Pseudo-differential operators on a manifold

Intuition Beginner

A pseudo-differential operator is a generalisation of a partial differential operator wide enough to swallow inverses, fractional powers, and parametrices that ordinary differentiation cannot reach. The classical Laplacian $-\Delta$ on ${\mathbb{R}}^n$ is a differential operator of order $2$. Its inverse $(1-\Delta {)}^{-1}$ is not a differential operator — it cannot be written as a finite combination of derivatives and multiplications — but it is a perfectly good pseudo-differential operator of order $-2$. The price of admission is that the operator is defined by a Fourier-side recipe, not by a finite differential expression.

Think of multiplication on the Fourier side. The Fourier transform turns differentiation in the $j$-th coordinate into multiplication by $i{\xi }_j$, so $-\Delta$ becomes multiplication by $|\xi {|}^2$, and $(1-\Delta {)}^{-1}$ becomes multiplication by $(1+|\xi {|}^2{)}^{-1}$. The first is a polynomial in $\xi$; the second is a smooth function of $\xi$ that decays at infinity. A pseudo-differential operator allows the Fourier-side multiplier to be any smooth function $a(x,\xi )$ with controlled growth in $\xi$, allowed to depend on the position $x$ as well.

Why bother? Because the right inverses, parametrices, and propagators of partial differential equations are usually pseudo-differential, not differential. Asking "what is the inverse of the Helmholtz operator?" forces you out of the differential category and into the pseudo-differential one. The framework also gives a clean principal-symbol invariant that controls regularity and propagation of singularities, and it generalises from ${\mathbb{R}}^n$ to a smooth manifold by patching local symbol-calculus calculations together.

Visual Beginner

Picture a smooth manifold $M$ on the horizontal axis and its cotangent fibre $T^{\ast }M$ rising above each point. A pseudo-differential operator is encoded by a function $a(x,\xi )$ on $T^{\ast }M$, called its symbol. The level sets of the symbol fibre over the manifold, and the operator acts by integrating against the symbol on the Fourier side at each point.

The integer attached to the operator is its order: a polynomial symbol of degree $m$ gives an operator of order $m$, and a symbol that decays like $|\xi {|}^{-m}$ gives an operator of negative order $-m$. The order is what distinguishes a Laplacian (order $2$) from its inverse (order $-2$) from a smoothing convolution (order $-\infty$). The principal symbol, the leading-order piece of $a(x,\xi )$ in $\xi$, is a function on $T^{\ast }M$ that survives the diffeomorphism patching.

Worked example Beginner

Compute the symbol of the Helmholtz parametrix $(1-\Delta {)}^{-1}$ on ${\mathbb{R}}^n$.

Step 1. The Laplacian acts on Schwartz functions by adding up the second derivatives in each coordinate, with an overall minus sign: $-\Delta u(x)=-(u_{x_1{x}_1}+\cdots +u_{x_n{x}_n})$. On the Fourier side, each coordinate derivative becomes multiplication by $i{\xi }_j$, so $-\Delta$ becomes multiplication by $|\xi {|}^2={\xi }_1^2+\cdots +{\xi }_n^2$.

Step 2. The Helmholtz operator $1-\Delta$ becomes multiplication by $1+|\xi {|}^2$ on the Fourier side. This is a polynomial of degree $2$ in $\xi$, so the operator has order $2$.

Step 3. To invert, divide: the operator $(1-\Delta {)}^{-1}$ becomes multiplication by $(1+|\xi {|}^2{)}^{-1}$ on the Fourier side. This is no longer a polynomial, but it is a smooth function of $\xi$ that decays like $|\xi {|}^{-2}$ at infinity. So the operator has order $-2$.

Step 4. Verify by direct computation. For a Schwartz function $u$, write $\hat{u}(\xi )$ for its Fourier transform. Then $((1-\Delta )v)(x)=u(x)$ becomes $(1+|\xi {|}^2)\hat{v}(\xi )=\hat{u}(\xi )$ on the Fourier side, giving $\hat{v}(\xi )=(1+|\xi {|}^2{)}^{-1}\hat{u}(\xi )$. The inverse Fourier transform recovers $v(x)$ as the inverse-Fourier-transform of the product $(1+|\xi {|}^2{)}^{-1}\hat{u}(\xi )$.

Step 5. Recognise this construction as the action of a pseudo-differential operator with symbol $a(x,\xi )=(1+|\xi {|}^2{)}^{-1}$. The symbol does not depend on $x$ here (the Helmholtz operator has constant coefficients), but the general definition allows it to. The principal symbol of $(1-\Delta {)}^{-1}$, taking the leading-order behaviour as $|\xi |$ grows large, is $|\xi {|}^{-2}$.

What this tells us: the inverse of a differential operator is generally not a differential operator, but it is a pseudo-differential operator. The order of the inverse is the negative of the order of the original. The symbol calculus turns operator inversion into the much easier task of inverting a function on the Fourier side, which is what makes the framework so useful.

Check your understanding Beginner

Formal definition Intermediate+

Let $X\subseteq {\mathbb{R}}^n$ be open. Recall the Fourier transform convention $$ \hat{u}(\xi) = \int_{\mathbb{R}^n} e^{-i x \cdot \xi} u(x) , dx, \qquad u(x) = (2\pi)^{-n} \int e^{i x \cdot \xi} \hat u(\xi) , d\xi, $$ for $u\in \mathcal{S}({\mathbb{R}}^n)$, extended to tempered distributions by duality.

Definition (Hörmander symbol class $S_{1,0}^m$). For $m\in \mathbb{R}$, the symbol class $S_{1,0}^m(X\times {\mathbb{R}}^n)$ consists of smooth functions $a:X\times {\mathbb{R}}^n\to \mathbb{C}$ such that for every compact $K\subset X$ and all multi-indices $\alpha ,\beta$, $$ |\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| \leq C_{K, \alpha, \beta} (1 + |\xi|)^{m - |\beta|}, \qquad x \in K, \xi \in \mathbb{R}^n. $$ The integer $m$ is the order of the symbol. Differentiation in $\xi$ lowers the order by one; differentiation in $x$ does not change the order. The space $S^{-\infty }={\bigcap }_m{S}_{1,0}^m$ consists of smoothing symbols that decay rapidly in $\xi$ together with all derivatives, uniformly in $x$ on compacts.

Definition (quantisation; $\Psi$DO on ${\mathbb{R}}^n$). For $a\in S_{1,0}^m(X\times {\mathbb{R}}^n)$, the pseudo-differential operator $a(x,D):C_c^{\infty }(X)\to C^{\infty }(X)$ is defined by $$ a(x, D) u(x) = (2\pi)^{-n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} a(x, \xi) \hat u(\xi) , d\xi, \qquad u \in C_c^\infty(X). $$ The integral converges absolutely on $C_c^{\infty }$ because $\hat{u}$ is Schwartz, and the polynomial growth of $a$ is absorbed by the rapid decay of $\hat{u}$. The operator extends by continuity to $a(x,D):{\mathcal{E}}^{\prime }(X)\to {\mathcal{D}}^{\prime }(X)$ (compactly supported distributions to general distributions). The space of all such operators of order $m$ is ${\Psi }^m(X)$.

A $\Psi$DO is properly supported if both $a(x,D)$ and its formal adjoint $a(x,D{)}^{\ast }$ map $C_c^{\infty }$ into $C_c^{\infty }$, equivalently, if the Schwartz kernel of $a(x,D)$ has proper support over the diagonal. Properly supported $\Psi$DOs can be composed; arbitrary $\Psi$DOs cannot.

Definition (principal symbol). The principal symbol of $a\in S_{1,0}^m$ is the equivalence class ${\sigma }_m(a)=[a]\in S_{1,0}^m/S_{1,0}^{m-1}$. The principal-symbol map ${\sigma }_m:{\Psi }^m(X)/{\Psi }^{m-1}(X)\to S_{1,0}^m(X\times {\mathbb{R}}^n)/S_{1,0}^{m-1}(X\times {\mathbb{R}}^n)$ is a linear isomorphism, induced by the quantisation. For a classical symbol $a\sim {\sum }_{j\ge 0}{a}_{m-j}$ with $a_{m-j}$ positively homogeneous of degree $m-j$ in $\xi$ for $|\xi |\ge 1$, the principal symbol is represented by the homogeneous top term $a_m$, regarded as a function on $T^{\ast }X\setminus 0$.

Definition ($\Psi$DO on a manifold). Let $M$ be a smooth manifold. A linear operator $A:C_c^{\infty }(M)\to C^{\infty }(M)$ is a pseudo-differential operator of order $m$ if in every coordinate chart $(U,\kappa )$ on $M$, the pushed-forward operator ${\kappa }_{\ast }(A{|}_U\circ {\kappa }^{\ast })$ is a ${\Psi }^m$-operator on the open subset $\kappa (U)\subset {\mathbb{R}}^n$ as defined above, and if the operator is pseudolocal: $\mathrm{singsupp}(Au)\subseteq \mathrm{singsupp}(u)$ for every $u\in {\mathcal{E}}^{\prime }(M)$. The class is denoted ${\Psi }^m(M)$. The chart-independence of the order is established by the diffeomorphism-invariance of $S_{1,0}^m$ under the cotangent action $(x,\xi )\mapsto ({\kappa }^{-1}(x),(D\kappa {)}^{\top }\xi )$; the principal symbol transforms as a function on $T^{\ast }M\setminus 0$.

Notation and conventions

• $S_{1,0}^m$: the Hörmander symbol class with $(\rho ,\delta )=(1,0)$, the classical case. The general class $S_{\rho ,\delta }^m$ allows $|{\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a|\le C(1+|\xi |{)}^{m-\rho |\beta |+\delta |\alpha |}$ with $0\le \delta <\rho \le 1$; the $\Psi$DO algebra is well-behaved when $\rho >\delta$ (Hörmander 1965).
• ${\Psi }^m(X)$, ${\Psi }^m(M)$: properly supported $\Psi$DOs of order $m$ on an open Euclidean set $X$ or on a manifold $M$.
• ${\sigma }_m(A)$: the principal symbol of $A\in {\Psi }^m$, regarded as an element of $S^m/S^{m-1}$ or as a homogeneous function on $T^{\ast }M\setminus 0$.
• $a\#b$: the symbol of the composition $a(x,D)\circ b(x,D)$ in the symbol calculus, with asymptotic expansion $a\#b\sim {\sum }_{\alpha }(i^{-|\alpha |}/\alpha !){\partial }_{\xi }^{\alpha }a\cdot {\partial }_x^{\alpha }b$.
• ${\Psi }^{-\infty }(M)={\bigcap }_m{\Psi }^m(M)$: smoothing operators, those with smooth Schwartz kernel on $M\times M$. They send distributions to smooth functions.
• Sign convention: $\hat{u}(\xi )=\int e^{-ix\cdot \xi }u(x)dx$. The opposite sign $e^{+ix\cdot \xi }$ used in some physics references flips the sign of $\xi$ throughout and reverses the role of advanced/retarded fundamental solutions; consistency must be checked when reading sources.

Counterexamples to common slips

• The symbol class $S_{1,0}^m$ is not the set of positively-homogeneous functions of degree $m$ in $\xi$. Homogeneous functions are singular at $\xi =0$; symbols are smooth there. The connection is via classical symbols, where the homogeneous top term is the principal symbol modulo lower order.
• The order of a $\Psi$DO is not the highest derivative appearing. For a differential operator the two agree, but the order of $(1-\Delta {)}^{-1}$ is $-2$, not $0$ — there are no derivatives at all in its closed-form action, but the symbol decays like $|\xi {|}^{-2}$.
• The principal symbol is not the leading $|\xi |$-power coefficient of $a$ — it is the equivalence class modulo $S_{1,0}^{m-1}$. Two symbols with the same homogeneous top term are equivalent at the principal level even if they differ by lower-order non-homogeneous contributions.
• The composition $A\circ B$ of $\Psi$DOs need not be a $\Psi$DO unless one is properly supported. The kernel of $A\circ B$ involves an integration over the full manifold, which may diverge without proper support.

Key theorem with proof Intermediate+

Theorem (composition law; Hörmander Vol. III §18.1 Theorem 18.1.8 ^{[Hörmander Vol. III §18.1]}). Let $a\in S_{1,0}^m(X\times {\mathbb{R}}^n)$ and $b\in S_{1,0}^{m^{\prime }}(X\times {\mathbb{R}}^n)$, and let $a(x,D)$ and $b(x,D)$ be the associated properly supported $\Psi$DOs. Then $c(x,D):=a(x,D)\circ b(x,D)$ is a $\Psi$DO of order $m+m^{\prime }$, with symbol $c\in S_{1,0}^{m+m^{\prime }}$ admitting the asymptotic expansion $$ c(x, \xi) \sim \sum_{\alpha} \frac{i^{-|\alpha|}}{\alpha!} , \partial_\xi^\alpha a(x, \xi) \cdot \partial_x^\alpha b(x, \xi), $$ meaning that for every $N\ge 1$, $$ c(x, \xi) - \sum_{|\alpha| < N} \frac{i^{-|\alpha|}}{\alpha!} \partial_\xi^\alpha a(x, \xi) \cdot \partial_x^\alpha b(x, \xi) \in S^{m + m' - N}{1, 0}. $$ *The principal symbols multiply: $\sigma{m + m'}(c) = \sigma_m(a) \cdot \sigma_{m'}(b)$.*

Proof. The argument runs in three steps: write the composition as an iterated oscillatory integral, perform stationary phase in the intermediate Fourier variable to produce the asymptotic expansion, and check the remainder belongs to the predicted symbol class.

Step 1: oscillatory-integral representation. For $u\in C_c^{\infty }(X)$, the composition is $$ (a(x, D) \circ b(x, D)) u(x) = (2\pi)^{-n} \int e^{i x \cdot \xi} a(x, \xi) \widehat{b(\cdot, D) u}(\xi) , d\xi. $$ Use the formula $\widehat{f\cdot g}(\xi )=(2\pi {)}^{-n}(\hat{f}\ast \hat{g})(\xi )$ on the inner Fourier transform: $b(x,D)u(x)=(2\pi {)}^{-n}\int e^{ix\cdot \eta }b(x,\eta )\hat{u}(\eta )d\eta$, so $$ \widehat{b(\cdot, D) u}(\xi) = (2\pi)^{-n} \int e^{-i x \cdot \xi} \int e^{i x \cdot \eta} b(x, \eta) \hat u(\eta) , d\eta , dx = \int \widetilde{b}(\xi - \eta, \eta) \hat u(\eta) , d\eta, $$ where $\tilde{b}(\zeta ,\eta )=(2\pi {)}^{-n}\int e^{-ix\cdot \zeta }b(x,\eta )dx$ is the partial Fourier transform of $b$ in the first variable. Combining and changing the order of integration, $$ c(x, \xi) = (2\pi)^{-n} \int \int e^{i x \cdot (\eta - \xi)} a(x, \xi) e^{i x \cdot \xi} \widetilde{b}(\xi - \eta, \eta) , d\eta , d\zeta $$ collapses (after a change of variables $\zeta =\xi -\eta$) to the double oscillatory integral $$ c(x, \xi) = \iint e^{-i y \cdot \zeta} a(x, \xi + \zeta) b(x + y, \xi) , dy , d\zeta, $$ absolutely convergent in the sense of oscillatory integrals (Hörmander Vol. I §7.8).

Step 2: stationary phase in $(y,\zeta )$. The phase $\varphi (y,\zeta )=-y\cdot \zeta$ has its unique critical point at $(y,\zeta )=(0,0)$, with non-degenerate Hessian $$ \phi'' = \begin{pmatrix} 0 & -I \ -I & 0 \end{pmatrix}, $$ signature $0$, and determinant $(-1{)}^n$. By the method of stationary phase (Hörmander Vol. I §7.7), the integral admits the asymptotic expansion $$ \iint e^{-i y \cdot \zeta} a(x, \xi + \zeta) b(x + y, \xi) , dy , d\zeta \sim \sum_{\alpha} \frac{1}{\alpha!} (-i \partial_\zeta)^\alpha (i \partial_y)^\alpha [a(x, \xi + \zeta) b(x + y, \xi)]\bigg|{y = 0, \zeta = 0}, $$ where the factor $(-i{)}^{|\alpha |}{i}^{|\alpha |}=1$ from the operator $(-i \partial\zeta) (i \partial_y)$hasbeenabsorbed,andacarefultrackingofthestationary-phasenormalisationproducestheprefactor$i^{-|\alpha|} / \alpha!$. Evaluating the derivatives at the critical point gives $$ \frac{i^{-|\alpha|}}{\alpha!} \partial_\xi^\alpha a(x, \xi) \cdot \partial_x^\alpha b(x, \xi), $$ the predicted summand.

Step 3: remainder estimate. The stationary-phase theorem gives a quantitative remainder. After taking $N$ terms of the expansion, the remainder $$ r_N(x, \xi) = c(x, \xi) - \sum_{|\alpha| < N} \frac{i^{-|\alpha|}}{\alpha!} \partial_\xi^\alpha a \cdot \partial_x^\alpha b $$ satisfies the symbol bound $|{\partial }_x^{\beta }{\partial }_{\xi }^{\gamma }{r}_N(x,\xi )|\le C_{\beta \gamma }(1+|\xi |{)}^{m+m^{\prime }-N-|\gamma |}$, which is exactly the membership $r_N\in S_{1,0}^{m+m^{\prime }-N}$. The constants $C_{\beta \gamma }$ depend on the seminorms of $a$ and $b$ in the relevant symbol classes and on compact subsets of $X$, but not on $\xi$.

Taking $N\to \infty$ produces the asymptotic series. The principal-symbol identity ${\sigma }_{m+m^{\prime }}(c)={\sigma }_m(a)\cdot {\sigma }_{m^{\prime }}(b)$ follows by reading off the $\alpha =0$ term and recognising that all higher-$\alpha$ terms have order at most $m+m^{\prime }-1$. $\square$

Bridge. The composition law builds toward every operator-algebraic argument in microlocal analysis. The asymptotic series is the central computational tool — it lets one read off the principal symbol of a parametrix from the principal symbols of the operator and target without ever leaving the symbol calculus. This pattern appears again in 02.14.01 (wave-front set), where the pseudolocality and elliptic-regularity statements both depend on the composition law applied to a parametrix construction, and in 03.12.16 (Poincaré duality), where the chart-by-chart gluing of local microlocal data mirrors the chart-by-chart gluing of cohomology classes on a manifold. The foundational insight is that operators of the form "differentiation plus lower-order things" can be inverted exactly modulo smoothing operators, and the inversion is performed at the symbol level — a polynomial-in-$\xi$ symbol like $|\xi {|}^2$ inverts to a smooth-in-$\xi$ symbol $|\xi {|}^{-2}$ outside a compact neighbourhood of the origin, smoothly extended to all of $T^{\ast }M$ by adding a ${\Psi }^{-\infty }$ correction. The principal symbol controls the leading behaviour; the asymptotic expansion controls the corrections term by term. Putting these together produces the elliptic-parametrix construction, the elliptic-regularity theorem $\mathrm{WF}(u)=\mathrm{WF}(Au)$ on $\{p_m\ne 0\}$, and Hörmander's propagation-of-singularities theorem along the Hamiltonian flow of the principal symbol. The composition law is the algebraic backbone of all of this; everything else is symbol calculation on top of it.

Exercises Intermediate+

Advanced results Master

Theorem (elliptic parametrix; Hörmander Vol. III §18.1 Theorem 18.1.24 ^{[Hörmander Vol. III §18.1]}). *Let $A\in {\Psi }^m(M)$ be elliptic on $T^{\ast }M\setminus 0$, meaning ${\sigma }_m(A)(x,\xi )\ne 0$ for all $(x,\xi )\in T^{\ast }M\setminus 0$. Then there exists a properly supported $B\in {\Psi }^{-m}(M)$ such that $AB-I\in {\Psi }^{-\infty }(M)$ and $BA-I\in {\Psi }^{-\infty }(M)$. The operator $B$ is unique modulo ${\Psi }^{-\infty }(M)$.*

The parametrix construction is the engine of elliptic regularity: it converts the abstract statement "$A$ is invertible up to compact errors" into the concrete statement "$A^{-1}$ exists modulo a smoothing operator, with explicit principal symbol $1/{\sigma }_m(A)$." On a closed manifold, smoothing operators are compact between Sobolev spaces, so $A$ is Fredholm with index $\dim \ker A-\dim \mathrm{coker}A$, an integer invariant. The Atiyah-Singer index theorem identifies this integer with a topological expression involving the principal symbol of $A$ and the characteristic classes of $M$ ^{[Hörmander Vol. III]}.

Theorem ($L^2$-continuity, Calderón-Vaillancourt; Hörmander Vol. III §18.6 ^{[Hörmander Vol. III §18.6]}). Let $A\in {\Psi }_{1,0}^0({\mathbb{R}}^n)$. Then $A$ extends to a bounded operator $A:L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)$, with operator norm bounded in terms of finitely many seminorms of the symbol.

The Calderón-Vaillancourt theorem gives the foundational continuity result of the $\Psi$DO calculus on Hilbert space. Combined with the composition law and the order count, it yields $A:H^s({\mathbb{R}}^n)\to H^{s-m}({\mathbb{R}}^n)$ bounded for every $A\in {\Psi }_{1,0}^m$ and $s\in \mathbb{R}$. On a closed manifold, the corresponding statement $A:H^s(M)\to H^{s-m}(M)$ uses the manifold Sobolev spaces defined chart-by-chart ^[Taylor].

Theorem (sharp Gårding inequality; Hörmander Vol. III §18.6 ^{[Hörmander Vol. III §18.6]}). Let $A\in {\Psi }_{1,0}^m(M)$ have real principal symbol ${\sigma }_m(A)$ with ${\sigma }_m(A)(x,\xi )\ge 0$ for $|\xi |$ large. Then for every $ϵ>0$ there is a constant $C_{ϵ}$ such that $$ \mathrm{Re}, \langle A u, u \rangle_{L^2} \geq -\epsilon |u|{H^{(m - 1)/2}}^2 - C\epsilon |u|_{H^{(m - 2)/2}}^2 $$ for every $u\in C_c^{\infty }(M)$.

The sharp Gårding inequality refines the elementary Gårding inequality (which bounds below by $-C\Vert u{\Vert }_{H^{(m-1)/2}}^2$ without the $ϵ$ improvement) and is the foundational positivity statement of microlocal analysis. It underwrites uniqueness theorems for elliptic boundary problems, semiclassical lower bounds for Schrödinger eigenvalues, and the existence of self-adjoint extensions in spectral theory.

Theorem (Weyl quantisation as an alternative; Shubin §3 ^[Shubin]). In addition to the Kohn-Nirenberg quantisation $a(x,D)u(x)=(2\pi {)}^{-n}\int e^{ix\cdot \xi }a(x,\xi )\hat{u}(\xi )d\xi$, the Weyl quantisation $$ a^w(x, D) u(x) = (2\pi)^{-n} \iint e^{i (x - y) \cdot \xi} a\left(\frac{x + y}{2}, \xi\right) u(y) , dy , d\xi $$ sends symbols to operators with the symmetry property $(a^w(x, D))^ = (\bar a)^w(x, D)$—realsymbolsgiveformallyself-adjointoperators.Thetwoquantisationsdifferbylower-orderterms:$a(x, D) - a^w(x, D) \in \Psi^{m - 1}$. The principal symbol is the same.*

Weyl quantisation is the natural quantisation in quantum mechanics, where the position-momentum symmetry of the phase-space variables suggests symmetrising the symbol over the position factor. The two quantisations agree at the principal-symbol level, so the symbol calculus, ellipticity, and propagation theorems are insensitive to the choice. Lower-order corrections matter for semiclassical asymptotics and for the precise form of subleading terms in spectral expansions.

Theorem (Hörmander's hypoelliptic and subelliptic classes; Hörmander Vol. III §22 ^{[Hörmander Vol. III]}). Beyond the elliptic class $S_{1,0}^m$, the more general Hörmander classes $S_{\rho ,\delta }^m$ with $0\le \delta <\rho \le 1$ retain the symbol calculus (composition law, principal symbol, parametrix construction) and accommodate operators that are merely hypoelliptic. The boundary case $\rho =1/2,\delta =1/2$ is the Beals-Fefferman class, useful for subelliptic problems on stratified manifolds and for the analysis of the $\bar{\partial }$-Neumann problem on strongly pseudoconvex domains.

The general Hörmander classes give the flexibility needed for problems where ellipticity fails but a weaker regularity-improving estimate holds. Hypoelliptic operators of principal type (real principal symbol, vanishing on a hypersurface where the Hamiltonian field is transverse) and operators of subprincipal type (Hörmander's notion based on Hamilton-Jacobi data) are all accommodated; the relevant symbol calculus is the one with $(\rho ,\delta )=(1/2,1/2)$ or $(1,1/2)$ as needed.

Synthesis. The pseudo-differential calculus is the algebraic backbone of microlocal analysis: it is the operator algebra in which every PDE-theoretic regularity question can be posed and resolved at the symbol level. The central insight is that the symbol class $S_{1,0}^m$ on $T^{\ast }M$ is preserved under composition modulo lower-order terms, and the principal symbol — the leading $|\xi |$-power piece, regarded as a function on $T^{\ast }M\setminus 0$ — is a complete invariant up to lower order. This is exactly the same local-to-global principle as in 02.14.01 (wave-front set), where the conic-decay characterisation of microlocal smoothness is preserved under the cotangent-lift action and pieced together on a manifold by chart-independent gluing. The two frameworks fit together: the wave-front set tracks the singularities of distributions, and the $\Psi$DO calculus tracks the action of operators on those singularities, with pseudolocality bounding $\mathrm{WF}(Au)\subseteq \mathrm{WF}(u)$ and elliptic regularity refining this to $\mathrm{WF}(u)=\mathrm{WF}(Au)$ away from the characteristic variety $\{p_m=0\}$. Putting these together produces every classical regularity theorem of linear elliptic PDE — Sobolev regularity, smoothness up to the boundary in the elliptic case, Fredholmness on closed manifolds, the index theorem — and underwrites the Hörmander propagation-of-singularities theorem for real-principal-type operators, which carries wave-front-set singularities along the Hamiltonian flow of the principal symbol on the characteristic variety. The bridge is the recognition that an operator is encoded by its symbol, the symbol calculus is functorial under composition, and the symbol-side computation is enormously easier than the operator-side one.

The $\Psi$DO calculus identifies several operator-theoretic notions that look distinct at first inspection. The differential operators of order $m$ form a subalgebra of ${\Psi }^m(M)$ consisting of symbols polynomial in $\xi$; the Calderón-Zygmund integral operators of order $0$ form another subalgebra consisting of symbols that depend only on $\xi /|\xi |$; the convolution operators on ${\mathbb{R}}^n$ with $L^1$ kernel form a subalgebra of ${\Psi }^{-\infty }$. The $\Psi$DO calculus unifies all three: each is a symbol class with its own regularity behaviour, and the composition law is the same in all cases. The Atiyah-Singer index theorem, in turn, identifies the analytical index of an elliptic ${\Psi }^m$ operator on a closed manifold with a topological expression in the principal symbol and the characteristic classes of the manifold, giving a complete correspondence between operator-theoretic and topological invariants. Pseudo-differential operators with operator-valued symbols generalise to vector-bundle-valued operators on Hermitian bundles, and the resulting symbol calculus underlies the Boutet de Monvel calculus for boundary problems on manifolds with boundary, the Connes-Moscovici noncommutative-geometric calculus on foliations and noncommutative algebras, and the algebraic-QFT programme on curved spacetimes where the Klein-Gordon parametrix is constructed by a pseudo-differential approximation of the geometric square-root operator. Each of these specialisations is controlled by the same composition law, the same principal-symbol invariant, and the same elliptic-parametrix construction with which we began.

Full proof set Master

Theorem (composition law), proof. Given in the Intermediate-tier section: oscillatory-integral representation of the composition, stationary-phase asymptotic expansion in the intermediate Fourier variable, and the remainder estimate $r_N\in S_{1,0}^{m+m^{\prime }-N}$ via the stationary-phase remainder bound. Full argument in Hörmander Vol. III §18.1 Theorem 18.1.8 ^{[Hörmander Vol. III §18.1]}. $\square$

Proposition (smoothing operators send distributions to smooth functions). If $R\in {\Psi }^{-\infty }(M)$ and $u\in {\mathcal{E}}^{\prime }(M)$, then $Ru\in C^{\infty }(M)$.

Proof. The Schwartz kernel $K_R(x,y)$ of $R$ is smooth on $M\times M$ because $R\in {\bigcap }_m{\Psi }^m$ implies the symbol decays faster than every polynomial-reciprocal in $\xi$, and the inverse Fourier transform of such a symbol is smooth in the $x$-variable. Pair against $u\in {\mathcal{E}}^{\prime }(M)$: $Ru(x)=⟨u_y,K_R(x,y)⟩$, the pairing of $u$ with the smooth function $y\mapsto K_R(x,y)$ for fixed $x$. Smoothness in $x$ follows from differentiation under the duality pairing, which is permitted because $K_R(x,y)$ is smooth in $(x,y)$ and $u$ is compactly supported. $\square$

Proposition (parametrix for $(1-\Delta )$). Define $b(\xi )=(1-\chi (\xi ))/(1+|\xi {|}^2)$ with $\chi \in C_c^{\infty }({\mathbb{R}}^n)$, $\chi =1$ near $0$. Then $b\in S_{1,0}^{-2}$ and $b(D)\circ (1-\Delta )-I\in {\Psi }^{-\infty }$.

Proof. For the symbol class membership: $b$ is smooth (the cutoff $\chi$ makes the apparent singularity at $\xi =0$ vanish), and for $|\xi |\ge 2$, $b(\xi )=1/(1+|\xi {|}^2)$ has $|{\partial }_{\xi }^{\beta }b(\xi )|\le C_{\beta }(1+|\xi |{)}^{-2-|\beta |}$ by direct computation. For $|\xi |\le 2$, $b$ and all its derivatives are bounded. So $b\in S_{1,0}^{-2}({\mathbb{R}}^n)$.

For the parametrix identity: $(1+|\xi {|}^2)\cdot b(\xi )=1-\chi (\xi )$. So $b(D)\circ (1-\Delta )$ has symbol $1-\chi (\xi )$, and $b(D)\circ (1-\Delta )-I$ has symbol $-\chi (\xi )$, a compactly supported smooth function of $\xi$, hence an element of ${\bigcap }_m{S}_{1,0}^m=S^{-\infty }$. The corresponding operator is in ${\Psi }^{-\infty }$. $\square$

Proposition (existence of asymptotic sum). Given a sequence $(a_j{)}_{j\ge 0}$ with $a_j\in S_{1,0}^{m_j}$ and $m_0>m_1>m_2>\cdots \to -\infty$, there exists $a\in S_{1,0}^{m_0}$ such that for every $N$, $a-{\sum }_{j<N}{a}_j\in S_{1,0}^{m_N}$.

Proof. (Borel-type construction; Hörmander Vol. I §1.2.) Pick a cutoff $\chi \in C_c^{\infty }({\mathbb{R}}^n)$ with $\chi =0$ on $|\xi |\le 1$ and $\chi =1$ on $|\xi |\ge 2$. Choose a sequence $t_j\to \infty$ rapidly enough that for each $j$ and each multi-index pair $(\alpha ,\beta )$ with $|\alpha |+|\beta |\le j$, the function $\chi (\xi /t_j)a_j(x,\xi )$ has symbol seminorms in $S_{1,0}^{m_j}$ bounded by $2^{-j}$ on the compact sets indexing the seminorms. The sum $$ a(x, \xi) = \sum_j \chi(\xi / t_j) a_j(x, \xi) $$ converges in $C^{\infty }$ by the seminorm bound, lies in $S_{1,0}^{m_0}$ (the worst term is $j=0$), and satisfies $a-{\sum }_{j<N}{a}_j={\sum }_{j\ge N}\chi (\xi /t_j)a_j$, whose every term lies in $S_{1,0}^{m_j}\subseteq S_{1,0}^{m_N}$ for $j\ge N$. The total lies in $S_{1,0}^{m_N}$. $\square$

Theorem (existence of elliptic parametrix). *Let $A\in {\Psi }^m(M)$ be elliptic on $T^{\ast }M\setminus 0$. Then there exists $B\in {\Psi }^{-m}(M)$ properly supported with $AB-I,BA-I\in {\Psi }^{-\infty }(M)$.*

Proof. Iterative construction at the symbol level. First approximation: let $b_0(x,\xi )=\chi (\xi )/{\sigma }_m(A)(x,\xi )$ where $\chi \in C^{\infty }(T^{\ast }M)$ vanishes near the zero section and equals $1$ for $|\xi |$ large. Then $b_0\in S_{1,0}^{-m}$ and by the composition law, $A\circ b_0(x,D)=I+R_1$ with $R_1\in {\Psi }^{-1}$. Iterate: define $b_{j+1}=-b_0\cdot {\sigma }_{-1}(R_1\circ b_j(x,D))/{\sigma }_m(A)$, an element of $S_{1,0}^{-m-j-1}$, such that $A\circ (b_0+b_1+\cdots +b_j)(x,D)=I+R_{j+1}$ with $R_{j+1}\in {\Psi }^{-j-1}$. By the asymptotic-sum construction, there exists $b\in S_{1,0}^{-m}$ with $b-{\sum }_{j<N}{b}_j\in S_{1,0}^{-m-N}$. Then $A\circ b(x,D)-I\in {\Psi }^{-N}$ for every $N$, hence in ${\Psi }^{-\infty }$. The reverse identity $b(x,D)\circ A-I\in {\Psi }^{-\infty }$ follows by the same construction on the left or by uniqueness modulo ${\Psi }^{-\infty }$. Properly supported representative is obtained by composing with a properly supported smoothing cutoff. $\square$

Theorem (elliptic regularity in wave-front-set form). *Let $A\in {\Psi }^m(M)$ be elliptic at $(x_0,{\xi }_0)\in T^{\ast }M\setminus 0$. Then $(x_0,{\xi }_0)\in \mathrm{WF}(u)$ iff $(x_0,{\xi }_0)\in \mathrm{WF}(Au)$.*

Proof. Forward: pseudolocality of $A$ gives $\mathrm{WF}(Au)\subseteq \mathrm{WF}(u)$, sharpened to a microlocal statement via the conic-decay characterisation of the wave-front set (Exercise 6 from unit 02.14.01 generalises). Backward: by Exercise 7 from this unit, microlocal parametrix $B\in {\Psi }^{-m}$ exists at $(x_0,{\xi }_0)$ with $BA-I\in {\Psi }^{-\infty }$ on a conic neighbourhood, so $u=B(Au)-(BA-I)u$ has wave-front set at $(x_0,{\xi }_0)$ controlled by $B(Au)$, hence by $Au$. The two inclusions give the identity. $\square$

Connections Master

• Wave-front set of a distribution 02.14.01. The $\Psi$DO calculus is the operator-side companion of the wave-front-set calculus on the distribution side: pseudolocality bounds the wave-front set of the action $Au$ by that of $u$, and elliptic regularity refines the bound to an equality on the non-characteristic set $\{{\sigma }_m(A)\ne 0\}$. The propagation-of-singularities theorem of Hörmander 1971 takes the next step: for real-principal-type operators, the wave-front set is transported along the Hamiltonian flow of the principal symbol on the characteristic variety. The interplay between the symbol calculus and the wave-front-set calculus is exactly the content of microlocal analysis.

• Smooth manifold 03.02.01. The intrinsic definition of ${\Psi }^m(M)$ requires the smooth-manifold structure to define charts, transition diffeomorphisms, and the cotangent bundle $T^{\ast }M$ on which the principal symbol lives. The chart-independence of the principal symbol uses the cotangent-lift action $(x,\xi )\mapsto (\kappa (x),(D\kappa {)}^{-\top }\xi )$ of a diffeomorphism, and Exercise 8 establishes that the principal symbol transforms as a function on $T^{\ast }M\setminus 0$ under this action. The order, the symbol class, and the smoothing ideal ${\Psi }^{-\infty }$ are all intrinsic to $M$.

• Hilbert space 02.11.08. The Calderón-Vaillancourt theorem extends every ${\Psi }_{1,0}^0$ operator to a bounded operator on $L^2$, and the symbol calculus gives the order-shifted continuity ${\Psi }^m:H^s\to H^{s-m}$. This is the foundational continuity result of the $\Psi$DO theory and is what makes the calculus a quantitative tool for elliptic regularity in Sobolev spaces, not just a microlocal-singularity bookkeeping device. The Hilbert-space framework also gives the spectral theory of elliptic self-adjoint $\Psi$DOs on closed manifolds, including the Weyl asymptotic law for eigenvalue counting and the heat-kernel expansion.

• Banach space 02.11.04. Symbols $a\in S_{1,0}^m$ form a Fréchet space under the family of seminorms ${\sup }_{x\in K}(1+|\xi |{)}^{|\beta |-m}|{\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a(x,\xi )|$ over compact $K$ and multi-indices $\alpha ,\beta$; this is the appropriate locally convex topology for the symbol calculus. The corresponding operator topology on ${\Psi }^m(M)$ is induced by the symbol topology and by the operator norm on Sobolev spaces. The constructive symbol-asymptotic-sum theorem is a Borel-type construction in the Fréchet-symbol topology.

Historical & philosophical context Master

Pseudo-differential operators were introduced in two parallel papers in 1965: Joseph Kohn and Louis Nirenberg, "An algebra of pseudo-differential operators," Comm. Pure Appl. Math. 18 (1965) 269–305 ^{[Kohn-Nirenberg 1965]}, and Lars Hörmander, "Pseudo-differential operators and non-elliptic boundary problems," Comm. Pure Appl. Math. 18 (1965) 501–517 ^{[Hörmander 1965 CPAM]}. Kohn-Nirenberg systematised the algebra of order-graded operators with the composition formula on ${\mathbb{R}}^n$, motivated by the $\bar{\partial }$-Neumann problem on strongly pseudoconvex domains and the corresponding subelliptic estimates. Hörmander introduced the symbol classes $S_{\rho ,\delta }^m$ in the same year, identifying $\rho >\delta$ as the regime in which the symbol calculus is well-behaved and constructing elliptic parametrices in the general class. Both papers built on earlier work of Calderón and Zygmund on singular integral operators (CPAM 1952, 1956) and on Mikhlin's multiplier theorem (Doklady 1956), recasting these as instances of a unified operator calculus on the Fourier side. The intrinsic theory on a manifold was developed by Hörmander in "Pseudo-differential operators and hypoelliptic equations," Proc. Symp. Pure Math. 10 (1967) 138–183, and definitively in his "Fourier integral operators I," Acta Math. 127 (1971) 79–183 ^{[Hörmander 1971]}, where the link to wave-front sets and the propagation-of-singularities theorem first appeared in modern form.

The applications of the $\Psi$DO calculus to PDE theory, spectral theory, and index theory expanded rapidly through the 1970s and 1980s. The Atiyah-Singer index theorem, originally proven in the cobordism formulation (Atiyah-Singer 1963 Bull. AMS 69, full proofs 1968–1971 in five Annals papers), was recast in $\Psi$DO language by Atiyah-Bott-Patodi 1973 and Seeley 1969 using the heat-kernel expansion of an elliptic operator. Michael Taylor's Pseudodifferential Operators, Princeton 1981 ^[Taylor], and M. A. Shubin's Pseudodifferential Operators and Spectral Theory, Springer 1987 ^[Shubin], consolidated the analytical theory into textbook form, with Taylor emphasising the manifold and boundary-problem aspects and Shubin the spectral-asymptotic and Weyl-quantisation aspects. Hörmander's four-volume The Analysis of Linear Partial Differential Operators, Springer 1983–1985, with Vol. III §18 the canonical reference for the manifold $\Psi$DO calculus, remains the definitive technical anchor. Christian Gérard's Microlocal Analysis of Quantum Fields on Curved Spacetimes, EMS 2019 ^[Gérard], applies the framework to algebraic quantum field theory on globally hyperbolic Lorentzian manifolds, with Chs. 2–3 developing the symbol calculus and Chs. 7–8 using it to construct Hadamard states via pseudo-differential approximation of the geometric square-root operator $\sqrt{-{\Delta }_h+m^2}$ on a Cauchy surface.

Bibliography Master

@article{KohnNirenberg1965,
  author  = {Kohn, J. J. and Nirenberg, L.},
  title   = {An algebra of pseudo-differential operators},
  journal = {Comm. Pure Appl. Math.},
  volume  = {18},
  year    = {1965},
  pages   = {269--305}
}

@article{Hormander1965CPAM,
  author  = {H{\"o}rmander, Lars},
  title   = {Pseudo-differential operators and non-elliptic boundary problems},
  journal = {Comm. Pure Appl. Math.},
  volume  = {18},
  year    = {1965},
  pages   = {501--517}
}

@article{Hormander1971FIO,
  author  = {H{\"o}rmander, Lars},
  title   = {Fourier integral operators. {I}},
  journal = {Acta Math.},
  volume  = {127},
  year    = {1971},
  pages   = {79--183}
}

@book{HormanderVolIII,
  author    = {H{\"o}rmander, Lars},
  title     = {The Analysis of Linear Partial Differential Operators, Vol. {III}: Pseudo-Differential Operators},
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}

@book{Taylor1981PDO,
  author    = {Taylor, Michael E.},
  title     = {Pseudodifferential Operators},
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  series    = {Princeton Mathematical Series},
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  year      = {1981}
}

@book{Shubin1987PDOSpectral,
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  title     = {Pseudodifferential Operators and Spectral Theory},
  publisher = {Springer-Verlag},
  year      = {1987}
}

@article{Seeley1969Complex,
  author  = {Seeley, R. T.},
  title   = {Complex powers of an elliptic operator},
  journal = {Proc. Symp. Pure Math. (AMS)},
  volume  = {10},
  year    = {1967},
  pages   = {288--307}
}

@article{AtiyahSinger1968I,
  author  = {Atiyah, M. F. and Singer, I. M.},
  title   = {The index of elliptic operators: {I}},
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  volume  = {87},
  year    = {1968},
  pages   = {484--530}
}

@book{Gerard2019Microlocal,
  author    = {G{\'e}rard, Christian},
  title     = {Microlocal Analysis of Quantum Fields on Curved Spacetimes},
  publisher = {European Mathematical Society},
  series    = {ESI Lectures in Mathematics and Physics},
  year      = {2019}
}

@article{CalderonVaillancourt1972,
  author  = {Calder{\'o}n, A. P. and Vaillancourt, R.},
  title   = {On the boundedness of pseudo-differential operators},
  journal = {J. Math. Soc. Japan},
  volume  = {23},
  year    = {1971},
  pages   = {374--378}
}

@article{Hormander1967Hypoelliptic,
  author  = {H{\"o}rmander, Lars},
  title   = {Pseudo-differential operators and hypoelliptic equations},
  journal = {Proc. Symp. Pure Math. (AMS)},
  volume  = {10},
  year    = {1967},
  pages   = {138--183}
}