 on semiclassical analysis.
Chapter 3: Pseudodifferential Operators
Pseudodifferential operators (abbreviated \(\Psi\)DOs) generalize differential operators by allowing symbols that are not polynomial in \(\xi\). They were introduced in the 1960s by Kohn-Nirenberg, Hörmander, and others to provide a flexible algebraic framework for studying elliptic PDE. The fundamental insight is that the class of pseudodifferential operators is closed under composition, taking adjoints, and — crucially — taking parametrices (approximate inverses) of elliptic operators.
3.1 Definition and Basic Properties
Definition 3.1.1 (Pseudodifferential Operator). An operator \(A : \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)\) is a pseudodifferential operator of order \(m\) and type \((\rho,\delta)\) if it has the form
\[
Au(x) = \mathrm{Op}(a)u(x) = (2\pi)^{-n} \int\!\!\!\int e^{i(x-y)\cdot\xi} a(x,\xi) u(y) \, dy \, d\xi
\]
for some symbol \(a \in S^m_{\rho,\delta}\). We write \(A \in \Psi^m_{\rho,\delta}(\mathbb{R}^n)\), or simply \(A \in \Psi^m\) when \((\rho,\delta) = (1,0)\). The class \(\Psi^{-\infty} = \bigcap_m \Psi^m\) consists of smoothing operators.
Proposition 3.1.2. Every \(A \in \Psi^{-\infty}(\mathbb{R}^n)\) has a Schwartz kernel \(K_A \in \mathcal{S}(\mathbb{R}^n \times \mathbb{R}^n)\); in particular, \(A\) maps \(\mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)\). Conversely, any operator with Schwartz kernel in \(\mathcal{S}\) belongs to \(\Psi^{-\infty}\).
Theorem 3.1.3 (Mapping Properties). If \(A \in \Psi^m_{\rho,\delta}(\mathbb{R}^n)\) with \(\delta < 1\), then:
(i) \(A : \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)\) is continuous.
(ii) \(A\) extends to a continuous map \(A : \mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)\).
(iii) \(A : H^s(\mathbb{R}^n) \to H^{s-m}(\mathbb{R}^n)\) is continuous for every \(s \in \mathbb{R}\).
(i) \(A : \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)\) is continuous.
(ii) \(A\) extends to a continuous map \(A : \mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)\).
(iii) \(A : H^s(\mathbb{R}^n) \to H^{s-m}(\mathbb{R}^n)\) is continuous for every \(s \in \mathbb{R}\).
Part (iii) says that a \(\Psi\)DO of order \(m\) “costs” exactly \(m\) derivatives in the Sobolev scale, which is the quantitative embodiment of the notion of order.
3.2 The Symbol Map
The assignment \(a \mapsto \mathrm{Op}(a)\) is not quite injective: two symbols that differ by an element of \(S^{-\infty}\) give rise to operators that differ by a smoothing operator. The quotient gives a well-defined map.
Theorem 3.2.1 (Symbol Map). The map \(\sigma : \Psi^m_{\rho,\delta} / \Psi^{-\infty} \to S^m_{\rho,\delta} / S^{-\infty}\) that assigns to an operator its symbol class (modulo \(S^{-\infty}\)) is a well-defined isomorphism. The principal symbol \(\sigma_m(A) \in S^m / S^{m-(\rho-\delta)}\) is an invariantly defined function on \(T^*\mathbb{R}^n\) (or more generally on the cotangent bundle of a manifold).
3.3 Composition of Pseudodifferential Operators
The closure of the pseudodifferential calculus under composition is its most powerful algebraic property.
Theorem 3.3.1 (Composition). Let \(A \in \Psi^{m_1}_{\rho,\delta}(\mathbb{R}^n)\) with symbol \(a\) and \(B \in \Psi^{m_2}_{\rho,\delta}(\mathbb{R}^n)\) with symbol \(b\), where \(\delta < \rho\). Then \(AB \in \Psi^{m_1+m_2}_{\rho,\delta}(\mathbb{R}^n)\) with symbol \(c \in S^{m_1+m_2}_{\rho,\delta}\) given by
\[
c(x,\xi) \sim \sum_\alpha \frac{1}{\alpha!} \partial_\xi^\alpha a(x,\xi) \cdot D_x^\alpha b(x,\xi).
\]
In particular, the principal symbol of the composition is the product of the principal symbols:
\[
\sigma_{m_1+m_2}(AB) = \sigma_{m_1}(A) \cdot \sigma_{m_2}(B).
\]
Proof (Sketch). The Schwartz kernel of \(AB\) is
\[
K_{AB}(x,y) = \int K_A(x,z) K_B(z,y) \, dz.
\]
Substituting the oscillatory integral representations and performing a stationary phase expansion in the intermediate variable \(z\) yields the asymptotic formula. The key identity is
\[
e^{-iz \cdot \eta} a(x,\xi+\eta) = \sum_{|\alpha| < N} \frac{(-iz)^\alpha}{\alpha!} \partial_\xi^\alpha a(x,\xi) \cdot e^{-iz \cdot \eta} + r_N(x,z,\xi,\eta),
\]
applied via Taylor expansion. The remainder \(r_N\) contributes a term in \(S^{m_1+m_2-N(\rho-\delta)}\), which can be made arbitrarily negative. \(\blacksquare\)
Remark 3.3.2. The formula \(c \sim \sum_\alpha \frac{1}{\alpha!} \partial_\xi^\alpha a \cdot D_x^\alpha b\) is a generalization of the Leibniz rule. When \(a\) and \(b\) are polynomials in \(\xi\) (i.e., the operators are differential operators), the sum is finite and reduces to the usual composition of differential operators.
3.4 Adjoint
Theorem 3.4.1 (Formal Adjoint). If \(A \in \Psi^m_{\rho,\delta}\) with symbol \(a\), then the formal \(L^2\)-adjoint \(A^*\) defined by \(\langle A^* u, v \rangle = \langle u, Av \rangle\) is a pseudodifferential operator \(A^* \in \Psi^m_{\rho,\delta}\) with symbol
\[
a^*(x,\xi) \sim \sum_\alpha \frac{1}{\alpha!} \partial_\xi^\alpha D_x^\alpha \overline{a(x,\xi)}.
\]
In particular, \(\sigma_m(A^*) = \overline{\sigma_m(A)}\).
Corollary 3.4.2. The principal symbol of a formally self-adjoint pseudodifferential operator is real-valued.
3.5 Elliptic Operators and Parametrices
Ellipticity is the key condition that allows one to “invert” a pseudodifferential operator modulo smoothing errors.
Definition 3.5.1 (Ellipticity). An operator \(A \in \Psi^m_{\rho,\delta}\) with symbol \(a\) is elliptic if there exist constants \(C, R > 0\) such that
\[
|a(x,\xi)| \geq C \langle \xi \rangle^m, \quad |\xi| \geq R.
\]
Equivalently, the principal symbol \(\sigma_m(A)(x,\xi) \neq 0\) for all \(x \in \mathbb{R}^n\) and \(\xi \neq 0\).
Example 3.5.2. The Laplacian \(\Delta\) has principal symbol \(-|\xi|^2\), which is elliptic. The operator \(I - \Delta\), with symbol \(1 + |\xi|^2\), is elliptic of order 2. The heat operator \(\partial_t - \Delta\), with symbol \(i\tau - |\xi|^2\) on \(\mathbb{R}^{n+1}\), is not elliptic (it vanishes when \(\tau = 0\) and \(\xi = 0\) in a degenerate way as a function on \(\mathbb{R}^{n+1}\)).
Theorem 3.5.3 (Parametrix Construction). Let \(A \in \Psi^m_{\rho,\delta}\) be elliptic with \(\delta < \rho\). Then there exists \(B \in \Psi^{-m}_{\rho,\delta}\) such that
\[
AB = I + R_1, \quad BA = I + R_2,
\]
where \(R_1, R_2 \in \Psi^{-\infty}\) are smoothing operators. The operator \(B\) is called a parametrix for \(A\).
Proof. We construct the symbol \(b\) of \(B\) by solving the asymptotic equation \(a \# b \sim 1\), where \(\#\) denotes the composition of symbols. Set \(b_0(x,\xi) = \chi(\xi) / a(x,\xi)\), where \(\chi \in C^\infty\) is a cutoff vanishing near the origin and equal to 1 for \(|\xi| \geq R\). Then \(b_0 \in S^{-m}_{\rho,\delta}\), and \(a \# b_0 = 1 + r_1\) with \(r_1 \in S^{-(\rho-\delta)}_{\rho,\delta}\).
We now set \(b_1 = -b_0 \cdot r_1 \in S^{-m-(\rho-\delta)}\) and iterate: having constructed \(b_0, \ldots, b_{N-1}\) such that \(a \# (b_0 + \cdots + b_{N-1}) = 1 + r_N\) with \(r_N \in S^{-N(\rho-\delta)}\), we set \(b_N = -b_0 \cdot r_N\). By Borel’s lemma (Theorem 2.3.2), there exists \(b \sim \sum_j b_j\) in \(S^{-m}\), and the corresponding operator \(B = \mathrm{Op}(b)\) satisfies \(AB = I + R_1\) with \(R_1 \in \Psi^{-\infty}\). A similar construction gives the left parametrix. \(\blacksquare\)
3.6 Elliptic Regularity
The parametrix immediately yields the fundamental regularity theorem for elliptic equations.
Theorem 3.6.1 (Elliptic Regularity). Let \(A \in \Psi^m\) be elliptic. If \(Au = f\) with \(f \in H^s_{\mathrm{loc}}\), then \(u \in H^{s+m}_{\mathrm{loc}}\). In particular, if \(f \in C^\infty\), then \(u \in C^\infty\).
Proof. Let \(B\) be a parametrix for \(A\), so \(BA = I + R\) with \(R \in \Psi^{-\infty}\). Then \(u = Bf - Ru\). Since \(f \in H^s_{\mathrm{loc}}\) and \(B \in \Psi^{-m}\), we have \(Bf \in H^{s+m}_{\mathrm{loc}}\). Since \(R\) is smoothing, \(Ru \in C^\infty\). Hence \(u = Bf - Ru \in H^{s+m}_{\mathrm{loc}}\). \(\blacksquare\)
Remark 3.6.2. This proof is remarkably clean compared to the classical proof of elliptic regularity via difference quotients or Schauder estimates. The parametrix method reduces regularity to a purely algebraic statement about symbol composition. This is the power of the pseudodifferential calculus.
3.7 The Calderón-Vaillancourt Theorem
Not every \(\Psi\)DO of order 0 is bounded on \(L^2\): this requires some care with the symbol class.
Theorem 3.7.1 (Calderón-Vaillancourt). If \(a \in S^0_{\rho,\rho}(\mathbb{R}^n \times \mathbb{R}^n)\) with \(0 \leq \rho < 1\), then \(\mathrm{Op}(a) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)\) is bounded, with
\[
\|\mathrm{Op}(a)\|_{L^2 \to L^2} \leq C \sum_{|\alpha+\beta| \leq N} \sup_{x,\xi} |\partial_\xi^\alpha \partial_x^\beta a(x,\xi)|,
\]
where \(N\) depends only on \(n\) and \(\rho\).
Proof (Sketch for \(\rho = 0\)). The case \(\rho = 0\) is the most classical. The symbol \(a \in S^0_{0,0}\) satisfies \(|\partial_\xi^\alpha \partial_x^\beta a| \leq C_{\alpha\beta}\) for all \(\alpha, \beta\). One decomposes the operator using a partition of unity in phase space (a Gabor-type decomposition) and estimates each piece. The Cotlar-Stein almost-orthogonality lemma then yields the \(L^2\)-boundedness. Specifically, if \(A = \sum_j A_j\) where the pieces satisfy
\[
\|A_i^* A_j\| \leq c(i-j)^2, \quad \|A_i A_j^*\| \leq c(i-j)^2,
\]
with \(\sum_j c(j) < \infty\), then \(\|A\| \leq \sum_j \sqrt{c(j)}\). \(\blacksquare\)
Remark 3.7.2. The theorem fails for \(\rho = 1\): there exist symbols in \(S^0_{1,1}\) whose quantizations are unbounded on \(L^2\). This is another manifestation of the pathology of the \((1,1)\) class.
3.8 Gårding’s Inequality
Gårding’s inequality is the pseudodifferential version of coercivity, fundamental for energy estimates.
Theorem 3.8.1 (Sharp Gårding Inequality). Let \(A \in \Psi^m_{1,0}(\mathbb{R}^n)\) with symbol \(a\) satisfying \(\operatorname{Re} a(x,\xi) \geq 0\) for all \(x,\xi\). Then there exists \(C > 0\) such that
\[
\operatorname{Re} \langle Au, u \rangle \geq -C\|u\|_{H^{(m-1)/2}}^2
\]
for all \(u \in \mathcal{S}(\mathbb{R}^n)\).
Theorem 3.8.2 (Fefferman-Phong Inequality). Under the stronger hypothesis that \(a(x,\xi) \geq 0\) (real non-negative symbol of order \(m\)), one has the improved lower bound
\[
\operatorname{Re} \langle Au, u \rangle \geq -C\|u\|_{H^{(m-2)/2}}^2.
\]
This gains a full derivative over the sharp Gårding inequality and is optimal.
Remark 3.8.3. The Fefferman-Phong inequality, proved in 1978, is a deep result whose proof uses a delicate decomposition of phase space. It plays an important role in the theory of subelliptic estimates and in semiclassical analysis.
Chapter 4: Wavefront Sets and Microlocal Analysis
The wavefront set is the central concept of microlocal analysis. Introduced by Hörmander in 1970, it refines the classical notion of singular support by tracking not only where a distribution is singular but also in which codirections the singularity occurs. This refinement, living in the cotangent bundle rather than the base manifold, is the passage from local to microlocal analysis — and it transforms the study of PDE.
4.1 Singular Support
We begin with the classical notion, which the wavefront set refines.
Definition 4.1.1 (Singular Support). The singular support of a distribution \(u \in \mathcal{D}'(\Omega)\), denoted \(\operatorname{sing\,supp}(u)\), is the smallest closed set \(F \subseteq \Omega\) such that \(u|_{\Omega \setminus F} \in C^\infty(\Omega \setminus F)\). Equivalently,
\[
x_0 \notin \operatorname{sing\,supp}(u) \iff \exists \, \varphi \in C_c^\infty(\Omega) \text{ with } \varphi(x_0) \neq 0 \text{ and } \varphi u \in C^\infty(\Omega).
\]
Example 4.1.2. For the Dirac delta, \(\operatorname{sing\,supp}(\delta_0) = \{0\}\). For the Heaviside function \(H\), \(\operatorname{sing\,supp}(H) = \{0\}\) as well, since \(H\) is smooth away from the origin.
The singular support tells us where the singularity is, but not how it is oriented. Two distributions can have the same singular support but very different analytic behavior — for instance, \(\delta(x_1)\) (singular on the hyperplane \(\{x_1 = 0\}\)) and \(\delta(x_1) + \delta(x_2)\) have qualitatively different singularity structures that the singular support alone cannot distinguish.
4.2 The Wavefront Set
Definition 4.2.1 (Wavefront Set). Let \(u \in \mathcal{D}'(\Omega)\). A point \((x_0, \xi_0) \in \Omega \times (\mathbb{R}^n \setminus \{0\})\) is not in the wavefront set \(\mathrm{WF}(u)\) if there exist \(\varphi \in C_c^\infty(\Omega)\) with \(\varphi(x_0) \neq 0\) and an open cone \(\Gamma \subset \mathbb{R}^n \setminus \{0\}\) containing \(\xi_0\) such that for every \(N \in \mathbb{N}\),
\[
|\widehat{\varphi u}(\xi)| \leq C_N \langle \xi \rangle^{-N}, \quad \xi \in \Gamma.
\]
The wavefront set \(\mathrm{WF}(u) \subset T^*\Omega \setminus 0\) is the complement of the set of all such "microlocally regular" points.
The wavefront set is a closed conic subset of \(T^*\Omega \setminus 0\) (conic means invariant under positive scaling of the fiber variable \(\xi\)). It refines the singular support via the projection:
Proposition 4.2.2. The projection of \(\mathrm{WF}(u)\) onto the base \(\Omega\) equals \(\operatorname{sing\,supp}(u)\):
\[
\pi(\mathrm{WF}(u)) = \operatorname{sing\,supp}(u),
\]
where \(\pi : T^*\Omega \setminus 0 \to \Omega\) is the base projection.
Proof. If \(x_0 \notin \operatorname{sing\,supp}(u)\), then there exists \(\varphi\) with \(\varphi(x_0) \neq 0\) and \(\varphi u \in C^\infty_c\), so \(\widehat{\varphi u}\) is rapidly decreasing in all directions. Thus \((x_0, \xi_0) \notin \mathrm{WF}(u)\) for every \(\xi_0\), so \(x_0 \notin \pi(\mathrm{WF}(u))\). Conversely, if \(x_0 \notin \pi(\mathrm{WF}(u))\), one can cover \(S^{n-1}\) by finitely many open cones in which \(\widehat{\varphi u}\) is rapidly decreasing (after possibly shrinking the support of \(\varphi\)), yielding rapid decrease in all directions and hence \(\varphi u \in C^\infty\). \(\blacksquare\)
Example 4.2.3 (Wavefront Set of the Dirac Delta). For \(\delta_0 \in \mathcal{D}'(\mathbb{R}^n)\), we have \(\hat{\delta}_0 = 1\), which does not decay in any direction. Thus
\[
\mathrm{WF}(\delta_0) = \{(0,\xi) : \xi \in \mathbb{R}^n \setminus \{0\}\} = \{0\} \times (\mathbb{R}^n \setminus \{0\}).
\]
The singularity at the origin is "omnidirectional."
Example 4.2.4 (Wavefront Set of a Characteristic Function). Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set with smooth boundary. Then
\[
\mathrm{WF}(\mathbf{1}_\Omega) = \{(x,\xi) : x \in \partial\Omega, \; \xi \perp T_x(\partial\Omega), \; \xi \neq 0\} = N^*(\partial\Omega) \setminus 0,
\]
the conormal bundle of the boundary (minus the zero section). The singularity is concentrated on the boundary and points in the normal direction.
Example 4.2.5 (Wavefront Set of \(\delta(x_1)\)). Consider \(\delta(x_1) \in \mathcal{D}'(\mathbb{R}^n)\), the distribution defined by \(\langle \delta(x_1), \varphi \rangle = \int \varphi(0, x') \, dx'\). Its Fourier transform in \(x_1\) is constant, so the singularity is in the \(\xi_1\)-direction:
\[
\mathrm{WF}(\delta(x_1)) = \{(0,x'; \xi_1, 0) : x' \in \mathbb{R}^{n-1}, \; \xi_1 \neq 0\}.
\]
This is precisely the conormal bundle of the hyperplane \(\{x_1 = 0\}\).
4.3 Microlocal Regularity and Pseudolocality
Theorem 4.3.1 (Microlocal Regularity of \(\Psi\)DOs). If \(A \in \Psi^m_{\rho,\delta}\) with \(\delta < \rho\), then
\[
\mathrm{WF}(Au) \subseteq \mathrm{WF}(u)
\]
for every \(u \in \mathcal{S}'(\mathbb{R}^n)\) (or \(u \in \mathcal{E}'(\mathbb{R}^n)\)). That is, a pseudodifferential operator does not create new singularities.
Proof (Sketch). If \((x_0, \xi_0) \notin \mathrm{WF}(u)\), we must show \((x_0, \xi_0) \notin \mathrm{WF}(Au)\). Choose \(\varphi\) localizing near \(x_0\) and a symbol \(b \in S^0\) that is 1 near \((x_0, \xi_0)\) (in a conic sense) and supported where \(u\) is microlocally regular. The key is to decompose \(A = A_1 + A_2\) where \(A_1\) has symbol supported in the region where \(u\) is microlocally smooth, so \(A_1 u\) is smooth near \((x_0,\xi_0)\), and \(A_2\) has symbol vanishing near \((x_0,\xi_0)\). A careful estimation using the symbol calculus completes the argument. \(\blacksquare\)
Corollary 4.3.2 (Pseudolocality). For \(A \in \Psi^m_{\rho,\delta}\) with \(\delta < \rho\),
\[
\operatorname{sing\,supp}(Au) \subseteq \operatorname{sing\,supp}(u).
\]
4.4 Microlocal Elliptic Regularity
The wavefront set allows a refinement of elliptic regularity to the microlocal level.
Definition 4.4.1 (Characteristic Set). For \(A \in \Psi^m\) with principal symbol \(\sigma_m(A)\), the characteristic set is
\[
\operatorname{Char}(A) = \{(x,\xi) \in T^*\mathbb{R}^n \setminus 0 : \sigma_m(A)(x,\xi) = 0\}.
\]
The operator \(A\) is elliptic precisely when \(\operatorname{Char}(A) = \emptyset\).
Theorem 4.4.2 (Microlocal Elliptic Regularity). Let \(A \in \Psi^m\) and \(Au = f\). Then
\[
\mathrm{WF}(u) \subseteq \mathrm{WF}(f) \cup \operatorname{Char}(A).
\]
In words: the wavefront set of a solution \(u\) can only contain points that are either in the wavefront set of the right-hand side or in the characteristic set of the operator.
Proof. Suppose \((x_0,\xi_0) \notin \mathrm{WF}(f) \cup \operatorname{Char}(A)\). Since \((x_0,\xi_0) \notin \operatorname{Char}(A)\), the principal symbol \(\sigma_m(A)\) is nonzero at \((x_0,\xi_0)\). We can construct a microlocal parametrix \(B \in \Psi^{-m}\) that inverts \(A\) microlocally near \((x_0,\xi_0)\): \(BA = I + R\) where \(R\) is microsupported away from \((x_0,\xi_0)\). Then
\[
u = Bf - Ru + Ru - Ru = Bf - Ru
\]
microlocally near \((x_0,\xi_0)\). Since \((x_0,\xi_0) \notin \mathrm{WF}(f)\) and \(B\) preserves wavefront sets, \((x_0,\xi_0) \notin \mathrm{WF}(Bf)\). And \(R\) is smoothing near \((x_0,\xi_0)\), so \((x_0,\xi_0) \notin \mathrm{WF}(Ru)\). Hence \((x_0,\xi_0) \notin \mathrm{WF}(u)\). \(\blacksquare\)
Remark 4.4.3. When the operator is elliptic (\(\operatorname{Char}(A) = \emptyset\)), this reduces to \(\mathrm{WF}(u) \subseteq \mathrm{WF}(f)\), which refines the global statement \(\operatorname{sing\,supp}(u) \subseteq \operatorname{sing\,supp}(f)\). For non-elliptic operators (like the wave operator), the characteristic set is nonempty, and singularities can propagate along it — the subject of Section 4.6.
4.5 The Wavefront Set of a Product
One of the most important applications of the wavefront set is to the problem of multiplying distributions. As noted in Chapter 1, not all distributions can be multiplied; the wavefront set gives a precise criterion.
Theorem 4.5.1 (Hörmander's Criterion for Multiplication). Let \(u, v \in \mathcal{D}'(\Omega)\). If the following condition holds:
\[
(x, \xi) \in \mathrm{WF}(u) \text{ and } (x, \eta) \in \mathrm{WF}(v) \implies \xi + \eta \neq 0,
\]
then the product \(uv \in \mathcal{D}'(\Omega)\) is well-defined, and
\[
\mathrm{WF}(uv) \subseteq \{(x, \xi+\eta) : (x,\xi) \in \mathrm{WF}(u) \cup \{0\}, \; (x,\eta) \in \mathrm{WF}(v) \cup \{0\}, \; \xi+\eta \neq 0\}.
\]
Here \(\{0\}\) means we allow \(\xi = 0\) or \(\eta = 0\) (but not \(\xi + \eta = 0\)) in the union.
Remark 4.5.2. Hörmander's condition is a "no-cancellation" condition: the singular codirections of \(u\) and \(v\) at any common singular point must not be antipodal. This is violated for \(\delta_0 \cdot \delta_0\) (both wavefront sets are all of \(\{0\} \times (\mathbb{R}^n \setminus \{0\})\)), which indeed cannot be defined. It is satisfied for the product of \(\delta(x_1)\) with a distribution smooth in the \(x_1\) variable — the codirections are complementary.
Example 4.5.3. Let \(u = \delta(x_1) \in \mathcal{D}'(\mathbb{R}^2)\) and \(v = \delta(x_2) \in \mathcal{D}'(\mathbb{R}^2)\). Then \(\mathrm{WF}(u) \subset \{(\xi_1, 0)\}\)-directions and \(\mathrm{WF}(v) \subset \{(0, \xi_2)\}\)-directions. Hörmander's condition is satisfied (these are never antipodal), and the product \(uv = \delta(x_1)\delta(x_2) = \delta_0\) is well-defined.
4.6 Propagation of Singularities
For operators that are not elliptic, singularities can propagate. The fundamental theorem describes this propagation.
Definition 4.6.1 (Bicharacteristic Flow). Let \(P \in \Psi^m\) have real principal symbol \(p = \sigma_m(P)\). The Hamilton vector field of \(p\) is
\[
H_p = \sum_{j=1}^n \left(\frac{\partial p}{\partial \xi_j} \frac{\partial}{\partial x_j} - \frac{\partial p}{\partial x_j} \frac{\partial}{\partial \xi_j}\right).
\]
A bicharacteristic is an integral curve of \(H_p\) lying in \(\operatorname{Char}(P) = \{p = 0\}\). Null bicharacteristics are bicharacteristic curves on which \(p = 0\).
Theorem 4.6.2 (Hörmander's Propagation of Singularities). Let \(P \in \Psi^m\) have real principal symbol \(p\). If \(Pu = f\), then \(\mathrm{WF}(u) \setminus \mathrm{WF}(f)\) is a union of maximally extended null bicharacteristics of \(P\) contained in \(\operatorname{Char}(P)\).
Remark 4.6.3. In plainer language: wherever the wavefront set of \(u\) is not "explained" by the right-hand side \(f\), the singularities must propagate along the Hamiltonian flow of the principal symbol. For the wave operator \(P = D_t^2 - \Delta_x\) with principal symbol \(p = \tau^2 - |\xi|^2\), the null bicharacteristics are the lifts to \(T^*\mathbb{R}^{n+1}\) of light rays (straight lines traversed at speed 1). This theorem thus contains, as a special case, the classical fact that singularities of waves propagate along light rays — Huygens' principle in its microlocal incarnation.
Example 4.6.4 (Wave Equation). For the wave operator \(P = \partial_t^2 - \Delta_x\) on \(\mathbb{R}^{1+n}\), the principal symbol is \(p(t,x;\tau,\xi) = -\tau^2 + |\xi|^2\). The Hamilton equations are
\[
\dot{t} = -2\tau, \quad \dot{x} = 2\xi, \quad \dot{\tau} = 0, \quad \dot{\xi} = 0.
\]
On \(\operatorname{Char}(P) = \{\tau^2 = |\xi|^2\}\), with \(\tau = \pm|\xi|\), these are straight lines in space-time with velocity \(\dot{x}/\dot{t} = -\xi/\tau = \mp \xi/|\xi|\), i.e., unit speed propagation. The propagation of singularities theorem says that the wavefront set of a solution to the wave equation propagates along these rays.
Chapter 5: Fourier Integral Operators
Fourier integral operators (FIOs) generalize pseudodifferential operators by allowing the phase function to be more general than the linear phase \((x-y) \cdot \xi\). They were introduced by Hörmander in 1971 to provide a coordinate-invariant framework for the propagation of singularities and to construct parametrices for hyperbolic equations. FIOs encode canonical transformations (symplectomorphisms) of the cotangent bundle, linking the symplectic geometry of classical mechanics to the analysis of PDEs. The theory draws heavily on the work of Maslov, Hörmander, and Duistermaat-Guillemin.
5.1 Symplectic Geometry and Lagrangian Submanifolds
The natural geometric setting for FIOs is symplectic geometry.
Definition 5.1.1 (Symplectic Structure). The cotangent bundle \(T^*\mathbb{R}^n\) carries the canonical symplectic form \(\omega = \sum_{j=1}^n d\xi_j \wedge dx_j\). A submanifold \(\Lambda \subset T^*\mathbb{R}^n\) of dimension \(n\) is Lagrangian if \(\omega|_\Lambda = 0\).
Example 5.1.2. The zero section \(\{(x,0) : x \in \mathbb{R}^n\}\) is Lagrangian. The conormal bundle \(N^*S = \{(x,\xi) : x \in S, \; \xi \perp T_xS\}\) of any smooth submanifold \(S \subset \mathbb{R}^n\) is Lagrangian. The graph of any exact 1-form \(\xi = d\phi(x)\) is Lagrangian.
Definition 5.1.3 (Canonical Relation). A canonical relation from \(T^*\mathbb{R}^{n_2}\) to \(T^*\mathbb{R}^{n_1}\) is a Lagrangian submanifold \(C \subset T^*\mathbb{R}^{n_1} \times T^*\mathbb{R}^{n_2}\) with respect to the twisted symplectic form \(\omega_1 \oplus (-\omega_2)\). In particular, the graph of any canonical transformation (symplectomorphism) \(\chi : T^*\mathbb{R}^n \to T^*\mathbb{R}^n\) is a canonical relation.
5.2 Generating Functions and Phase Functions
Definition 5.2.1 (Non-degenerate Phase Function). A phase function is a smooth real-valued function \(\Phi(x,y,\theta)\) defined on \(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \times (\mathbb{R}^N \setminus \{0\})\) that is positively homogeneous of degree 1 in \(\theta\). It is non-degenerate if the differentials \(d(\partial\Phi/\partial\theta_1), \ldots, d(\partial\Phi/\partial\theta_N)\) are linearly independent on the critical set
\[
C_\Phi = \{(x,y,\theta) : \nabla_\theta \Phi = 0\}.
\]
Proposition 5.2.2. If \(\Phi\) is a non-degenerate phase function, then \(C_\Phi\) is a smooth manifold of dimension \(n_1 + n_2\), and the map
\[
C_\Phi \ni (x,y,\theta) \mapsto (x, \nabla_x\Phi; y, -\nabla_y\Phi) \in T^*\mathbb{R}^{n_1} \times T^*\mathbb{R}^{n_2}
\]
parametrizes a Lagrangian submanifold (with respect to the twisted symplectic form), which is the canonical relation associated to \(\Phi\).
Example 5.2.3. The phase function \(\Phi(x,y,\xi) = (x-y) \cdot \xi\) is non-degenerate, with critical set \(C_\Phi = \{x = y\}\) and associated canonical relation equal to the diagonal \(\{(x,\xi;x,\xi)\}\) — corresponding to the identity transformation. This is the phase of pseudodifferential operators.
5.3 Definition of Fourier Integral Operators
Definition 5.3.1 (Fourier Integral Operator). A Fourier integral operator of order \(m\) associated to a non-degenerate phase function \(\Phi(x,y,\theta)\) is an operator of the form
\[
Au(x) = \int\!\!\!\int e^{i\Phi(x,y,\theta)} a(x,y,\theta) u(y) \, dy \, d\theta,
\]
where \(a \in S^{m + n_1/2 - N/2}_{1,0}(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \times \mathbb{R}^N)\) is an amplitude, interpreted as an oscillatory integral. We write \(A \in I^m(C)\), where \(C\) is the canonical relation generated by \(\Phi\).
Remark 5.3.2. The order convention is set so that pseudodifferential operators of order \(m\) are FIOs of order \(m\) associated to the identity canonical relation (the diagonal in \(T^*\mathbb{R}^n \times T^*\mathbb{R}^n\)).
Theorem 5.3.3 (Wavefront Set of FIOs). If \(A \in I^m(C)\) and \(u \in \mathcal{E}'(\mathbb{R}^{n_2})\), then
\[
\mathrm{WF}(Au) \subseteq C \circ \mathrm{WF}(u) := \{(x,\xi) : \exists (y,\eta) \in \mathrm{WF}(u), \; (x,\xi;y,\eta) \in C\}.
\]
That is, the FIO transforms the wavefront set according to its canonical relation.
5.4 Composition with Pseudodifferential Operators
Theorem 5.4.1. If \(A \in I^m(C)\) is an FIO and \(P \in \Psi^k\) is a pseudodifferential operator, then \(PA \in I^{m+k}(C)\) — the composition is again an FIO associated to the same canonical relation, with order increased by \(k\). Similarly, \(AP' \in I^{m+k'}(C)\) for \(P' \in \Psi^{k'}\).
This means that the class of FIOs associated to a given canonical relation forms a module over the ring of pseudodifferential operators — a crucial structural property.
5.5 Egorov’s Theorem
Egorov’s theorem is the microlocal version of the statement that conjugation by a unitary operator associated to a canonical transformation implements that transformation at the symbol level.
Theorem 5.5.1 (Egorov's Theorem). Let \(\chi : T^*\mathbb{R}^n \to T^*\mathbb{R}^n\) be a canonical transformation, and let \(U \in I^0(\operatorname{Graph}(\chi))\) be an elliptic FIO of order 0 with parametrix \(U^{-1} \in I^0(\operatorname{Graph}(\chi^{-1}))\). Then for any \(A \in \Psi^m\), the conjugated operator
\[
B = U^{-1} A U
\]
is a pseudodifferential operator \(B \in \Psi^m\), and its principal symbol satisfies
\[
\sigma_m(B) = \sigma_m(A) \circ \chi.
\]
Proof (Sketch). By Theorem 5.4.1, \(AU \in I^m(\operatorname{Graph}(\chi))\), and then \(U^{-1}(AU) \in I^m(\operatorname{Graph}(\chi^{-1}) \circ \operatorname{Graph}(\chi)) = I^m(\Delta)\), where \(\Delta\) is the diagonal canonical relation. But FIOs associated to the diagonal are precisely pseudodifferential operators, so \(B \in \Psi^m\). The formula for the principal symbol follows from tracking the symbol through the composition. \(\blacksquare\)
Remark 5.5.2. Egorov's theorem is the rigorous version of the "quantum-classical correspondence" for a single observable: conjugation by the quantum propagator \(U\) corresponds to composition with the classical flow \(\chi\) at the level of principal symbols. This is a cornerstone of semiclassical analysis.
5.6 Application: Parametrix for the Wave Equation
We illustrate the power of FIO theory by constructing a parametrix for the wave equation.
Theorem 5.6.1 (Parametrix for the Wave Operator). Consider the Cauchy problem
\[
\begin{cases} (\partial_t^2 - \Delta) u = 0, \\ u(0,x) = f(x), \\ \partial_t u(0,x) = g(x). \end{cases}
\]
There exist Fourier integral operators \(E_0(t), E_1(t) \in I^0(\mathbb{R}^n)\) such that the solution is
\[
u(t,\cdot) = E_0(t)f + E_1(t)g + R(t)(f,g),
\]
where \(R(t)\) is a smoothing operator. The canonical relation of \(E_j(t)\) is the graph of the geodesic flow at time \(t\) (for the flat metric, this is the linear flow \((x,\xi) \mapsto (x + 2t\xi/|\xi|, \xi)\) on the characteristic set \(\{|\xi| = 1\}\), extended homogeneously).
Proof (Sketch). One constructs \(E_0(t)\) as an FIO with phase function
\[
\Phi_\pm(t,x,y,\xi) = (x - y) \cdot \xi \pm t|\xi|
\]
and amplitude \(a(t,x,y,\xi)\) determined by solving the transport equations obtained from applying the wave operator to the ansatz
\[
E_0(t)f(x) = \frac{1}{(2\pi)^n} \int\!\!\!\int \left(e^{i((x-y)\cdot\xi + t|\xi|)} a^+(t,x,\xi) + e^{i((x-y)\cdot\xi - t|\xi|)} a^-(t,x,\xi)\right) f(y) \, dy \, d\xi.
\]
The phase functions \(\Phi_\pm\) generate the forward and backward light cones. The transport equations, obtained by collecting terms of decreasing homogeneity in \(\xi\), can be solved iteratively, yielding an amplitude with an asymptotic expansion. The initial conditions \(u(0) = f\), \(\partial_t u(0) = g\) determine the split between the \(+\) and \(-\) contributions. \(\blacksquare\)
Chapter 6: Applications to PDE
With the machinery of pseudodifferential operators, wavefront sets, and Fourier integral operators in hand, we now demonstrate the power of microlocal analysis through applications to partial differential equations. These applications range from the foundational (elliptic regularity) to the sophisticated (propagation of singularities for the wave equation, analytic wavefront set). The microlocal perspective not only simplifies many classical arguments but also yields results that are inaccessible by other methods.
6.1 Elliptic Regularity Revisited
We have already proved elliptic regularity (Theorem 3.6.1) via parametrices. The microlocal viewpoint gives a sharper result.
Theorem 6.1.1 (Microlocal Elliptic Regularity, Refined). Let \(P \in \Psi^m(\mathbb{R}^n)\) be a classical pseudodifferential operator with principal symbol \(p_m\). If \(Pu = f\) and \((x_0, \xi_0) \notin \operatorname{Char}(P) \cup \mathrm{WF}(f)\), then \((x_0,\xi_0) \notin \mathrm{WF}(u)\). Furthermore, microlocally near \((x_0,\xi_0)\),
\[
u \equiv B f \pmod{C^\infty},
\]
where \(B\) is a microlocal parametrix with principal symbol \(1/p_m\).
This refined statement is immediately useful for equations that are elliptic only in some directions.
6.2 Hypoellipticity
An operator is hypoelliptic if it gains regularity, though perhaps not as much as an elliptic operator.
Definition 6.2.1. A linear differential operator \(P\) is hypoelliptic if \(\operatorname{sing\,supp}(Pu) = \operatorname{sing\,supp}(u)\) for all \(u \in \mathcal{D}'(\Omega)\). That is, \(Pu \in C^\infty(\Omega) \implies u \in C^\infty(\Omega)\).
Example 6.2.2. The heat operator \(P = \partial_t - \Delta_x\) on \(\mathbb{R}^{1+n}\) is hypoelliptic but not elliptic. Its characteristic set \(\operatorname{Char}(P) = \{\tau = 0, \xi = 0\}\) is nonempty (as a set in the projectivized cotangent bundle, it is the "direction" \(d\tau\)), yet solutions to \(Pu = 0\) are smooth. Hörmander proved that a constant-coefficient operator \(P(D)\) is hypoelliptic if and only if for every \(\alpha\), \(|P^{(\alpha)}(\xi)/P(\xi)| \to 0\) as \(|\xi| \to \infty\) (with \(\xi\) staying away from the zeros of \(P\)).
Theorem 6.2.3 (Hörmander's Sum of Squares). Let \(X_0, X_1, \ldots, X_r\) be smooth real vector fields on an open set \(\Omega \subseteq \mathbb{R}^n\), and consider the operator
\[
P = \sum_{j=1}^r X_j^2 + X_0 + c
\]
where \(c \in C^\infty(\Omega)\). If the Lie algebra generated by \(X_0, X_1, \ldots, X_r\) spans the tangent space at every point (the Hörmander bracket condition), then \(P\) is hypoelliptic.
Remark 6.2.4. This landmark theorem, proved by Hörmander in 1967 and partly motivating his later development of microlocal analysis, covers important operators in sub-Riemannian geometry and mathematical finance (e.g., the Kolmogorov equation). The proof involves subelliptic estimates that are most naturally expressed in the pseudodifferential framework.
6.3 The Heat Kernel via Pseudodifferential Methods
The pseudodifferential calculus provides an elegant construction of the heat kernel for elliptic operators.
Theorem 6.3.1. Let \(P \in \Psi^2(\mathbb{R}^n)\) be a classical, self-adjoint, elliptic operator that is positive (i.e., \(\langle Pu, u \rangle \geq c\|u\|_{H^1}^2 - C\|u\|_{L^2}^2\)). Then the heat semigroup \(e^{-tP}\) exists for \(t > 0\) as a smoothing operator, and its Schwartz kernel \(K_t(x,y)\) satisfies:
(i) \(K_t \in C^\infty(\mathbb{R}^+ \times \mathbb{R}^n \times \mathbb{R}^n)\),
(ii) \(K_t(x,y) \to \delta(x-y)\) as \(t \to 0^+\) in the sense of distributions,
(iii) For small \(t > 0\), \(K_t\) admits an asymptotic expansion on the diagonal: \[ K_t(x,x) \sim (4\pi t)^{-n/2} \sum_{j=0}^\infty a_j(x) t^j \quad \text{as } t \to 0^+, \] where the coefficients \(a_j(x)\) are determined by the symbol of \(P\).
(i) \(K_t \in C^\infty(\mathbb{R}^+ \times \mathbb{R}^n \times \mathbb{R}^n)\),
(ii) \(K_t(x,y) \to \delta(x-y)\) as \(t \to 0^+\) in the sense of distributions,
(iii) For small \(t > 0\), \(K_t\) admits an asymptotic expansion on the diagonal: \[ K_t(x,x) \sim (4\pi t)^{-n/2} \sum_{j=0}^\infty a_j(x) t^j \quad \text{as } t \to 0^+, \] where the coefficients \(a_j(x)\) are determined by the symbol of \(P\).
Remark 6.3.2. The diagonal expansion of the heat kernel is intimately connected to spectral geometry through the trace formula
\[
\operatorname{Tr}(e^{-tP}) = \int K_t(x,x) \, dx \sim (4\pi t)^{-n/2} \sum_{j=0}^\infty \left(\int a_j(x) \, dx\right) t^j.
\]
On a compact manifold, the left-hand side equals \(\sum_k e^{-t\lambda_k}\), where \(\lambda_k\) are the eigenvalues of \(P\). This connection between the heat trace and spectral data is the starting point for the Weyl law (Chapter 7) and the Atiyah-Singer index theorem.
6.4 Propagation of Singularities for the Wave Equation
We now give a detailed treatment of the propagation theorem for the wave equation, using the machinery developed in Chapters 4 and 5.
Theorem 6.4.1 (Propagation of Singularities for the Wave Equation). Let \(P = D_t^2 - \sum_{j,k} \partial_{x_j}(g^{jk}(x)\partial_{x_k})\) be a wave operator with smooth metric \(g\), and suppose \(Pu = f\) on \(\mathbb{R} \times \mathbb{R}^n\). Then:
(i) \(\mathrm{WF}(u) \setminus \mathrm{WF}(f) \subset \operatorname{Char}(P) = \{\tau^2 = |\xi|_g^2\}\),
(ii) \(\mathrm{WF}(u) \setminus \mathrm{WF}(f)\) is invariant under the null bicharacteristic flow, i.e., it is a union of maximally extended null bicharacteristics.
(i) \(\mathrm{WF}(u) \setminus \mathrm{WF}(f) \subset \operatorname{Char}(P) = \{\tau^2 = |\xi|_g^2\}\),
(ii) \(\mathrm{WF}(u) \setminus \mathrm{WF}(f)\) is invariant under the null bicharacteristic flow, i.e., it is a union of maximally extended null bicharacteristics.
Proof (Sketch). Part (i) is microlocal elliptic regularity (Theorem 4.4.2). For part (ii), the idea is to construct a microlocal energy estimate near the bicharacteristic. Let \(\gamma : [0,T] \to T^*(\mathbb{R}^{1+n}) \setminus 0\) be a null bicharacteristic of \(P\), and suppose \(\gamma(0) \notin \mathrm{WF}(u)\). We want to show \(\gamma(t) \notin \mathrm{WF}(u)\) for \(t \in [0,T]\).
\[
E(t) = \|B(t) D_t u\|_{L^2}^2 + \|B(t) \nabla_x u\|_{L^2}^2.
\]
One computes \(\frac{d}{dt}E(t)\) using the equation \(Pu = f\), the composition formula for \(\Psi\)DOs, and the positive commutator method (following Hörmander). The crucial point is that \([P, B(t)^*B(t)]\) has principal symbol \(H_p(|b(t)|^2)\), and by choosing \(B(t)\) to propagate along the flow, this commutator can be made non-positive (modulo lower-order terms). A Gronwall argument then yields \(E(t) = 0\) for all \(t\), hence \(\gamma(t) \notin \mathrm{WF}(u)\). \(\blacksquare\)
6.5 Duhamel’s Principle
Theorem 6.5.1 (Duhamel's Principle). Consider the inhomogeneous wave equation
\[
\begin{cases} (\partial_t^2 - \Delta) u = F(t,x), \quad t > 0, \\ u(0,x) = f(x), \quad \partial_t u(0,x) = g(x). \end{cases}
\]
If \(E_1(t)\) is the propagator for the homogeneous equation with data \(u(0) = 0\), \(\partial_t u(0) = g\) (so that \(u(t) = E_1(t)g\)), then the solution to the inhomogeneous problem is
\[
u(t) = E_0(t)f + E_1(t)g + \int_0^t E_1(t-s) F(s) \, ds.
\]
Remark 6.5.2. Duhamel's principle, combined with the FIO structure of \(E_0(t)\) and \(E_1(t)\), allows one to analyze the wavefront set of solutions to inhomogeneous wave equations. In particular, \(\mathrm{WF}(u)\) is determined by the wavefront sets of the initial data and forcing, transported along null bicharacteristics.
6.6 The FBI Transform
The FBI (Fourier-Bros-Iagolnitzer) transform is a tool for studying analytic regularity, bridging the gap between \(C^\infty\) microlocal analysis and analytic microlocal analysis.
Definition 6.6.1 (FBI Transform). The FBI transform of \(u \in \mathcal{E}'(\mathbb{R}^n)\) is
\[
Tu(x,\xi) = c_n \int_{\mathbb{R}^n} e^{i(x-y) \cdot \xi - |\xi|(x-y)^2/2} u(y) \, dy,
\]
where \(c_n\) is a normalization constant. This is a smooth function of \((x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n\).
Theorem 6.6.2. A point \((x_0,\xi_0) \notin \mathrm{WF}(u)\) if and only if \(|Tu(x,\xi)|\) decays rapidly as \(|\xi| \to \infty\) in a neighborhood of \((x_0, \xi_0/|\xi_0|)\) with \(|\xi| \to \infty\) in the direction \(\xi_0\).
6.7 Analytic Wavefront Set
Definition 6.7.1 (Analytic Wavefront Set). The analytic wavefront set \(\mathrm{WF}_A(u)\) is defined by: \((x_0, \xi_0) \notin \mathrm{WF}_A(u)\) if there exist \(\varepsilon > 0\) and a neighborhood \(U\) of \((x_0, \xi_0/|\xi_0|)\) such that
\[
|Tu(x,\xi)| \leq C e^{-\varepsilon |\xi|}
\]
for \((x, \xi/|\xi|) \in U\) and \(|\xi|\) sufficiently large.
Remark 6.7.2. The analytic wavefront set refines the \(C^\infty\) wavefront set: \(\mathrm{WF}_A(u) \subseteq \mathrm{WF}(u)\), and while \(\mathrm{WF}(u) = \emptyset\) means \(u \in C^\infty\), \(\mathrm{WF}_A(u) = \emptyset\) means \(u\) is real-analytic. The analytic wavefront set is relevant for the study of analytic hypoellipticity and for the analysis of operators with analytic coefficients. Its theory, developed by Bros-Iagolnitzer, Sjöstrand, and Kashiwara-Kawai, is considerably deeper than the \(C^\infty\) theory.
Chapter 7: Semiclassical Analysis
Semiclassical analysis studies the behavior of quantum-mechanical systems in the limit where Planck’s constant \(\hbar \to 0\) — or, mathematically, operators depending on a small parameter \(h\) that plays the role of \(\hbar\). This chapter introduces the semiclassical pseudodifferential calculus, where the symbol depends on \(h\) and the relevant notion of order tracks powers of \(h\) rather than growth in \(\xi\). Semiclassical analysis provides quantitative refinements of the qualitative results of microlocal analysis and has deep applications to spectral theory, scattering theory, and quantum ergodicity. The foundational contributions of Weyl, Egorov, Maslov, and more recently Zworski and Sjöstrand shape this field.
7.1 Semiclassical Pseudodifferential Operators
Definition 7.1.1 (Semiclassical Symbol Classes). For \(m \in \mathbb{R}\) and \(0 < h \leq h_0\), the semiclassical symbol class \(S^m_h(\mathbb{R}^{2n})\) consists of families \(\{a_h(x,\xi)\}_{h \in (0,h_0]}\) in \(C^\infty(\mathbb{R}^{2n})\) satisfying
\[
|\partial_x^\alpha \partial_\xi^\beta a_h(x,\xi)| \leq C_{\alpha,\beta} \langle \xi \rangle^{m - |\beta|}
\]
uniformly in \(h\). The class \(S^m_{\delta,h}\) allows \(h\)-dependent bounds:
\[
|\partial_x^\alpha \partial_\xi^\beta a_h(x,\xi)| \leq C_{\alpha,\beta} h^{-\delta(|\alpha|+|\beta|)} \langle \xi \rangle^{m - |\beta|}.
\]
Definition 7.1.2 (Semiclassical Quantization). The standard (Kohn-Nirenberg) semiclassical quantization of \(a \in S^m_h\) is
\[
\mathrm{Op}_h(a) u(x) = \frac{1}{(2\pi h)^n} \int\!\!\!\int e^{i(x-y)\cdot\xi/h} a(x,\xi) u(y) \, dy \, d\xi.
\]
This is related to the non-semiclassical quantization by the rescaling \(\xi \mapsto h\xi\): \(\mathrm{Op}_h(a) = \mathrm{Op}(a(x, h\cdot))\) after appropriate identification.
The factor of \(1/h\) in the phase reflects the fundamental semiclassical scaling: oscillations at frequency \(\sim 1/h\) in position space correspond to momenta of order 1 in the classical limit.
7.2 Weyl Quantization
The Weyl quantization, which evaluates the symbol at the midpoint, has special properties that make it the preferred quantization in semiclassical analysis.
Definition 7.2.1 (Weyl Quantization). The Weyl quantization of \(a \in S^m_h\) is
\[
\mathrm{Op}_h^w(a) u(x) = \frac{1}{(2\pi h)^n} \int\!\!\!\int e^{i(x-y)\cdot\xi/h} a\!\left(\frac{x+y}{2}, \xi\right) u(y) \, dy \, d\xi.
\]
Proposition 7.2.2 (Properties of Weyl Quantization).
(i) If \(a\) is real-valued, then \(\mathrm{Op}_h^w(a)\) is formally self-adjoint on \(L^2(\mathbb{R}^n)\).
(ii) Weyl and standard quantizations differ by lower-order terms: \(\mathrm{Op}_h^w(a) = \mathrm{Op}_h(a) + \mathcal{O}(h)\) as operators on \(L^2\).
(iii) The Weyl symbol of the composition \(\mathrm{Op}_h^w(a) \mathrm{Op}_h^w(b)\) is given by the Moyal product: \[ (a \#_h b)(x,\xi) = e^{\frac{ih}{2}(\partial_{\xi_a} \cdot \partial_{x_b} - \partial_{x_a} \cdot \partial_{\xi_b})} a(x_a,\xi_a) b(x_b, \xi_b) \Big|_{\substack{x_a = x_b = x \\ \xi_a = \xi_b = \xi}}. \] In particular, \(a \#_h b = ab + \frac{h}{2i}\{a,b\} + \mathcal{O}(h^2)\), where \(\{a,b\} = \sum_j (\partial_{\xi_j}a \, \partial_{x_j}b - \partial_{x_j}a \, \partial_{\xi_j}b)\) is the Poisson bracket.
(i) If \(a\) is real-valued, then \(\mathrm{Op}_h^w(a)\) is formally self-adjoint on \(L^2(\mathbb{R}^n)\).
(ii) Weyl and standard quantizations differ by lower-order terms: \(\mathrm{Op}_h^w(a) = \mathrm{Op}_h(a) + \mathcal{O}(h)\) as operators on \(L^2\).
(iii) The Weyl symbol of the composition \(\mathrm{Op}_h^w(a) \mathrm{Op}_h^w(b)\) is given by the Moyal product: \[ (a \#_h b)(x,\xi) = e^{\frac{ih}{2}(\partial_{\xi_a} \cdot \partial_{x_b} - \partial_{x_a} \cdot \partial_{\xi_b})} a(x_a,\xi_a) b(x_b, \xi_b) \Big|_{\substack{x_a = x_b = x \\ \xi_a = \xi_b = \xi}}. \] In particular, \(a \#_h b = ab + \frac{h}{2i}\{a,b\} + \mathcal{O}(h^2)\), where \(\{a,b\} = \sum_j (\partial_{\xi_j}a \, \partial_{x_j}b - \partial_{x_j}a \, \partial_{\xi_j}b)\) is the Poisson bracket.
Remark 7.2.3. Property (i) is the main advantage of Weyl quantization: real symbols give self-adjoint operators, which is physically natural since observables in quantum mechanics are represented by self-adjoint operators. Property (iii) shows that the commutator \([\mathrm{Op}_h^w(a), \mathrm{Op}_h^w(b)] = \frac{h}{i}\mathrm{Op}_h^w(\{a,b\}) + \mathcal{O}(h^2)\), which is the mathematical content of Dirac's correspondence principle.
Example 7.2.4. The semiclassical Schrödinger operator \(P_h = -h^2 \Delta + V(x)\) is the Weyl quantization of the classical Hamiltonian \(p(x,\xi) = |\xi|^2 + V(x)\). The semiclassical principal symbol is thus the total energy of the classical system.
7.3 Semiclassical Wavefront Set
Definition 7.3.1 (Semiclassical Wavefront Set). Let \(\{u_h\}_{h \in (0,h_0]}\) be a bounded family in \(L^2(\mathbb{R}^n)\). The semiclassical wavefront set \(\mathrm{WF}_h(u_h) \subset T^*\mathbb{R}^n\) is the complement of the set of points \((x_0,\xi_0)\) such that there exists \(a \in C_c^\infty(T^*\mathbb{R}^n)\) with \(a(x_0,\xi_0) \neq 0\) and \(\|\mathrm{Op}_h(a) u_h\|_{L^2} = \mathcal{O}(h^\infty)\).
Remark 7.3.2. Unlike the classical wavefront set (which lives in the cosphere bundle, being conic in \(\xi\)), the semiclassical wavefront set lives in the full cotangent bundle \(T^*\mathbb{R}^n\) and is not conic. It detects concentration of the family \(u_h\) at a specific point in phase space, not merely a direction. This reflects the uncertainty principle: at scale \(h\), one can localize in both position and momentum simultaneously (up to an \(h\)-neighborhood).
7.4 Egorov’s Theorem in the Semiclassical Regime
Theorem 7.4.1 (Semiclassical Egorov's Theorem). Let \(P_h = \mathrm{Op}_h^w(p)\) with real principal symbol \(p \in S^0_h\), and let \(U_h(t) = e^{-itP_h/h}\) be the semiclassical propagator. Let \(\Phi^t : T^*\mathbb{R}^n \to T^*\mathbb{R}^n\) be the Hamiltonian flow of \(p\). Then for any \(a \in S^0_h\),
\[
U_h(t)^* \mathrm{Op}_h^w(a) U_h(t) = \mathrm{Op}_h^w(a \circ \Phi^t) + \mathcal{O}(h) \quad \text{in } \mathcal{L}(L^2),
\]
for \(|t| \leq T\) (any fixed \(T\)). More precisely, the full symbol has an asymptotic expansion in powers of \(h\).
Proof (Sketch). Set \(A(t) = U_h(t)^* \mathrm{Op}_h^w(a) U_h(t)\). Then
\[
\frac{d}{dt} A(t) = \frac{i}{h} [P_h, A(t)].
\]
Writing \(A(t) = \mathrm{Op}_h^w(a_t)\), the Moyal product formula gives
\[
\frac{d}{dt} a_t = \{p, a_t\} + \mathcal{O}(h) = H_p(a_t) + \mathcal{O}(h),
\]
with initial condition \(a_0 = a\). To leading order, this is the transport equation along the Hamiltonian flow, solved by \(a_t^{(0)} = a \circ \Phi^t\). Higher-order corrections are obtained iteratively. \(\blacksquare\)
7.5 Quantum-Classical Correspondence
The semiclassical results above are manifestations of a broader principle: quantum mechanics converges to classical mechanics as \(h \to 0\).
Theorem 7.5.1 (Semiclassical Propagation of Wavefront Sets). Let \(u_h\) solve the semiclassical Schrödinger equation \(ih\partial_t u_h = P_h u_h\), where \(P_h = \mathrm{Op}_h^w(p)\) with real \(p\). Then the semiclassical wavefront set propagates according to the classical flow:
\[
\mathrm{WF}_h(u_h(t)) = \Phi^t(\mathrm{WF}_h(u_h(0))),
\]
where \(\Phi^t\) is the Hamiltonian flow of \(p\).
Remark 7.5.2. This theorem makes precise the statement that "quantum particles follow classical trajectories in the semiclassical limit." More precisely, the phase-space concentration of a quantum state, as measured by the semiclassical wavefront set, is transported by the classical Hamiltonian flow. This is the mathematical foundation of the WKB approximation and geometric optics.
7.6 The Weyl Law
The Weyl law describes the asymptotic distribution of eigenvalues of elliptic operators and is one of the crown jewels of semiclassical analysis.
Theorem 7.6.1 (Weyl Law). Let \(P_h = -h^2\Delta + V(x)\) on a compact Riemannian manifold \((M,g)\) of dimension \(n\), where \(V \in C^\infty(M)\). Let \(N(E, h) = \#\{\lambda_j(h) \leq E\}\) count the number of eigenvalues of \(P_h\) below \(E\). If \(E\) is a regular value of the classical Hamiltonian \(p(x,\xi) = |\xi|_g^2 + V(x)\), then
\[
N(E,h) = \frac{1}{(2\pi h)^n} \operatorname{Vol}\{(x,\xi) \in T^*M : p(x,\xi) \leq E\} + \mathcal{O}(h^{1-n}).
\]
Equivalently, on a compact manifold without potential, the eigenvalue counting function \(N(\lambda) = \#\{\lambda_j \leq \lambda\}\) of \(-\Delta\) satisfies
\[
N(\lambda) = \frac{\omega_n \operatorname{Vol}(M)}{(2\pi)^n} \lambda^{n/2} + \mathcal{O}(\lambda^{(n-1)/2})
\]
as \(\lambda \to \infty\), where \(\omega_n\) is the volume of the unit ball in \(\mathbb{R}^n\).
Proof (Sketch). The proof proceeds in several steps.
Step 1: Tauberian argument. The eigenvalue counting function is related to the trace of the heat kernel or the spectral projector. One introduces a smoothed counting function \(\tilde{N}(E,h) = \operatorname{Tr}(f((P_h - E)/\delta))\) for a smooth approximation \(f\) to the characteristic function of \((-\infty, 0]\).
\[ f(P_h) = \frac{1}{\pi} \int_{\mathbb{C}} \bar{\partial}\tilde{f}(z) (z - P_h)^{-1} \, dL(z), \]where \(\tilde{f}\) is an almost-analytic extension of \(f\). The resolvent \((z - P_h)^{-1}\) is a semiclassical pseudodifferential operator for \(z\) away from the spectrum.
\[ \operatorname{Tr}(f(P_h)) = \frac{1}{(2\pi h)^n} \int_{T^*M} f(p(x,\xi)) \, dx \, d\xi + \mathcal{O}(h^{1-n}). \]The leading term is exactly the phase-space volume below \(E\), and the error is controlled by the remainder in the stationary phase expansion.
Step 4: De-smoothing. A Tauberian theorem (or direct comparison argument) allows one to pass from the smoothed counting function back to the sharp one, at the cost of a controlled error. \(\blacksquare\)
Remark 7.6.2 (Historical Context). The leading term of the Weyl law was conjectured by Hermann Weyl in 1911 for the Dirichlet Laplacian on bounded domains. The proof of the sharp remainder \(\mathcal{O}(\lambda^{(n-1)/2})\) is due to Hörmander (1968), who used the wave equation and FIO methods — specifically, the relation between the wave trace \(\operatorname{Tr}(\cos(t\sqrt{-\Delta}))\) and the length spectrum of closed geodesics. The semiclassical version presented here, following Zworski, uses the resolvent approach.
Remark 7.6.3. The remainder estimate \(\mathcal{O}(\lambda^{(n-1)/2})\) is sharp in general (as shown by the round sphere, where the high multiplicities of eigenvalues cause concentration). However, under additional dynamical hypotheses — for instance, if the set of closed geodesics has measure zero (the generic case) — the remainder can be improved to \(o(\lambda^{(n-1)/2})\). This is the content of the Duistermaat-Guillemin theorem (1975).
7.7 Applications to Spectral Theory
We conclude with some further applications of semiclassical analysis to spectral theory.
Theorem 7.7.1 (Quantum Ergodicity, Shnirelman-Zelditch-Colin de Verdière). Let \((M,g)\) be a compact Riemannian manifold such that the geodesic flow on the unit cosphere bundle \(S^*M\) is ergodic. Let \(\{u_j\}_{j=1}^\infty\) be an orthonormal basis of eigenfunctions of \(-\Delta_g\) with eigenvalues \(\lambda_j \to \infty\). Then there exists a subsequence \(\{u_{j_k}\}\) of density 1 (meaning \(\#\{k : j_k \leq N\}/N \to 1\) as \(N \to \infty\)) such that for every \(a \in C^\infty(S^*M)\),
\[
\langle \mathrm{Op}(a) u_{j_k}, u_{j_k} \rangle \to \frac{1}{\operatorname{Vol}(S^*M)} \int_{S^*M} a \, d\mu_L
\]
as \(k \to \infty\), where \(\mu_L\) is the Liouville measure.
Remark 7.7.2. In physical terms, quantum ergodicity says that "almost all" eigenfunctions become equidistributed in phase space as the eigenvalue goes to infinity — the quantum analog of the ergodic hypothesis in classical mechanics. The question of whether all eigenfunctions equidistribute (the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak, 1994) remains one of the central open problems at the interface of analysis, number theory, and mathematical physics. It has been resolved in the arithmetic setting by Lindenstrauss (Fields Medal, 2010).
Theorem 7.7.3 (Semiclassical Trace Formula). Let \(P_h = -h^2\Delta + V\) on a compact manifold, and let \(\chi \in C_c^\infty(\mathbb{R})\). Then
\[
\operatorname{Tr}(\chi(P_h)) = \frac{1}{(2\pi h)^n} \int_{T^*M} \chi(p(x,\xi)) \, dx \, d\xi + \sum_{k=1}^\infty h^k \int_{T^*M} c_k(x,\xi) \chi(p(x,\xi)) \, dx \, d\xi + \mathcal{O}(h^\infty),
\]
where the coefficients \(c_k\) are determined by the symbol of \(P_h\). The leading term is the Weyl term, and the subleading corrections encode geometric and dynamical information about the classical system.
Remark 7.7.4. The trace formula connects the spectral side (eigenvalues of \(P_h\)) with the geometric/dynamical side (the Hamiltonian flow of \(p\) on \(T^*M\)). In its more refined versions (Gutzwiller trace formula, Selberg trace formula), it relates individual eigenvalues to periodic orbits of the classical system. This is one of the deepest and most active areas of contemporary mathematical physics.
7.8 Conclusion
The semiclassical perspective brings the entire subject full circle. The distributions and Sobolev spaces of Chapter 1 provided the analytic foundation; the symbol calculus of Chapters 2–3 gave us the algebraic tools; the wavefront set of Chapter 4 revealed the geometric structure of singularities; the FIOs of Chapter 5 encoded symplectic transformations; the PDE applications of Chapter 6 demonstrated the power of the framework; and now semiclassical analysis shows how all of these ideas fit together in the regime where analysis meets geometry and physics. The passage from local to microlocal — from base manifold to cotangent bundle — is not merely a technical refinement but a genuine change in perspective, one that has transformed the study of linear PDE and continues to drive research in spectral theory, mathematical physics, and beyond.
