Microlocal Analysis: Foundations¶

Job: define wavefront sets and the $\Psi$DO calculus sufficiently to state microlocal elliptic regularity and Hörmander’s propagation of singularities.

Core¶

Distributions (minimal package)¶

On open $X\subset\mathbb{R}^n$ (or a manifold), $\mathcal{D}'(X)=(C_c^\infty(X))'$. Pairing $\langle u,\varphi\rangle$. Operations: differentiation, multiplication by $C^\infty$, convolution (when supports allow), Fourier transform on $\mathcal{E}'$ (compactly supported distributions): $$ \hat u(\xi)=\langle u,e^{-ix\cdot\xi}\rangle,\qquad u\in\mathcal{E}'. $$ Sobolev: $H^s(\mathbb{R}^n)=\{u\in\mathcal{S}':\langle\xi\rangle^s\hat u\in L^2\}$, $\langle\xi\rangle=(1+|\xi|^2)^{1/2}$. Local $H^s_{\mathrm{loc}}$ via cutoffs.

Singular support: $\operatorname{sing\,supp}u=$ complement of the largest open set where $u$ is $C^\infty$.

Wavefront set¶

Definition (Hörmander). $(x_0,\xi_0)\in T^*X\setminus 0$ is not in $\mathrm{WF}(u)$ if there exist $\varphi\in C_c^\infty(X)$ with $\varphi(x_0)\neq 0$ and a conic neighborhood $\Gamma$ of $\xi_0$ such that for all $N$, $$ |\widehat{\varphi u}(\xi)|\le C_N\langle\xi\rangle^{-N},\qquad\xi\in\Gamma. $$ $\mathrm{WF}(u)$ is closed, conic in $\xi$. Properties:

Fact	Statement
Smoothness	$\mathrm{WF}(u)=\emptyset\Leftrightarrow u\in C^\infty$
Projection	$\pi_X(\mathrm{WF}(u))=\operatorname{sing\,supp}u$
Differentiation	$\mathrm{WF}(\partial^\alpha u)\subset\mathrm{WF}(u)$
Multiplication	$\mathrm{WF}(au)\subset\mathrm{WF}(u)$ for $a\in C^\infty$
Pullback (transversal)	Hörmander’s pullback theorem when $f^*\mathrm{WF}(u)$ avoids the conormal to $f$

Example. $u=\delta_0$ on $\mathbb{R}^n$: $\mathrm{WF}(\delta)=\{(0,\xi):\xi\neq 0\}$. Heaviside $H(x)$ on $\mathbb{R}$: $\mathrm{WF}(H)=\{(0,\xi):\xi\neq 0\}$.

Pseudodifferential operators¶

Symbol class $S^m(X\times\mathbb{R}^n)$: $a\in C^\infty$ with $$ \bigl|\partial_x^\alpha\partial_\xi^\beta a(x,\xi)\bigr|\le C_{\alpha\beta K}\langle\xi\rangle^{m-|\beta|}\quad(x\in K\Subset X). $$ (Kohn–Nirenberg; Hörmander $S^m_{1,0}$.) Quantization (left/Kohn–Nirenberg): $$ a(x,D)u=(2\pi)^{-n}\int e^{ix\cdot\xi}a(x,\xi)\hat u(\xi)\,d\xi $$ for $u\in\mathcal{S}$, extends to $\mathcal{S}'$ and microlocally on manifolds via charts.

Principal symbol $\sigma_m(A)=a_m$ of order $m$ operator; for differential $P=\sum_{|\alpha|\le m}a_\alpha(x)D^\alpha$, $$ \sigma_m(P)(x,\xi)=\sum_{|\alpha|=m}a_\alpha(x)\xi^\alpha. $$

Calculus (asymptotic). - Composition: $AB$ has symbol $a\#b\sim\sum\frac{1}{\alpha!}\partial_\xi^\alpha a\,D_x^\alpha b$. - Principal symbols multiply: $\sigma(AB)=\sigma(A)\sigma(B)$. - Formal adjoint: $\sigma(A^*)=\overline{\sigma(A)}$ mod lower order. - Elliptic: $\sigma_m(A)(x,\xi)\neq 0$ on $T^*X\setminus 0$ (or on a conic set for microlocal ellipticity). Elliptic $\Rightarrow$ parametrix $B$ of order $-m$ with $AB-I,BA-I$ smoothing (order $-\infty$).

Microlocal WF rule. For $A\in\Psi^m$, $\mathrm{WF}(Au)\subset\mathrm{WF}(u)$. If $A$ is microlocally elliptic on a conic set $\Gamma$, then $\mathrm{WF}(u)\cap\Gamma=\mathrm{WF}(Au)\cap\Gamma$ mod $\mathrm{WF}$ of errors.

Microlocal elliptic regularity¶

Theorem. Let $P\in\Psi^m(X)$, $u\in\mathcal{D}'(X)$. If $P$ is elliptic at $(x,\xi)\in T^*X\setminus 0$ and $(x,\xi)\notin\mathrm{WF}(Pu)$, then $(x,\xi)\notin\mathrm{WF}(u)$.

Global form. $P$ elliptic, $Pu\in C^\infty$ $\Rightarrow$ $u\in C^\infty$ (classical elliptic regularity). Sobolev: $Pu\in H^s_{\mathrm{loc}}$ $\Rightarrow$ $u\in H^{s+m}_{\mathrm{loc}}$.

Propagation of singularities (Hörmander)¶

Let $P$ have real principal symbol $p$ of real principal type ($dp\neq 0$ on $p=0$ in $T^*X\setminus 0$, or $H_p$ not radial). Bicharacteristic strips: integral curves of the Hamilton vector field $$ H_p=\sum_i\Bigl(\frac{\partial p}{\partial\xi_i}\partial_{x_i}-\frac{\partial p}{\partial x_i}\partial_{\xi_i}\Bigr) $$ in the characteristic set $p^{-1}(0)\subset T^*X\setminus 0$.

Theorem (Hörmander). $\mathrm{WF}(u)\setminus\mathrm{WF}(Pu)$ is invariant under the bicharacteristic flow of $H_p$. Equivalently: if a bicharacteristic segment avoids $\mathrm{WF}(Pu)$ and one point of it avoids $\mathrm{WF}(u)$, the whole segment avoids $\mathrm{WF}(u)$.

Hyperbolic model. $P=\partial_t^2-\Delta_x$, $p=\tau^2-|\xi|^2$; null bicharacteristics project to light rays at speed $1$. Singularities of solutions of $Pu=0$ propagate along those rays.

See it¶

Parametrix for elliptic $P$. Construct $q\sim\sum_{j\ge 0}q_{-m-j}$ with $q_{-m}=1/p$ on $|\xi|\ge 1$ and transport equations for lower terms from $p\#q\sim 1$. Then $Q=\mathrm{Op}(q)$ satisfies $PQ-I\in\Psi^{-\infty}$. Apply to $Pu=f$: $u=Qf\bmod C^\infty$ microlocally where $P$ is elliptic.

Product of Heavisides. $u=H(x_1)H(x_2)$ on $\mathbb{R}^2$ has wavefront set along the conormal to the edges and an extra component at the corner — illustrates that $\operatorname{sing\,supp}$ alone misses directional structure needed for multiplication theorems.

Check¶

Show that $\mathrm{WF}(\delta_0)=\{(0,\xi):\xi\neq 0\}$ using the definition.

$\widehat{\varphi\delta}=\varphi(0)$ is constant for $\varphi(0)\neq 0$, hence not rapidly decreasing in any conic set of $\xi$; every direction at $x=0$ is in $\mathrm{WF}$. For $x_0\neq 0$ take $\varphi$ supported away from $0$: $\varphi\delta=0$, so no WF there.

State microlocal elliptic regularity for $P=-\Delta+V$ on $\mathbb{R}^n$ and deduce classical Weyl lemma for weak solutions of $\Delta u=0$.

$\sigma_2(-\Delta+V)=|\xi|^2\neq 0$ for $\xi\neq 0$, so elliptic everywhere. If $\Delta u=0$ in $\mathcal{D}'$, then $\mathrm{WF}(u)=\emptyset$ $\Rightarrow$ $u\in C^\infty$. (Analytic hypoellipticity upgrades to real-analytic for constant-coeff Laplace.)

Pitfalls¶

Order vs principal type: elliptic is order $m$ invertible principal symbol; real principal type is about $H_p\neq 0$ on char set — hyperbolic, not elliptic.
Complex principal symbols (e.g. $\bar\partial$, heat): propagation is more subtle (analytic WF, FBI transform); classical $C^\infty$ WF propagation as above needs real $p$.
Smoothing operators ($\Psi^{-\infty}$) can have empty WF of the Schwartz kernel off the diagonal zero section — they erase singularities completely.
Manifolds: symbols live in $T^*X$; quantization requires densities or half-densities for clean $L^2$ theory.

Refs¶

Hörmander, ALPDO I (distributions, WF) and III ($\Psi$DO, propagation)
Grigis–Sjöstrand, Microlocal Analysis for Differential Operators
Taylor, Pseudodifferential Operators
Abels, Pseudodifferential and Singular Integral Operators

Next¶

Microlocal overview · Fourier integral operators · PDE foundations · Functional analysis · Fourier analysis · Differential geometry · Elliptic PDE — pseudodifferential operators and elliptic regularity · Function spaces — Sobolev spaces and the microlocal elliptic regularity · FA / oscillatory & Carleson