Introduction - Mathematicsrbm/18-155-F15/PseudodifferentialOperators.pdfthe original papers, on pseudodi erential operators. Calder on and Zygmund (singular integral operators) [1,

PSEUDODIFFERENTIAL OPERATORS

Abstract. A brief treatment of classical pseudodifferential operators on Rn

intended to be reasonably complete but approachable and enough to get elliptic

regularity.

Thanks to Vishesh Jain, Mark Sellke and Ethan Jaffe for many comments, sug-gestions and corrections.

Introduction

These pages should ultimately be incoporated into a new version of the ‘GraduateAnalysis’ notes for 18.155 at MIT. The only ‘innovation’ here, relative to traditionaltreatments, is the use of expansions in Hermite functions to define the operatorsand prove L2 (i.e. Sobolev) boundedness.

The material here can be found in many places although it may be difficultinitially to see the relationships between the different treatments. First there arethe original papers, on pseudodifferential operators.

• Calderon and Zygmund (singular integral operators) [1, 2].• Seeley in [9] and [10].• Kohn and Nirenberg [8].• Hormander [3] treats the ‘standard’ calculus and the later paper [4] extends

this greatly.

Then there are various newer sources:-

• Hormander’s treatise, Vol 3 [6]• Lectures by Mark Joshi [7]• Michael Taylor’s book [12]• Misha Shubin’s book [11]• My microlocal analysis notes

If P (x,D) is a differential operator with smooth coefficients, say bounded withall derivatives, on Rn,

(0.1) P (x,D)u(x) =∑|α|≤m

Pα(x)Dαu(x), u ∈ S(Rn), P (x, ξ) =∑|α|≤m

Pα(x)ξα

then it can be written in terms of the Fourier transform of u as

(0.2) P (x,D)u(x) = (2π)−n∫Rn

eix·ξP (x, ξ)u(ξ)dξ.

This a rather ‘fake’ formula in the sense that it only works because we can de-compose the integral into a finite sum of terms, using the fact that P (x, ξ) is apolynomial in ξ, so that it becomes the inverse Fourier transform

(0.3) P (x,D)u(x) =∑|α|≤m

Pα(x)(2π)−n∫Rn

eix·ξξαu(ξ)dξ

and we recover (0.1). Nevertheless the idea is that we can interpret, and manipulate,‘improper’ (usually called oscillatory) integrals of the type (0.2) directly leading to

1

2 PSEUDODIFFERENTIAL OPERATORS

the theory of pseudodifferential operators. In particular such an integral makessense for the space of ‘symbols of order m’ which is usually taken to mean thespace of smooth functions on Rn × Rn which satisfy estimates (which we haveencountered before)

(0.4) |∂αx ∂βξ a(x, ξ)| ≤ Cα,β(1 + |ξ|)m−|β|.

The most important class of operators correspond to classical symbols which arediscussed below, but the term means that they have asymptotic expansions inhomogeneous functions in ξ as |ξ| → ∞.

Still the theory passes through the more general symbols (0.4) (or even moregeneral ones) and one reason for this is that the oscillatory integrals are definedusing density arguments from rapidly decreasing functions and, just as for boundedcontinuous functions on R, rapidly decreasing functions are not dense in classicalsymbols in the natural topology but are dense (with loss of ε in the order) in thesymbol topology given by the constants in (0.4).

In this brief introduction to pseudodifferential operators I want to avoid thisdiscussion and get to the local theory as quickly as possible. The approach is basedon the observation that the formula (0.3) ‘works’ because the symbol P (x, ξ) isin the case of a differential operator a finite sum of products of functions of x(the coefficients) and functions of ξ (giving monomial differentiatial operators); thegeneral case is handled below by expansion in terms of such products.

To do this I first limit attention to symbols with rapidly decreasing coefficients– i.e. which are Schwartz in the x variable; one disadvantage is that the identityis thereby excluded from the ‘algebra’ but this is not an issue for local regularityquestions. Then the simple idea is that one can expand these functions in terms ofthe Hermite functions eγ ∈ S(Rn), the eigenfunctions of the harmonic oscillator,

(0.5) a(x, ξ) =∑γ

eγ(x)aγ(ξ), aγ(ξ) =

∫Rn

a(x, ξ)eγ(x)dx.

This series converges rapidly and the coefficients aγ(ξ) are classical symbols. Thenthe operator can be defined directly as in (0.3)

(0.6) a(x,D)u(x) =∑γ

(2π)−neγ(x)

∫Rn

eix·ξaγ(ξ)u(ξ)dξ

=∑γ

eγ(x)aγ(D)u, aγ(D)u = Aγ ∗ u

where Aγ = aγ is the sort of convolution operator we dealt with earlier. The series(0.6) converges rapidly as a series of bounded operators on Sobolev spaces so theseoperators are easily defined and bounded. The crucial theorem is that the productof two such operators is of the same form, that the operators form an algebra.

Really for no particular reason I elected to initially define the algebra of psue-dodifferential operators with the coefficients on the right. This may be confusingbut is a decision easily reversed and of no particular consequence. Indeed one ofthe basic result is that one gets the same class of operators either way – this isdiscussed below.

A word about the Schwartz kernel theorem. This is not used here but therelationship between operators and kernels is used. Consider the space of continuous

PSEUDODIFFERENTIAL OPERATORS 3

linear maps (with the weak topology on S ′(Rp))

(0.7) S(Rn) −→ S ′(Rp) = B(S(Rn);S ′(Rp))

(here we only consider p = n and I suppose the B is slightly bogus). There is anatural bijection

(0.8) B(S(Rn);S ′(Rp))←→ S ′(Rp+n) Schwartz’ kernel Theorem

which is written formally

(0.9) (Aφ)(ψ) =

∫ψ(x)A(x, y)φ(y), φ ∈ S(Rn), ψ ∈ S(Rp)

where the actual relationship between operator A and kernel A (why waste a letterwhen we should think of them as the same thing) is given by the pairing

(0.10) (Aφ)(ψ) = (A(x, y), ψ(x)φ(y)).

Here I have added the variables formally because it is usual to ‘reverse’ them in thisway so that this looks like the integral operator (0.9). The harder part of Schwartzkernel theorem is to show that each continuous operator corresponds to a kernel.The other direction is easier and what we really use is the fact that the kernel,once it is shown to exist, is unique. This follows from the formula (0.10) whichdetermines the operator from the kernel and conversely shows that the pairing ofthe kernel with finite sums of products of Schwartz functions in the two variables isfixed by the operator. These finite sums are dense in S(Rp+n) (‘completed tensorproduct’) so the kernel is determined by the operator. The remaining issue in theproof of the kernel theorem is the continuity of the pairing of the putative kerneland the elements of the finite tensor product.

For differential operators with Schwartz coefficients the kernel is of the form

(0.11)∑|α|≤m

Pα(x)(Dαδ)(x− y)

so is supported on the diagonal. Our pseudodifferential operators are really char-acterized by their kernels which are singular (and only in a special ‘conormal’ way)at the diagonal away from which they are given by a Schwartz function.

Exercise 1. Write out the relationship between the coefficients in the left and rightforms of a differential operator

(0.12) P (x,D)u(x) =∑|α|≤m

Pα(x)Dαxu(x) =

∑|α|≤m

Dαx (Qα(x)u(x)).

1 Harmonic oscillatorRBM:Add sketch of alternate

as suggested by VisheshThe eigenbasis for the harmonic oscillator, which I will call the Hermite basis,gives an expansion for any element of φ ∈ S(Rn) as a series

(1.1) φ =∑γ∈Nn

0

(φ, eγ)eγ

which is the Fourier-Bessel expansion for the orthonormal basis of eigenfunctionsin L2(Rn). It has the desirable property of converging in S(Rn) precisely whenφ ∈ S(Rn).


This is used below in showing that certain spaces are ‘completed tensor products’.Consider C∞([0, 1];S(R2n)), consisting of those f ∈ C∞([0, 1]× Rn) which satisfy

(1.2) sup |yβDαyD

kt f(t, y)| <∞, ∀ k, α, β.

If we expand in the Hermite basis then we get

(1.3) f(t, y) =∑γ

fγ(t)eγ(y), fγ ∈ C∞([0, 1]), eγ ∈ S(Rn)

converging in C∞([0, 1];S(Rn)).

Such an expansion is used below in the treatment of pseudodifferential operators.We proceed to derive the appropriate properties of the Hermite basis. You can

safely skip this if you believe that for the Hermite basis, in each dimension, that(1.1) holds and if φ ∈ S(Rn) then

(1.4) |(φ, eγ)| ≤ CN (1 + |γ|)−N , |xβDαx eγ(x)| ≤ C ′k(1 + |γ|)M+k, |α|+ |β| ≤ k

where CN depends on some norm on φ, M depends only on n and C ′k depends onn and k.

The one-dimensional harmonic oscillator can be written

(1.5) H = (d

dx+ x)(− d

dx+ x)− 1 = − d2

dx2+ x2.

So we proceed to invert H + 1 by inverting the two ordinary differential operators

(1.6) A =d

dx+ x, C = A∗ = − d

dx+ x.

Both are bounded operators

(1.7) A,C : H1iso(R) = H1(R) ∩ 〈x〉−1L2(R) −→ L2, 〈x〉 = (1 + |x|2)

12 .

The creation operator, C, is injective, since its null space on all distributions is

spanned by ex2/2. If g ∈ C∞c (R) satisfies

(1.8)

∫e−x

2/2g(x)dx = 0 then defining

ECg = −ex2/2

∫ x

−∞e−s

2/2g(s)ds = ex2/2

∫ ∞x

e−s2/2g(s)ds ∈ C∞c (R)

given an operator satisfying CECg = g.

Since e−s2/2 is montonic decreasing near infinity it follows that if g ∈ S(R) sat-

isfies the same constraint then the integrals are bounded by CN |x|−N exp(−x2/2)near infinity. The equation then gives corresponding decay of the derivative anddifferentiating shows iteratively that

(1.9) EC : g ∈ S(R);

∫e−x

2/2g(x)dx = 0 −→ S(R), CEC = Id .

If v ∈ S(R) then

(1.10)

∫|Cv(x)|2dx =

∫R(|dvdx|2 + x2|v|2)dx−

∫Rxd|v|2

dxdx

=

∫R

(|dvdx|2 + x2|v|2 + |v|2)dx


is a Hilbert norm on H1iso(R). Thus extending by continuity gives a bounded linear

operator

(1.11) EC : g ∈ L2(R);

∫e−x

2/2g(x)dx = 0 −→ H1iso(R)

and so

(1.12) C : H1iso(R) −→ L2(R) is injective with range

(exp(−x2/2)

)⊥.

In particular, C is Fredholm.The dual of H1

iso(R) with respect to the L2 pairing is

(1.13) H−1iso (R) = H−1(R) + 〈x〉L2(R)

although this is not used below. Passing to the adjoint it follows that

(1.14) A : L2(R) −→ H−1iso (R) is surjective with Nul(A) = sp(e−x2/2).

A right inverse from C∞c (R) to S(R) is given by

(1.15) EAf(x) = u(x) = e−x2/2

∫ x

−∞es

2/2f(s)dx ∈ C∞(R), Au = f.

In x > a, supp(f) ⊂ [−a, a], u = ce−x2/2, c =

∫R e

s2/2f(s), so indeed

EA : C∞c (R) −→ S(R) ⊂ H1iso(R).

For u ∈ S(R) integration by parts gives

(1.16)

∫R|Au|2dx =

∫R

(|dudx|2 + x2|u|2)dx+

∫Rxd|u2|dx

dx

=

∫R(|dudx|2 + x2|u|2 − |u|2)dx

=⇒ ‖EAf‖2H1iso≤ ‖f‖2L2 + 2‖EAf‖2L2 .

Combining (1.12) and (1.14) it follows that

Lemma 1. The shifted harmonic oscillator

(1.17) H + 1 = AC : H1iso(R) −→ H−1iso (R) is an isomorphism

with inverse which restricts to a compact self-adjoint operator

(1.18) (H + 1)−1 : L2(R) −→ H1iso(R) → L2(R).

Proof. The open mapping theorem shows that (H + 1)−1 is a bounded linear mapfrom H−1iso (R) to H1

iso(R). That (H+ 1)−1 restricts to a compact operator on L2(R)follows from the compactness of the inclusion of H1

iso(R) in L2(R).

If we modify the generalized inverse EA to have range in the orthocomplementof the null space of A :

(1.19) EAf = EAf −1

π(f, exp(−x2/2) exp(−x2/2), EA : S(R) −→ S(R)

then (H+1)−1 = ECEA on S(R). This shows that the H+1 is a bijection on S(R)and the evident symmetry of H + 1 on S(R) :

(1.20) ((H + 1)φ, ψ) = (φ, (H + 1)ψ) =⇒(f, (H + 1)−1g) = ((H + 1)−1f, g), f = (H + 1)φ, g = (H + 1)−1ψ ∈ S(R)


transfers to the symmetry of (H + 1)−1 on the dense subspace S(R) from whichself-adjointness follows.

Thus L2(R) has an orthonormal basis of eigenfunctions of (H+1)−1. If ψ ∈ L2(R)and (H + 1)−1ψ = λψ with λ > 0 then ψ ∈ H1

iso(R) and (H + 1)ψ = λ−1ψ is aneigenfunction for H + 1. The commutation relation, valid initially for the action onS(R)

[H + 1, C] = 2C =⇒ [(H + 1)−1, C] = −2(H + 1)−1C(H + 1)−1 on H1iso(R)

using the density of S(R) in H1iso(R) which follows from the properties of EC . Thus

an eigenfunction ψ satisfies

(1.21) (H + 1)−1Cψ = C(H + 1)−1ψ − 2(H + 1)−1C(H + 1)−1ψ =⇒

(H + 1)−1Cψ =λ

1 + 2λCψ.

So Cψ is also an eigenfunction and in particular Cψ ∈ H1iso(R). This argument can

therefore be repeated and shows that Ckψ ∈ H1iso(R) satisfies

(1.22) (H + 1)Ckψ = (1

λ+ 2k)Ckψ, ∀ k

and these are all non-trivial eigenfunctions which are orthogonal in L2(R).The analogous commutation relation for the annihilation operator [H + 1, A] =

−2A shows that (1− 2λ)(H + 1)−1Aψ = λAψ, so unless 1− 2λ = 0, Aψ ∈ H1iso(R)

is again an eigenfunction. This can be iterated to show that Akψ ∈ H1iso(R) is an

eigenfunction

(H + 1)Akψ = (1

λ− 2k)Akψ,

unless 2kλ = 1 in which case Akψ = 0 since (H + 1)−1 is injective. In fact thismust occur, i.e. λ = 1/2j for some j, since otherwise, for sufficiently large j, Ajψis an eigenfunction of (H + 1)−1 with a negative eigenvalue. However integrationby parts shows that

(1.23) ((H + 1)u, u) ≥ 0 ∀ u ∈ S(R).

The operator ECEA = (H + 1)−1 : S(R) −→ S(R) so taking u = ECEAf showsthat ((H + 1)−1f, f) ≥ 0 for f ∈ C∞c (R). It follows that (H + 1)−1 is non-negativeso such a negative eigenvalue cannot occur. So indeed the eigenvalues of (H + 1)−1

are just (2 + 2k)−1, of multiplicity 1 with eigenspace the multiples of the Hermitefunction sp(Ck exp(−x2/2)), k ∈ N0.

Proposition 1. The eigenfunction expansion

(1.24) u =∑k

(u, ek)ek

converges in Hsiso(R) for any s ∈ R, where as spaces

(1.25)Hs

iso(R) = Hs(R) ∩ 〈x〉−sL2(R), s ≥ 0, Hsiso(R) = Hs(R) + 〈x〉−sL2(R), s ≤ 0.

Furthermore the L2 normalized Hermite functions are such that for any k contin-uous seminorm ‖ · ‖ on S(R) there is a bound

(1.26) ‖ej‖ ≤ CN (1 + j)N+1 ∀ j


and for any N there is a continuous norm ‖ · ‖(N) on S(R) such that

(1.27) |(u, ej)| ≤ (1 + j)−N‖u‖(N), ∀ φ ∈ S(R);

these estimates are uniform on compact subsets of S(R).

Proof. The rapid decay of the coefficients in (1.27) follows from the fact that‖u‖(N) = ‖(H + 1)Nu‖L2 is a continuous seminorm on S(R) so

(1.28) (2j + 2)N |(u, ej)| = |(u, (H + 1)Nej)|= |(H + 1)Nu, ej)| ≤ ‖(H + 1)Nu‖L2 = ‖u‖(N)

is bounded and uniformly so on compact sets.For any eigenfunction ACej = (2j + 1)ej so ‖Cej‖L2 = (ACej , ej) = (2j + 1)

from which it follows that

(1.29) ej+1 = (2j + 1)−12Cej , ej = (2j − 1)−1Aej+1

and

(1.30) ‖ej‖H1 ≤ (2j + 1) =⇒ sup |ej(x| ≤ C(2j + 1)

by Sobolev embedding.The finite sums of the supremum norms on Dkej and xkej , k ≤ N, give a

complete set of on S(R), i.e. bound and other continuous norm. These can beexpressed in terms of the creation and annihilation operators

(1.31)dk

dxk= (A− C)k/2k, xk = (A+ C)k/2k

and expanding these out and using (1.29) to evaluate the products of the A and Cand to bound the Hermit functions shows that

(1.32) |dkejdxk|+ |xkej | ≤ C(k)(2j + 2k)k max

|l|≤ksup |ej+l| ≤ C(k)(2j + 2k)k+1.

This gives the polynomial bound (1.26).It follows by combining these two estimates that the series (1.24) is absolutely

summable, and hence converges with respect to, any continuous norm on S(R).Conversely of course if the series converges in this sense then the sum is in S(R)and it is just the Fourier-Bessel series for the sum.

The higher dimensional case follows by easy computation:

Proposition 2. The Hermite functions for the harmonic oscillator on Rn give acomplete orthonormal basis

(1.33) eγ(x) = eγ1(x1) . . . eγn(xn)

of L2(Rn) which are polynomially bounded with respect to any continuous seminormon S(Rn) :

(1.34) ‖eγ‖ ≤ C(1 + |γ|)N

and for which the Fourier-Bessel coefficients are uniformly rapidly decaying

(1.35) φγ = (φ, eγ) =⇒ |φγ | ≤ CN (1 + |γ|)−N

on compact sets of S(Rn), ensuring the convergence of (1.1) in S(Rn).


2 Symbols and quantization

Recall that we constructed a parameterix for an elliptic operator P (D) withconstant coefficients and used it to prove local elliptic regularity. Ellipticity impliesthat the characteristic polynomial P (ξ) has real zeros (if any) only in a compact setand if we choose an appropriate χ ∈ C∞c (R) which is equal to one in a neighbourhoodof the zeros then

(2.1) a(ξ) =1− χ(ξ)

P (ξ)∈ SM (Rn), M = −m.

Technically we define a to be zero where the numerator vanishes.The symbol space of order M is defined by estimates on the growth of the

derivatives like those of a polynomial

(2.2) supξ

(1 + |ξ|)−M+|α||Dαξ a(ξ)| <∞ ∀ α.

Using an iterative formula for the derivatives we showed that a ∈ S−m(Rn). Weshowed earlier that if a ∈ SM (Rn) then its inverse Fourier transform, A ∈ S ′(Rn),

A = a, satisfies

(2.3) singsupp(A) ⊂ 0, (1− χ(x))A(x) ∈ S(Rn), A ∈ H−M−s(Rn), s > n/2

where χ ∈ C∞c (Rn) is again equal to 1 near 0. We use this below in the strongerform (which follows from the proof) that

(2.4) SM (Rn) 3 a −→ (1− χ)A ∈ S(R), A = a,

F : SM (Rn) −→ H−M−s(Rn) are continuous.

The distribution χA was used as a convolution operator to deduce smoothness ofsolutions to P (D)u = f – in particular to show that singsupp(u) = singsupp(f),which is one form of elliptic regularity.

This convolution operator has kernel χ(x− y)A(x− y) ∈ S ′(R2n) which is onlysingular at the diagonal, x = y, and which we have arranged to be supported closeto the diagonal as well. The leading estimate in (2.2) shows the boundedness ofthese operators on Sobolev spaces:

(2.5) a ∈ SM (Rn) =⇒ a(D) = A∗ : Ht(Rn) −→ Ht−M (Rn), ∀ t ∈ R.

The convenient notation a(D) for this convolution operator extends the standardnotation for constant coefficient differential operators P (D) where P = P (ξ) ∈Sk(Rn) is a polynomial. Then, for the a in (2.1), when P is elliptic,

P (D)a(D) = a(D)P (D) = Id−e(D)

where e(D) arises from e ∈ S(Rn) (in fact compactly supported) and so is a smooth-ing operator. This is what we want to extend to variable coefficient elliptic differen-tial operators. The pseudodifferential operators used to carry this out correspondto a smooth family of such convolution operators and the main problem is just tomake sense of this statement.

As noted above, we could work with the symbol spaces ‘with bounds’ SM (Rn)and it is conventional to do so for several, good, reasons. Some of these disappearwith the approach I am taking here, so instead I concentrate on the smaller space of‘classical’ symbols – with which we are really more familiar. Replacing these by thefull symbol spaces involves little more than change of notation. An example of a


classical symbol is precisely a in (2.1). It differs from a general symbol in the senseof (2.2) by ‘having an asymptotic expansion in smooth homogeneous functions atinfinity’. Rather than discuss the usual formulation of this concept let us look at ain (2.1) in terms of the radial compactification of Rn.

Recall that we discussed this compactification earlier, by identifying Rn withthe hyperplane ηn+1 = 1 in Rn+1 and then taking the stereographic projection tothe half-sphere (not the full sphere minus a point although we did that two). Thismeans considering the bijection

(2.6) F : Rn 3 ξ 7−→ η =(ξ, 1)

〈ξ〉∈ Sn ∩ ηn+1 > 0, 〈ξ〉 = (1 + |ξ|2)

12 .

Since it is topologically, and differentially, a ball let me write the closure of theimage as Bn, even though it is the half-sphere Sn ∩ ηn+1 ≥ 0.

Exercise 2. Write out an explicit ‘flattening’ of the half-sphere to a ball, making surethat you do not inadvertently introduce a square-root singularity at the boundary.

We can use F to identify a function f on Rnξ with the function (F−1)∗f on

the interior of Bnη . For instance (F−1)∗〈ξ〉−1 = ηn+1|Bn is smooth right up to theboundary where it vanishes simply – it is a defining function for the boundary ofBn. The point of all this discussion is that with a in (2.1),

(F−1)∗a = ηmn+1b, b ∈ C∞(Bn) = C∞(Rn+1)∣∣Bn .

That is, transferred to the ball, the upper half-sphere, it becomes a smooth functionwhich vanishes to precisely order m at the boundary (in this case m is an integerbut we do not assume this in general).

Definition 1. The space of classical symbols of real order s is defined to be

(2.7) Sscl(Rn) = F ∗(η−sn+1C∞(Bn)

)⊂ Ss(Rn).

The sign reversal in the order here is just the usual historical baggage. In par-ticular

(2.8) Sscl(Rn) ⊂ Ss+kcl (Rn), k ∈ N0

but there is no such inclusion for non-integral k.Really of course (2.7) is by way of a theorem since it claims that the estimates

(2.2) hold with M = s for all a ∈ Sscl(Rn). If a ∈ Sscl(Rn) then a = F ∗(ηsn+1b),b ∈ C∞(Bn), where b is certainly bounded, so |a| ≤ C〈ξ〉s which is the top estimatein (2.2). The (infinitely many) other estimates reduce to the same bound afterapplication of any number (repeated arbitrarily) of the vector fields ξi∂ξj .

Lemma 2. There are vector fields on Bn (smooth first order differential operatorswithout constant terms) Vij such that ξi∂ξj (F ∗a) = F ∗(Vija) and these vector fieldsare tangent to the boundary of Bn, meaning that Vijηn+1 = ηn+1eij , eij ∈ C∞(Bn).

Proof. For the moment left as an exercise.

This shows that the assertion in (2.7) does indeed hold.One small advantage of considering classical symbols is that we are already

pretty familiar with spaces of smooth functions such as C∞(Bn) although there arethings to check to feel confident about it. To define C∞(Bn) there are two obviouschoices, we can say u ∈ C∞(Bn) if u = u

∣∣Bn for some u ∈ C∞(Rn) or we can

just demand that u is smooth in the interior of Bn (which remember is the closed


ball) with all derivatives uniformly bounded. By integration the second definitionmeans all derivatives are continuous up to the boundary. If we take the second,apparently weaker definition, then the topology is given by the supremum normson the derivatives and there is a nice theorem of Seeley that says there exists acontinuous linear extension map

(2.9) E : C∞(Bn) −→ u ∈ C∞c (Rn), supp u ⊂ K, Eu∣∣Bn = u

where we can take K to be the ball of radius 2 for instance. So these two definitionsare the same. If you use Borel’s Lemma you can see that the restriction map issurjective but Seeley gives quite a bit more.

So, it is perhaps a relief to know that the space of pseudodifferential operatorsΨm

cl,S(Rn), as defined below, is actually identified linearly, and topologically, withthe space

(2.10) 〈ξ〉mF ∗C∞(Bn;S(Rn)),

so ultimately with

(2.11) C∞(Bn;S(Rn)).

The idea of ‘quantization’ is to turn a function, such as one of these symbols, intoan operator.

Recall that a space like (2.11) is straightforward to define starting from thedefinition of C∞(Bn). We can start with the continuous maps from the ball tothe Frechet space S(Rn) forming C(Bn;S(Rn)). Then consider once-differentiableelements, so the difference quotients defined in the interior of the ball converge(in the topology of S(Rn)) pointwise and extend to define the derivatives whichare required to be in C(Bn;S(Rn)). This definition can then be iterated to defineinfinite differentiability and the space (2.11)

Exercise 3. Show that the space C∞(Bn;S(Rn)) is naturally identified with thesubspace of C∞ functions on the interior of Bnη × Rny for which

(2.12) sup〈y〉N |DαηD

βy (η, y)| <∞ ∀ N, α, β.

You might recall, or check, that S(Rn) is also represented by a simple space

under radial compactification. Namely S(Rn) = F ∗C∞(Bn) where C∞(Bn) is thespace of smooth functions on Rn with support in the (closed) ball Bn. These canalso be identified with the subspace of C∞(Bn) with all derivatives vanishing at theboundary – the existence of an extension map from this subspace is then clear! Wecan similarly define C∞(Bn;S(Rn)) and identify it with S(R2n).

It is convenient to introduce a notion of ‘support’ for our classical symbols,which distinguishes where an element of (2.11) is locally equal to an element of

C∞(Bn;S(Rn)). Namely we define the cone-support, as usual through its comple-ment

(2.13) For a ∈ C∞(Bnη ;S(Rn)y) conesupp(a) = (η, x) ∈ Sn−1 × Rn

∃ a ∈ C∞(Bn;S(Rn)) s.t. a = a near (ξ, x)

We can think of the cone support as represnting a cone in (Rnξ \ 0) × Rny , conicin the first variable, such that on the complementary open cone F ∗a is rapidly


decaying. Since the ‘comparison’ functions C∞(Bn;S(Rn)) form an ideal, and asfor the usual support of smooth functions

(2.14)conesupp(ab) ⊂ conesupp(a) ∩ conesupp(b),

conesupp(a) = ∅ ⇐⇒ a ∈ C∞(Bn;S(Rn))

although this last result is not quite obvious (you can see it by checking that the

absence of a point ξ, x) from conesupp(a) means that all derivatives of a restricted

to Sn−1 × Rn vanish near (ξ, x) so if conesupp(a) is empty these boundary valuesare all zero, from here it is not so hard).

Exercise 4. One can refine the definition of the cone support to be ‘uniform nearinfinity’ in the second variable. Define the uniform cone support uconesupp byworking on the space Bn×Bn in which the symbols are identified as C∞(Bn; C∞(Bn))using radial compactification in the second variable and the ‘trivial’ symbols areC∞(Rn; C∞(Bn). Then define uconesupp(a) to be the closed subset of Sn−1 × Bn)on the complement of which a symbol is locally, on Bn × Bn, equal to a ‘trivial’symbol. Over the interior of the ball in the second variable, which is to say over Rnin that variable the definitions are the same. For classical symbols with Schwartzcoefficients, which is what we are discussing here, you can check that

(2.15) uconesupp(a) = conesupp(a) in Sn−1 × Bn.Familiarity with the topology here allows us to check easily that taking the

Hermite expansion in the second set of variables(2.16)

C∞(Bn;S(Rn)) 3 a(η, y) =∑γ

aγ(η)eγ(y), aγ =

∫a(·, y)eγ(y)dy ∈ C∞(Bn)

gives a series which converges in C∞(Bn;S(Rn)).

Lemma 3. For any continuous norm on C∞(Bn) and any N ∈ N there is a constantCN such that the coefficients in (2.16) satisfy

(2.17) ‖aγ‖ ≤ CN (1 + |γ|)−N .Proof. This follows directly from Proposition 2. Namely, as norms fixing the topol-ogy of C∞(Bn) we can take the supremum over Bn of the derivatives up to multi-order k. By definition, each of these derivatives defines an element of C(Bn;S(Rn))which therefore has compact image in S(Rn) so the estimates (2.17) follows from(1.35).

Each of the terms in (2.16) is already identified with an operator, since the

‘coefficients’ aγ ∈ S(Rn) are multipliers on Hs(Rn) for any s.

Proposition 3. For any a ∈ Smcl,S(Rn;Rn) = F ∗x−mn+1C∞(Bn;S(Rn), the series

(2.16) converges in B(Hs(Rn);Hs−m(Rn) for each s to define a bounded operatorwhich we write as

(2.18) a(Dy, y)u =∑γ

aγ(D)(eγ(y)u(y)).

Somewhat perversely the ‘coefficients’ are written on the right here. We will seebelow that these operators restrict to continuous linear maps S(Rn) −→ S(Rn),so the operators on the Sobolev spaces are all determined by continuous extensionand it is not necessary to distinguish then by the space on which they act.


Exercise 5. Show the boundedness of the operators or order m ≤ 0 on Lp(Rn)for 1 < p < ∞ and on the Holder spaces Ck,γ(Rn) for 0 < t < 1, k ∈ N0. Thehard work, really the Marcinkiewicz multiplier theorem in the first case and lesshard in the second (see [5, §7.9]) is to show the boundedness in this sense of theconvolution operators aγ(D) in terms of a continuous seminorm on S0(Rn). Forclassical symbols boundedness can be reduced to the homogenous case and theestimates in [5, §4.5] can be used instead.

Definition 2. We define the space Ψmcl,S(Rn) as consisting of the operators on

Sobolev spaces given by the sum (2.18) for a classical symbol a, so as the image ofa ‘right quantization’ map

(2.19) qR : Smcl,S(Rn;S(Rn)) −→ Ψmcl,S(Rn).

The right and ‘R’ refers to fact that the coefficients are put on the right. Notethat each of the summands in (2.18) has as Schwartz kernel

Bγ(x, y) = Aγ(x− y)eγ(y).

The estimates (2.17) and the continuity in (2.4) show the convergence of

(2.20) B(x, y) =∑γ

Bγ(x, y) =∑γ

Aγ(x− y)eγ(y) ∈ S ′(R2n),

In fact if we write z = x− y and set

(2.21) A(z, y) = B(z + y, y) =∑γ

Aγ(z)eγ(y) converges in

S(Rny ;H−m−s(Rn)), s > n/2,

and (1− χ(z))A(z, y) ∈ S(R2n).

Here χ ∈ C∞c (Rn) is identically equal to 1 near 0.We define a support set for these operator in terms of the symbol:-

(2.22) WF′(qR(a)) = conesupp(a) ⊂ Sn−1 × Rn.

This is also called the ‘operator wavefront set’ and has the property

(2.23) WF′(qR(a)) = ∅ =⇒ a ∈ C∞(Rn;S(Rn)) =⇒ B ∈ S(R2n)

where B is the Schwartz kernel of qR(a) given by (2.20).

Exercise 6. If you did Exercise 4 then you can define

(2.24) WF′S(qR(a)) = uconesupp(a) ⊂ Sn−1 × Bn.

This definition of Ψmcl,S(Rn(=) involves a degree of ‘handism’ – which I hope is

not a real word. Namely, We can do exactly the same and define another operator,from the same symbol a, by ‘left quantization’

(2.25) a(x,D)u =∑γ

eγ(x)(aγ(D)u).

In general a(x,D) 6= a(D, y). As we show below, the space of operators obtainedthis way is indeed the same as (2.18). That is (2.25) also defines a linear bijection

(2.26) qL : Smcl,S(Rn;Rn) −→ Ψmcl,S(Rn)

i.e. it has the same range.


There are other quantizations (especially ‘Weyl quantization’, see below). Moregenerally we can consider

(2.27) Smcl,S(Rn;R2n) = F ∗(η−sn+1C∞(Bn;S(R2n

(x,y)))

where we allow the ‘symbol’ to depend on both x and y and also ξ. Then theHermite expansion for functions on R2n gives

(2.28) Smcl,S(Rn;R2n) 3 b(x, y, ξ) =∑γ,γ′

eγ(x)bγ,γ′(ξ)eγ′(y)

where the convergence is strong enough that the operator

(2.29) b(x,D, y) =∑γ,γ′

eγ(x)(bγ,γ′(D)(eγ′(·)u(·)))

converges in B(Hs(Rn);Hs−m(Rn)). We can write this ‘general’ quantization mapas

(2.30) q : Smcl,S(Rn;R2n) −→ Ψmcl,S(Rn)

because it again has the same range.The Schwartz kernel of q(a) is given the series

(2.31) B(x, y) =∑γ,γ′

eγ′Aγ′,γ(x− y)eγ(y) ∈ S ′(R2n)

since this is true for the indivdual terms in (2.29) and the resulting series convergesin S ′(R2n). If we set x = z + y it follows that

(2.32) A(z, y) = B(z + y, y) ∈ S(Rny ;H−m−s(Rnz ), s > n/2

just as for the kernels right quantized operators.Left and right quantization are both bijections, but from this formula it is clear

that q is not.

Exercise 7. Find an explict symbol in the null space of q.

Exercise 8. If a ∈ Smcl (Rn) and e ∈ S(Rn) show that the kernel

(2.33) A(x− y)e(x+ y

2)

defines a bounded operator Hs(Rn) −→ Hs−m(Rn) for any s ∈ R – for instancedivide A(z) into a part A1 with compact support in |z| < 1 and a Schwarz part A2

and write(2.34)

A(x− y)e(x+ y

2) = A1(x− y)φ(x− y)e(

x+ y

1) +A2(x− y)(1− φ(x− y))e(

x+ y

2)

for an appropriate cutoff φ ∈ C∞c (Rn); then apply q to the first part and observe thatthe second part is in S(R2n). Show that this operator is independent of the choice ofcut-off and estimate the norm in terms of a seminorm on a(ξ)e(X) ∈ Smcl,S(Rn;Rn).Conclude that the Hermite expansion of symbols leads to a Weyl quantization mapqW with values in the bounded operators just as for qR and qL. Use the discussionin §5 to show that this is another bijection

(2.35) qW : Smcl,S(Rn;Rn) −→ Ψmcl,S(Rn)

which has the useful property (among others) that (qW (a))∗ = qW (a).


3 Algebra of pseudodifferential operators

Let me collect a substantial, although not exhaustive, list of properties of thespace Ψm

cl,S(Rn), given for definiteness sake, as above, as the image of the map

qR in (2.19). Note that the subscript cl refers to the ‘classical’ symbols and theS is there because the coefficients are, by assumption, in S(Rn). This space hascertain defects, in particular Id /∈ Ψ0

cl,S(Rn) because of the assumed rapid decayof the coefficients. There are many variants of these spaces but we have to startsomewhere.

Theorem 1. The spaces Ψmcl,S(Rn) have the following properties:

(P1) (Boundedness) The elements of Ψmcl,S(Rn) define bounded operators

(3.1) A : Hs(Rn) −→ Hs−m(Rn) ∀ s ∈ R.

For each j and A ∈ Ψmcl,S(Rn), xjA ∈ Ψm

cl,S(Rn), the commutator [xj , A] ∈Ψm−1

cl,S (Rn) and in consequence

(3.2)A : 〈a〉tHs(Rn) −→ 〈x〉−tHs−m(Rn) ∀ t, s

A : S(Rn) −→ S(Rn), A : S ′(Rn) −→ S ′(Rn).

(P2) (Left-right reduction) The maps qL and qR are injective and qR, qL and qhave the same range for each m ∈ R, Futhermore

(3.3) qR(a) = qL(b), a, b ∈ C∞(Bn;S(Rn)) =⇒ conesupp(a) = conesupp(b).

(P3) (Symbol map) For any m ∈ R, Ψm−1cl,S (Rn) ⊂ Ψm

cl,S(Rn) and there is a shortexact sequence

(3.4) Ψm−1cl,S (Rn) −→ Ψm

cl,S(Rn)σm−→ |ξ|mC∞(Sn−1;S(Rn))

where |ξ|mC∞(Sn−1;S(Rn)) should be interpreted as the space of those el-ements of C∞(Rn \ 0;S(Rn)) which are homogeneous of degree m in thefirst n variables. So σm is surjective and has null space exactly Ψm−1

cl,S (Rn).

(P4) (Multiplicativity) For any m and m′, in terms of operator composition

(3.5)Ψm

cl,S(Rn) Ψm′

cl,S(Rn) ⊂ Ψm+m′

cl,S (Rn),

σm+m′(AB) = σm(A)σm′(B), WF′(AB) ⊂WF′(A) ∩WF′(B).

(P5) (Adjoints) Taking adjoints with respect to Lebesgue measure,

(3.6)Ψm

cl,S(Rn) 3 A 7−→ A∗ ∈ Ψmcl,cS(Rn)

is an isomorphism and σm(A∗) = σm(A), WF′(A∗) = WF′(A).

(P6) (Asymptotic completeness) For any sequence Ak ∈ Ψm−kcl,S (Rn), k ∈ N0,

there exists A ∈ Ψmcl,S(Rn) such that

(3.7) A−N∑k=0

Ak ∈ Ψm−N−1cl,S (Rn) ∀ N ∈ N0.

This relationship is written

(3.8) A ∼N∑k=0

Ak =⇒WF′(A) ⊂⋃k

WF′(Ak).


(P7) (Pseudo/Microlocality) For any A ∈ Ψ∗cl,Rn(Rn),

(3.9)singsupp(Au) ⊂ singsupp(u), ∀ u ∈ S ′(Rn),

WF(Au) ⊂WF(u) ∩WF′(A), ∀ u ∈ S ′(Rn)

where the second of these implies the first.(P8) (Residual Algebra)

(3.10)⋂k∈N0

Ψm−kcl,S (Rn) = Ψ−∞S (Rn) ≡ S(R2n) = A ∈ Ψm

cl,S(Rn); WF′(A) = ∅

is the space of integral operators with Schwartz kernels

Bu(x) =

∫Rn

B(x, y)u(y)dy, B ∈ S(R2n) and B, B∗ : S ′(Rn) −→ S(Rn).

4 Elliptic regularity

Rather than proceeding directly to check all these properties, let’s see what wecan achieve by using them. Suppose that

(4.1) P (x,D) =∑|α|≤m

pα(x)Dα, pα ∈ C∞(Ω),

is a differential operator with smooth coefficients in some open set Ω ⊂ Rn. This isnot in our algebra unless the coefficients happen to be Schwartz functions on thewhole of Rn. Still, we are interested in the regularity of solutions to

(4.2) P (x,D)u = f, f ∈ C−∞(Ω), u ∈ C−∞(Ω).

Ellipticity for P in Ω means precisely that

(4.3) Pm(x, ξ) =∑|α|=m

pα(x)ξα 6= 0 on Ω× Rn \ 0.

Theorem 2. [Elliptic Regularity] If P as in (4.1) is elliptic in Ω in the sense of(4.3) then

(4.4)singsupp(u) = singsupp(P (x,D)u),

WF(u) = WF(P (x,D)u), ∀ u ∈ C−∞(Ω).

Proof. The inclusions

(4.5)singsupp(u) ⊃ singsupp(P (x,D)u),

WF(u) ⊃WF(P (x,D)u), ∀ u ∈ C−∞(Ω)

are consequences of the elementary properties of singsupp and WF (even if P (x,D)is not elliptic) so it is the reverse that requires proof. Moreover, these are localstatements in the sense that we just need to show

(4.6) x /∈ singsupp(P (x,D)u) =⇒ x /∈ singsupp(u)

and similarly for WF . So we can look near a particular point x ∈ Ω.Here is one way to see this using the properties of the pseudodifferential algebra

listed above

Lemma 4. Assuming that P (x,D) is elliptic:

(4.7)If φ ∈ C∞c (Ω) and ψ ∈ C∞c (Ω), supp(φ) ⊂ ψ = 1 then ∃ A ∈ Ψ−mcl,S(Rn),

WF′(A) ⊂ Sn−1 × supp(φ) s.t. A P (x,D)ψ = φ Id−E, E ∈ Ψ−∞S (Rn).


Notice that P (x,D)ψ ∈ Ψmcl,S(Rn) since now the coefficients are compactly sup-

ported in Ω, so the composition in (4.7) is defined and covered by (3.5).So, suppose we have obtained such an A where we choose φ to be identically 1

near x. We can choose another ψ′ ∈ C∞(Ω) (this is being a bit pedantic) so thatψψ′ = ψ and then we see that

(4.8) P (x,D)ψu = P (x,D)ψ(ψ′u) = φf + g, f = P (x,D)u,

supp(g) ⊂ φ 6= 1 =⇒ x /∈ supp(g),

and where everything is now in C−∞c (Ω) ⊂ S ′(Rn). So we can apply A on the leftand see that

(4.9)AP (x,D)ψ(ψ′u) = Aφf +Ag = φu− E(ψ′u) =⇒

φu = A(φf) +Ag + E(ψ′u).

The last term is in S(Rn) by (P8). Since x /∈ supp(g), and in particular x /∈singsupp(g), pseudolocality shows that x /∈ singsupp(Ag). Thus indeed we get (4.6)since x /∈ singsupp(P (x,D)u) implies x /∈ singsupp(φf), f = P (x,D)u, so x /∈singsupp(A(φf)) and then x /∈ singsuppφu and so finally x /∈ singsuppu sinceφ(x) 6= 0.

The proof for WF is the same, using microlocality instead of pseudolocality.

This shows the efficacy of the calculus of pseudodifferential operators, althoughyou might rightly object that the proof is a little clumsy because of the need toinsert all these cutoffs. This is indeed the case and we can get rid of that annoyanceby working with pseudodifferential operators on the open set Ω; this just requiresa bit more machinery, but really only about supports and such.

So to the

Proof of Lemma 4. Directly from the definitions(4.10)

φ Id ∈ Ψ0cl,S(Rn), σ0(φ Id) = φ(x),

P (x,D)ψ ∈ Ψmcl,S(Rn), σm(P (x,D)ψ) = Pm(x, ξ)ψ(x) =

∑|α|=m

pα(x)ψ(x)ξα.

Now we are looking for A ∈ Ψ−mcl,S(Rn) and for any choice

(4.11) AP (x,D)ψ ∈ Ψ0cl,S(Rn), σ(AP (x,D)ψ) = σ−m(A)Pm(x, ξ)ψ(x).

Since P (x,D) is elliptic, the function

(4.12) a0 =φ(x)

Pm(x, ξ)ψ(x)∈ |ξ|−mC∞(Sn−1;S(Rn))

(with the usual caveat that the function is taken to be zero outside the support ofthe numerator and remembering that supp(φ) is contained in the set where ψ = 1).So this is a legitimate choice for the symbol, being a smooth function homogeneousof degree −m as claimed. Set

A0 = qR(a0(1− µ(ξ))

where µ(ξ) ∈ C∞c (Rn) is equal to 1 in a neighbourhood of 0 – here we are usingthe surjectivity of the symbol map in (3.4) in a slightly stronger sense since we also


have WF′(A0) ⊂ Sn−1 ×K where K = supp(φ). With this initial choice

(4.13) E1 = A0P (x,D)ψ − φ Id ∈ Ψ0cl,S(Rn), σ0(E1) = a0Pmψ − φ = 0 =⇒E1 ∈ Ψ−1cl,S(Rn) and WF′(E1) ⊂ Sn−1 ×K.

Then we can continue by induction to construct a succession of operators Aj j = 0,. . . N so that

(4.14) Aj ∈ Ψ−m−jcl,S (Rn), WF′(Aj) ⊂ Sn−1 ×K s.t. N∑j=0

Aj

P (x,D)ψ = φ Id +EN+1,

EN+1 ∈ Ψ−N−1cl,S (Rn), WF′(EN−1) ⊂ Sn−1 ×K.

Then we can choose AN+1 = qR(aN+1)

(4.15) aN+1 =σ−N−1(EN+1)

Pm(x, ξ)∈ |ξ|−m−N−1C∞(Sn−1;S(Rn))

=⇒WF′(AN+1) ⊂ Sn−1 ×K.

This makes sense because of the ellipticity and the last condition in (4.14) whichensures that WF′(EN+1) ⊂ Sn−1 ×K.

Now we use the asymptotic completeness result to show that we can choose one

(4.16) A ∼∑j

A−j , A ∈ Ψ−mcl,S(Rn).

The inductive arrangement means that

(4.17) AP (x,D)ψ − φ Id ∈ Ψ−Ncl,S(Rn) ∀ N

which implies, by the last property above, that it is a smoothing operator, withkernel in S(R2n).

Note that we do not really need all the properties listed above to prove ellipticregularity. For instance, it is enough to stop at a sufficiently large N in the con-struction above, depending on the regularity of the distribution u involved. Still,it is neater to have one operator which works for all – just as it is convenient todispense with the cutoffs as we can do with a bit of work.

Exercise 9. Carry through the microlocal version of this construction. That is, if

Pm(x, ξ) 6= 0 for some x ∈ Ω and ξ ∈ Sn−1 show that there exists A ∈ Ψ−mcl,S(Rn),

ψ, φ ∈ C∞c (Ω) such that

(4.18) AP (x,D)ψ = φ Id−E, (x, ξ) /∈WF(E), φ(x) 6= 0.

Denoting the set of such ‘elliptic points’ of P as Ell(P ) ⊂ Sn−1 × Ω conclude thatfor any u ∈ C−∞(Ω),

(4.19) WF(u) ∩ Ell(P ) ⊂WF(P (x,D)u) ⊂WF(u).


5 Proof of properties of pseudodifferential operators

We start with (P1). Since yj is a multiplier on S(Rn) it follows immediately thatqR(a)yj = qR(ayj) ∈ Ψm

cl,S(Rn); this applies to any such multiplier, i.e. any smoothfunction of slow growth so for instance

(5.1) a(D, y)〈y〉M ∈ Ψmcl,S(Rn) ∀ M.

For a convolution operator a(D) where a ∈ Smcl (Rn) we know that

[xj , A] = xja(D)− a(D)yj = b(D), b(ξ) = i∂ξja(ξ).

Since ∂ξj : Smcl,S(Rn;Rn) −→ Sm−1cl,S (Rn;Rn), the convergence of the operator series

(2.18) shows that

(5.2) [xj , a(D, y)] = b(D, y), b(ξ, y) = i∂ξja(ξ, y).

Thus xja(D, y) ∈ Ψmcl,S(Rn) as well and, iterating, it follows that xαa(D, y) ∈

Ψmcl,S(Rn). Thus

(5.3) xαa(D, y)〈y〉M : Hs(Rn) −→ Hs−m(Rn), ∀ α, M, s.

This shows the first part of (3.2) and the second part follows from the fact that

(5.4) S(Rn) =⋂m

〈x〉−mHm(Rn), S(Rn) =⋃m

〈x〉−mHm(Rn).

The injectivity of qR in (P2) is a weak form of the Schwartz kernel theorem, herejust amounting to the fact that a can be recovered from the kernel of the operatorqR(a). The kernel of qR(a) is given by B in (2.20). Using (2.21) we can recover theexpansion of the symbol by taking the Hermite expansion of A(z, y) in y and thenthe Fourier transform of the resulting distribution.

From this the exactness of the symbol sequence in (3.4) follows with σm(qR(a)) =|ξ|mam, am = xma

∣∣Sn−1×Rn being the restriction of a ∈ x−mn+1C∞(Bn;S(Rn)) to the

boundary. That is to say, after removal of a factor of η−mn+1, (3.4) just arises fromthe exact ‘restriction sequence’

ηn+1C∞(Bn;S(Rn) −→ C∞(Bn;S(Rn) −→ C∞(Sn−1;S(Rn).

Exercise 10. (Maybe on a second reading ...) Show that for the more generaloperators

(5.5) ΨmS (Rn) = qR(SmS (Rn;Rn)

defined by ‘non-classical symbols’ the discussion above carries over except that theprincipal symbol sequence has to be interpreted in terms of a quotient

(5.6) Ψm−1S (Rn) −→ Ψm

S (Rn)σm−→ (SmS /S

m−1S )(Rn;Rn)

Go through the rest of the proofs checking what happens in the non-classical case.

Next consider the ‘residual algebra’ defined to be

(5.7)Ψ−∞S (Rn) =

⋂k

Ψm−kcl,S (Rn), m ∈ R =⇒ Ψ−∞S (Rn) =

⋂M

ΨMcl,S(Rn)

B : Ψ−∞S (Rn)←→ S(R2n)

where B is the map to the Schwartz kernel. Here the first part is the definitionand the claim is that the result is independent of m hence the second part holds.So, fix m and consider qR(a) ∈ Ψ−∞S (Rn). The symbol map, applied iteratively,


shows that all terms in the Taylor series of a at the boundary of Bn vanish. Thusa ∈ C∞(Bn;Rn). This in turn means that F ∗a ∈ S(R2n) and so the kernel of qR(a)is given by (2.20) where the series now converges in S(R2n). Conversely, takinga kernel B ∈ S(R2n) and writing it as B(x, y) = A(x − y, y) with A ∈ S(R2n

the inverse Fourier transform in z of A(z, y) is a ‘trivial’ symbol F ∗a with a ∈C∞(Bn;S(Rn)) such that qR(a) has kernel B. This proves (5.7) and hence (P8).

Asymptotic completeness as in (P6) is then a form of Borel’s Lemma. Namely,

If Ak = qR(ak) ∈ Ψm−kcl,S (Rn) then xmn+1ak ∈ xkn+1C∞(Bn;S(Rn)). An appropriate

form of Borel’s Lemma shows that there exists b ∈ C∞(Bn;S(Rn)) such that b −N∑k=0

xmn+1ak ∈ xN+1n+1 C∞(Bn;S(Rn)). Then A = qR(x−mn+1b) satisfies (3.7). From the

discussion of the residual algebra above, the difference of two such ‘asymptoticsums’ is in Ψm−N

cl,S (Rn) for all N and hence in Ψ−∞S (Rn).Next consider the quantization map q with the objective being to show that it

has range contained in that of qR; this is really the crucial result here. The map qturns a symbol F ∗a, a ∈ x−mn+1CI(Bn;S(R2n) into an operator q(a) by expanding

it in Hermite series on R2n

(5.8) a(x, y, ξ) =∑γ,γ′

eγ(x)aγ,γ′(ξ)eγ′(y).

Then each term becomes the operator

(5.9) u 7−→ q(eγaγ,γ′eγ′)u = eγ(x)(Aγ,γ′∗)(eγ′u).

Then we use the convergence of (5.8) to sum (5.9) over γ, γ′ in B(Hs, Hs−m) forany s. The continuity of this operator can be expressed by

(5.10) ‖q(a)‖B(Hs,Hs−m) ≤ ‖a‖s,m

for some continuous seminorm ‖ ·‖s,m on x−mn+1C∞(Bn;S(R2n). The Schwartz kernelis given by the corresponding series (2.31).

Given a ∈ Smcl,S(Rn;R2n) we ‘prepare’ it by writing

(5.11) a = a1 + a2, F∗ai(x, ·, y) = A(x, ·, y), F ∗a(x, ·, y) = A(x, ·, y),

A1(x, z, y) = φ(z)A(x, z, y), φ ∈ C∞c (Rn), φ(z) = 1 in |z| < 1.

That is, treating x and y as parameters we divide a into a part for which F ∗a1has Fourier transform compactly supported near 0 in the dual variable z. Now, theproperties of symbols imply that A2 ∈ S(R3n) is Schwartz in all variables – being aSchwartz function of z with values in the Schwartz functions in the other variables.From

The Hermite expansion respects this deivision so

(5.12) q(a) = q(a1) + q(a2), q(a2) ∈ Ψ−∞S (Rn)

where the last statement follows from the formula for the Schwartz kernel and (5.7).The continuity of (5.10) and the fact that it is explicitly defined on products

e(x)a(ξ)f(y), e, f ∈ S(Rn), a ∈ C∞(Bn) means that we can use any expansion asin (5.8):

(5.13) a(x, ξ, y) =∑γ,γ′

lγ(x)aγ,γ′(ξ)rγ′(y)


provided it converges in x−mn+1C∞(Bn;S(R2n) and has the correct limit. In particularthis shows that

(5.14) [xi, q(a)] = q((xi − yi)a) = q(−i∂ξja), ∂ξja ∈ x−m+1n+a C∞(Bn;S(R2n)

comes from a symbol one order lower. Iterating this statement we see that

(5.15) q((xi − yi)αa) = q((−i∂ξ)αa), ∂αξ a ∈ x−m+|α|n+a C∞(Bn;S(R2n)

We showed that q(a2) ∈ Ψ−∞S (Rn) so are reduced to examing q(a1) for the firstpart a1 in (5.11); let me again denote this a to simplify the notation but now weknow that it is ‘prepared’ in the sense that

(5.16) q(a(·, x, y)) = q(a(·, x, y)φ(x− y)), φ ∈ C∞c (Rn)

since this formula holds for the Schwartz kernels.Consider Taylor’s formula with remainder for g(·, x, y) = ηmn+1b(·, x, y) (just re-

moving the leading factor to make everything smooth) around x = y :

(5.17) g(·, x, y) =∑|α|≤N

(x− y)αgα(·, y) +∑

|α|=N+1

(x− y)αhα(·, x, y).

This identity only holds in |x− y| < 2. If we insert the cutoff we see that(5.18)

a(·, x, y)φ(x−y) =∑|α|≤N

(x−y)αaα(·, y)φ(x−y)+∑

|α|=N+1

(x−y)αbα(·, x, y)φ(x−y)

where aα = η−mn+1gγ , bγ = η−mn+1hγ have the leading powers restored. Now all terms

are in the symbol space Smcl,S(Rn;R2n) and the identity (5.18) holds globally.

The identities (5.16) and (5.15) show that

(5.19) q(a) = q(φ(x− y)a) =∑|α|≤N

q((−i∂ξ)αaα(·, y)φ(x− y))

+∑

|α|=N+1

q((−i∂ξ)αbα(·, x, y)φ(x− y))

for any N. As noted above, the Schwartz kernel an operator such as

q((−i∂ξ)αaα(·, y)φ(x− y)

can be computed from any convergent expansion (5.13) for its symbol. Set

(5.20) fα(·, y) = (−i∂ξ)αaα(·, y) ∈ Smcl,S(Rn;Rn).

Then taking the expansion of fα(·, y) as usual in a ‘single’ Hermite expansion andmultiplying by φ(x− y)

(5.21) fα(·, y)φ(x− y)) =∑γ

fα,γ(·)eγ(y)φ(x− y)

converges in Smcl,S(Rn;R2n). We can further expand each eγ(y)φ(x−y)] =∈ S(R2n)

in a double Hermite series and get a rapidly converging expansion as in (5.13) withmore indices. Summing the series for the Schwartz kernel over this series first wefind that the kernel of fα(·, y)φ(x− y)) is

(5.22) Bα(x, y) =∑γ

Eα,γ(x− y)eγ(y)φ(x− y), Eα,γ = fα,γ .


However, the preparation of the symbol above means that all the Eα,γ(z) are sup-ported in |z| < 1 so in fact

(5.23) q(fα(·, y)φ(x− y)) = qR(fα(·, y))

by equality of the corresponding Schwartz kernels.Thus, for a prepared symbol the sum on the right in (5.19) is of the form

(5.24)∑|α|≤N

Aα, Aα = qR(aα) ∈ Ψm−|α|cl,S (Rn).

So now we may invoke the asymptotic summability result shown above to find oneA ∈ Ψm

cl,S(Rn) such that

(5.25) A ∼∑|α|

Aα.

It follows now from (5.19) again that for all N

(5.26) E = q(a)−A = q(aφ(x− y))−A = A(N)

+∑

|α|=N+1

q((−i∂ξ)αbα(·, x, y)φ(x− y)), A(N) ∈ Ψm−N−1cl,S (Rn).

Here there is a single operator, so with a fixed Schwartz kernel, expressed in termsof symbols all of order m−N − 1. The regularity of the kernels in (2.32), togetherwith the fact that they are all supported in |x− y| < 1, shows that E has kernel inS(R2n) finally we have shown that

(5.27) q(a) = qR(a)

and the range of q is contained in the range of qR.The converse, that the range of qR is contained in that of q, is also readily

checked by a similar argument. Namely ‘preparing’ the symbol a ∈ Smcl,S(Rn;Rn)

as in (5.11) allows us to insert a cutoff φ(x−y) to see that qR(a) = q(φ(x−y)a)+ewhere e is residual. Thus it is enough to show that any residual operator is inthe range of q which (is not very important and) follows from the fact that any

B ∈ S(Rn) (its kernel) is the restriction to z = x− y of B(x, z, y) ∈ S(R3n) and so

is the kernel of q(b) where b ∈ C∞(Bn;S(R2n) is obtained from B by inverse Fouriertransform in z.

Directly from the definitions taking adjoints, with respect to Lebesgue measureas usual, interchanges qR and qL :

(5.28) (qR(a))∗ = qL(a) =⇒ σm(qR)∗) = σm(qR(a))

which is (P5). Using this we conclude that the range of qR is contained in that ofqL, so these are equal.

The fundamental property is multiplicativity but this now follows easily. Namelyif A ∈ Ψm

cl,S(Rn), B ∈ Ψm′

cl,S(Rn) the composite A B is certainly well-defined as an

operator on S(Rn). Moreover we now know that we can write A = qL(a), B = qR(b)and then from the definitions (using the series expansions and the composition ofconvolution operators in the middle)

(5.29) A B = q(a(·, x)b(·, y)) ∈ Ψm+m′

cl,S (Rn) =⇒ σm+m′(A B) = σm(A)σm′(B).

This is (P4) with the formula for the principal symbol following fromThis, I think, only leaves (P7) which is a good exercise!


6 Diffeomorphism-invariance

For any open set Ω ⊂ Rn we can consider the

(6.1) Ψmcl,c(Ω) ⊂ Ψm

cl,S(Rn)

with the defining support property

(6.2) A ∈ Ψmcl,S(Rn) s.t. A, A∗ : S(Rn) −→ C∞c (Ω) ⊂ S(Rn).

Lemma 5. The defining condition (6.2) that A ∈ Ψmcl,c(Ω) is equivalent to the

condition that the Schwartz kernel B ∈ S ′(R2n) of A have compact support withinthe product

(6.3) supp(B) b Ω× Ω.

Proof.

Proposition 4. If F : Ω −→ Ω′ is a diffeomorphism between open subsets of Rnwith inverse G : Ω′ −→ Ω then

(6.4)AFu = F ∗(AG∗u), u ∈ C∞c (Ω) defines a bijection

Ψmcl,c(Ω

′) 3 A 7−→ AF ∈ Ψmcl,c(Ω).

7 Localization

We define the large space Ψmcl (Ω) as consisting of continuous linear operators

(7.1) A : C∞c (Ω) −→ C∞(Ω) s.t. Aφ =∑i,j

χiAijχj , Aij ∈ Ψmcl,S(Rn)

for one (and hence any) partition of unity χi ∈ C∞c (Ω) for Ω with locally finitesupports (meaning that for any K b Ω there are only finitely many j such thatsupp(χj) ∩K 6= ∅.

Ψmcl,c(Ω) ⊂ Ψm

cl,P(Ω) ⊂ Ψmcl (Ω)

References

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[2] , On singular integrals, American Journal of Mathematics 78 (1956), no. 2, 289309.[3] L. Hormander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501–517.

[4] , The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32

(1979), 359–443.[5] , The analysis of linear partial differential operators, vol. 1, Springer-Verlag, Berlin,

Heidelberg, New York, Tokyo, 1983.

[6] , The analysis of linear partial differential operators, vol. 3, Springer-Verlag, Berlin,Heidelberg, New York, Tokyo, 1985.

[7] Mark Joshi, Lectures on pseudo-differential operators, arXiv:math/9906155.

[8] J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operators, Comm. PureAppl. Math. 18 (1965), 269–305.

[9] Richard S. Palais, Seminar on the Atiyah-Singer index theorem, With contributions by M. F.

Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of MathematicsStudies, No. 57, Princeton University Press, Princeton, N.J., 1965. MR 33 #6649

[10] R. T. Seeley, Refinement of the functional calculus of Calderon and Zygmund, Nederl. Akad.Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 521–531. MR 37 #2040

[11] M.A. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin-

Heidelberg-New York, 1987, (Nauka, Moscow, 1978).[12] Michael E. Taylor, Pseudodifferential operators, Princeton University Press, Princeton, N.J.,

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Documents

Introduction - Mathematicsrbm/18-155-F15/PseudodifferentialOperators.pdfthe original papers, on pseudodi erential operators. Calder on and Zygmund (singular integral operators) [1,