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, suggestions and corrections.
Introduction
These pages should ultimately be incoporated into a new version of the ‘Graduate
Analysis’ notes for 18.155 at MIT. The only ‘innovation’ here, relative to traditional
treatments, is the use of expansions in Hermite functions to define the operators
and prove L2 (i.e. Sobolev) boundedness.
The material here can be found in many places although it may be difficult
initially to see the relationships between the different treatments. First there are
the original papers, on pseudodifferential operators.
• Calderón and Zygmund (singular integral operators) [1, 2].
• Seeley in [9] and [10].
• Kohn and Nirenberg [8].
• Hörmander [3] treats the ‘standard’ calculus and the later paper [4] extends
this greatly.
Then there are various newer sources:• Hörmander’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 with
all derivatives, on Rn ,
X
X
(0.1) P (x, D)u(x) =
Pα (x)Dα u(x), u ∈ S(Rn ), P (x, ξ) =
Pα (x)ξ α
|α|≤m
|α|≤m
then it can be written in terms of the Fourier transform of u as
Z
−n
(0.2)
P (x, D)u(x) = (2π)
eix·ξ P (x, ξ)û(ξ)dξ.
Rn
This a rather ‘fake’ formula in the sense that it only works because we can decompose the integral into a finite sum of terms, using the fact that P (x, ξ) is a
polynomial in ξ, so that it becomes the inverse Fourier transform
Z
X
(0.3)
P (x, D)u(x) =
Pα (x)(2π)−n
eix·ξ ξ α û(ξ)dξ
Rn
|α|≤m
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 makes
sense for the space of ‘symbols of order m’ which is usually taken to mean the
space of smooth functions on Rn × Rn which satisfy estimates (which we have
encountered before)
|∂xα ∂ξβ a(x, ξ)| ≤ Cα,β (1 + |ξ|)m−|β| .
(0.4)
The most important class of operators correspond to classical symbols which are
discussed below, but the term means that they have asymptotic expansions in
homogeneous functions in ξ as |ξ| → ∞.
Still the theory passes through the more general symbols (0.4) (or even more
general ones) and one reason for this is that the oscillatory integrals are defined
using density arguments from rapidly decreasing functions and, just as for bounded
continuous functions on R, rapidly decreasing functions are not dense in classical
symbols in the natural topology but are dense (with loss of in the order) in the
symbol topology given by the constants in (0.4).
In this brief introduction to pseudodifferential operators I want to avoid this
discussion and get to the local theory as quickly as possible. The approach is based
on the observation that the formula (0.3) ‘works’ because the symbol P (x, ξ) is
in 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); the
general 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 identity
is thereby excluded from the ‘algebra’ but this is not an issue for local regularity
questions. Then the simple idea is that one can expand these functions in terms of
the Hermite functions eγ ∈ S(Rn ), the eigenfunctions of the harmonic oscillator,
Z
X
eγ (x)aγ (ξ), aγ (ξ) =
a(x, ξ)eγ (x)dx.
(0.5)
a(x, ξ) =
Rn
γ
This series converges rapidly and the coefficients aγ (ξ) are classical symbols. Then
the operator can be defined directly as in (0.3)
Z
X
(0.6) a(x, D)u(x) =
(2π)−n eγ (x)
eix·ξ aγ (ξ)û(ξ)dξ
γ
Rn
=
X
eγ (x)aγ (D)u, aγ (D)u = Aγ ∗ u
γ
where Âγ = 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 these
operators are easily defined and bounded. The crucial theorem is that the product
of 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 psuedodifferential operators with the coefficients on the right. This may be confusing
but is a decision easily reversed and of no particular consequence. Indeed one of
the basic result is that one gets the same class of operators either way – this is
discussed below.
A word about the Schwartz kernel theorem. This is not used here but the
relationship between operators and kernels is used. Consider the space of continuous
PSEUDODIFFERENTIAL OPERATORS
3
linear maps (with the weak topology on S 0 (Rp ))
(0.7)
S(Rn ) −→ S 0 (Rp ) = B(S(Rn ); S 0 (Rp ))
(here we only consider p = n and I suppose the B is slightly bogus). There is a
natural bijection
(0.8)
B(S(Rn ); S 0 (Rp )) ←→ S 0 (Rp+n ) Schwartz’ kernel Theorem
which is written formally
Z
(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 letter
when 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 this
way so that this looks like the integral operator (0.9). The harder part of Schwartz
kernel 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) which
determines the operator from the kernel and conversely shows that the pairing of
the kernel with finite sums of products of Schwartz functions in the two variables is
fixed by the operator. These finite sums are dense in S(Rp+n ) (‘completed tensor
product’) so the kernel is determined by the operator. The remaining issue in the
proof of the kernel theorem is the continuity of the pairing of the putative kernel
and the elements of the finite tensor product.
For differential operators with Schwartz coefficients the kernel is of the form
X
(0.11)
Pα (x)(Dα δ)(x − y)
|α|≤m
so is supported on the diagonal. Our pseudodifferential operators are really characterized 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 right
forms of a differential operator
X
X
(0.12)
P (x, D)u(x) =
Pα (x)Dxα u(x) =
Dxα (Qα (x)u(x)).
|α|≤m
|α|≤m
1 Harmonic oscillator
The 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
X
(1.1)
φ=
(φ, eγ )eγ
γ∈Nn
0
which is the Fourier-Bessel expansion for the orthonormal basis of eigenfunctions
in L2 (Rn ). It has the desirable property of converging in S(Rn ) precisely when
φ ∈ S(Rn ).
RBM:Add sketch of alternate
as suggested by Vishesh
4
PSEUDODIFFERENTIAL OPERATORS
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
sup |y β Dyα Dtk f (t, y)| < ∞, ∀ k, α, β.
(1.2)
If we expand in the Hermite basis then we get
X
(1.3) f (t, y) =
fγ (t)eγ (y), fγ ∈ C ∞ ([0, 1]), eγ ∈ S(Rn )
γ
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 , |xβ Dxα eγ (x)| ≤ Ck0 (1 + |γ|)M +k , |α| + |β| ≤ k
where CN depends on some norm on φ, M depends only on n and Ck0 depends on
n and k.
The one-dimensional harmonic oscillator can be written
d
d
d2
(1.5)
H=(
+ x)(−
+ x) − 1 = − 2 + x2 .
dx
dx
dx
So we proceed to invert H + 1 by inverting the two ordinary differential operators
(1.6)
A=
d
d
+ x, C = A∗ = −
+ x.
dx
dx
Both are bounded operators
1
1
A, C : Hiso
(R) = H 1 (R) ∩ hxi−1 L2 (R) −→ L2 , hxi = (1 + |x|2 ) 2 .
(1.7)
The creation operator, C, is injective, since its null space on all distributions is
2
spanned by ex /2 . If g ∈ Cc∞ (R) satisfies
Z
2
(1.8)
e−x /2 g(x)dx = 0 then defining
Z x
Z ∞
2
2
2
2
EC g = −ex /2
e−s /2 g(s)ds = ex /2
e−s /2 g(s)ds ∈ Cc∞ (R)
−∞
x
given an operator satisfying CEC g = g.
2
Since e−s /2 is montonic decreasing near infinity it follows that if g ∈ S(R) satisfies 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 and
differentiating shows iteratively that
Z
2
(1.9)
EC : {g ∈ S(R); e−x /2 g(x)dx = 0} −→ S(R), CEC = Id .
If v ∈ S(R) then
Z
Z
Z
d|v|2
dv
dx
(1.10)
|Cv(x)|2 dx = (| |2 + x2 |v|2 )dx − x
dx
R
R dx
Z
dv
= (| |2 + x2 |v|2 + |v|2 )dx
dx
R
PSEUDODIFFERENTIAL OPERATORS
5
1
is a Hilbert norm on Hiso
(R). Thus extending by continuity gives a bounded linear
operator
Z
2
1
(1.11)
EC : {g ∈ L2 (R); e−x /2 g(x)dx = 0} −→ Hiso
(R)
and so
(1.12)
1
C : Hiso
(R) −→ L2 (R) is injective with range exp(−x2 /2)
⊥
.
In particular, C is Fredholm.
1
The dual of Hiso
(R) with respect to the L2 pairing is
−1
Hiso
(R) = H −1 (R) + hxiL2 (R)
(1.13)
although this is not used below. Passing to the adjoint it follows that
(1.14)
−1
A : L2 (R) −→ Hiso
(R) is surjective with Nul(A) = sp(e−x
/2
).
Cc∞ (R)
to S(R) is given by
Z x
2
−x2 /2
ẼA f (x) = u(x) = e
es /2 f (s)dx ∈ C ∞ (R), Au = f.
A right inverse from
(1.15)
2
−∞
In x > a, supp(f ) ⊂ [−a, a], u = ce
−x2 /2
,c=
R
R
es
2
/2
f (s), so indeed
1
ẼA : Cc∞ (R) −→ S(R) ⊂ Hiso
(R).
For u ∈ S(R) integration by parts gives
Z
Z
Z
d|u2 |
du 2
2
2
2
x
dx
|Au| dx = (| | + x |u| )dx +
dx
R
R dx
Z R
du
(1.16)
= (| |2 + x2 |u|2 − |u|2 )dx
R dx
=⇒ kẼA f k2H 1 ≤ kf k2L2 + 2kẼA f k2L2 .
iso
Combining (1.12) and (1.14) it follows that
Lemma 1. The shifted harmonic oscillator
(1.17)
−1
1
H + 1 = AC : Hiso
(R) −→ Hiso
(R) is an isomorphism
with inverse which restricts to a compact self-adjoint operator
(1.18)
1
(H + 1)−1 : L2 (R) −→ Hiso
(R) ,→ L2 (R).
Proof. The open mapping theorem shows that (H + 1)−1 is a bounded linear map
−1
1
from Hiso
(R) to Hiso
(R). That (H + 1)−1 restricts to a compact operator on L2 (R)
1
follows from the compactness of the inclusion of Hiso
(R) in L2 (R).
If we modify the generalized inverse ẼA to have range in the orthocomplement
of the null space of A :
1
(1.19)
EA f = EA f − (f, exp(−x2 /2) exp(−x2 /2), EA : S(R) −→ S(R)
π
then (H + 1)−1 = EC EA 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)−1 g) = ((H + 1)−1 f, g), f = (H + 1)φ, g = (H + 1)−1 ψ ∈ S(R)
6
PSEUDODIFFERENTIAL OPERATORS
transfers to the symmetry of (H + 1)−1 on the dense subspace S(R) from which
self-adjointness follows.
Thus L2 (R) has an orthonormal basis of eigenfunctions of (H +1)−1 . If ψ ∈ L2 (R)
1
and (H + 1)−1 ψ = λψ with λ > 0 then ψ ∈ Hiso
(R) and (H + 1)ψ = λ−1 ψ is an
eigenfunction for H + 1. The commutation relation, valid initially for the action on
S(R)
1
[H + 1, C] = 2C =⇒ [(H + 1)−1 , C] = −2(H + 1)−1 C(H + 1)−1 on Hiso
(R)
1
using the density of S(R) in Hiso
(R) which follows from the properties of EC . Thus
an eigenfunction ψ satisfies
(1.21)
(H + 1)−1 Cψ = C(H + 1)−1 ψ − 2(H + 1)−1 C(H + 1)−1 ψ =⇒
(H + 1)−1 Cψ =
λ
Cψ.
1 + 2λ
1
So Cψ is also an eigenfunction and in particular Cψ ∈ Hiso
(R). This argument can
1
therefore be repeated and shows that C k ψ ∈ Hiso
(R) satisfies
1
+ 2k)C k ψ, ∀ k
λ
and these are all non-trivial eigenfunctions which are orthogonal in L2 (R).
The analogous commutation relation for the annihilation operator [H + 1, A] =
1
−2A shows that (1 − 2λ)(H + 1)−1 Aψ = λAψ, so unless 1 − 2λ = 0, Aψ ∈ Hiso
(R)
k
1
is again an eigenfunction. This can be iterated to show that A ψ ∈ Hiso (R) is an
eigenfunction
1
(H + 1)Ak ψ = ( − 2k)Ak ψ,
λ
unless 2kλ = 1 in which case Ak ψ = 0 since (H + 1)−1 is injective. In fact this
must occur, i.e. λ = 1/2j for some j, since otherwise, for sufficiently large j, Aj ψ
is an eigenfunction of (H + 1)−1 with a negative eigenvalue. However integration
by parts shows that
(H + 1)C k ψ = (
(1.22)
((H + 1)u, u) ≥ 0 ∀ u ∈ S(R).
(1.23)
The operator EC EA = (H + 1)−1 : S(R) −→ S(R) so taking u = EC EA f shows
that ((H + 1)−1 f, f ) ≥ 0 for f ∈ Cc∞ (R). It follows that (H + 1)−1 is non-negative
so 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 Hermite
function sp(C k exp(−x2 /2)), k ∈ N0 .
Proposition 1. The eigenfunction expansion
X
(1.24)
u=
(u, ek )ek
k
s
Hiso
(R)
converges in
for any s ∈ R, where as spaces
(1.25)
s
s
Hiso
(R) = H s (R) ∩ hxi−s L2 (R), s ≥ 0, Hiso
(R) = H s (R) + hxi−s L2 (R), s ≤ 0.
Furthermore the L2 normalized Hermite functions are such that for any k continuous seminorm k · k on S(R) there is a bound
(1.26)
kej k ≤ CN (1 + j)N +1 ∀ j
PSEUDODIFFERENTIAL OPERATORS
7
and for any N there is a continuous norm k · k(N ) on S(R) such that
|(u, ej )| ≤ (1 + j)−N kuk(N ) , ∀ φ ∈ S(R);
(1.27)
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
kuk(N ) = k(H + 1)N ukL2 is a continuous seminorm on S(R) so
(1.28)
(2j + 2)N |(u, ej )| = |(u, (H + 1)N ej )|
= |(H + 1)N u, ej )| ≤ k(H + 1)N ukL2 = kuk(N )
is bounded and uniformly so on compact sets.
For any eigenfunction ACej = (2j + 1)ej so kCej kL2 = (ACej , ej ) = (2j + 1)
from which it follows that
1
ej+1 = (2j + 1)− 2 Cej , ej = (2j − 1)−1 Aej+1
(1.29)
and
kej kH 1 ≤ (2j + 1) =⇒ sup |ej (x| ≤ C(2j + 1)
(1.30)
by Sobolev embedding.
The finite sums of the supremum norms on Dk ej and xk ej , k ≤ N, give a
complete set of on S(R), i.e. bound and other continuous norm. These can be
expressed in terms of the creation and annihilation operators
dk
= (A − C)k /2k , xk = (A + C)k /2k
dxk
and expanding these out and using (1.29) to evaluate the products of the A and C
and to bound the Hermit functions shows that
(1.31)
|
(1.32)
dk ej
| + |xk ej | ≤ C(k)(2j + 2k)k max sup |ej+l | ≤ C(k)(2j + 2k)k+1 .
dxk
|l|≤k
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 a
complete orthonormal basis
(1.33)
2
eγ (x) = eγ1 (x1 ) . . . eγn (xn )
n
of L (R ) which are polynomially bounded with respect to any continuous seminorm
on S(Rn ) :
(1.34)
keγ k ≤ 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 ).
8
PSEUDODIFFERENTIAL OPERATORS
2 Symbols and quantization
Recall that we constructed a parameterix for an elliptic operator P (D) with
constant coefficients and used it to prove local elliptic regularity. Ellipticity implies
that the characteristic polynomial P (ξ) has real zeros (if any) only in a compact set
and if we choose an appropriate χ ∈ Cc∞ (R) which is equal to one in a neighbourhood
of the zeros then
1 − χ(ξ)
(2.1)
a(ξ) =
∈ S M (Rn ), M = −m.
P (ξ)
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)
sup(1 + |ξ|)−M +|α| |Dξα a(ξ)| < ∞ ∀ α.
ξ
Using an iterative formula for the derivatives we showed that a ∈ S −m (Rn ). We
showed earlier that if a ∈ S M (Rn ) then its inverse Fourier transform, A ∈ S 0 (Rn ),
 = a, satisfies
(2.3)
singsupp(A) ⊂ {0}, (1 − χ(x))A(x) ∈ S(Rn ), A ∈ H −M −s (Rn ), s > n/2
where χ ∈ Cc∞ (Rn ) is again equal to 1 near 0. We use this below in the stronger
form (which follows from the proof) that
(2.4) S M (Rn ) 3 a −→ (1 − χ)A ∈ S(R), Â = a,
F : S M (Rn ) −→ H −M −s (Rn ) are continuous.
The distribution χA was used as a convolution operator to deduce smoothness of
solutions 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 0 (R2n ) which is only
singular at the diagonal, x = y, and which we have arranged to be supported close
to the diagonal as well. The leading estimate in (2.2) shows the boundedness of
these operators on Sobolev spaces:
(2.5)
a ∈ S M (Rn ) =⇒ a(D) = A∗ : H t (Rn ) −→ H t−M (Rn ), ∀ t ∈ R.
The convenient notation a(D) for this convolution operator extends the standard
notation for constant coefficient differential operators P (D) where P = P (ξ) ∈
S k (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 smoothing operator. This is what we want to extend to variable coefficient elliptic differential operators. The pseudodifferential operators used to carry this out correspond
to a smooth family of such convolution operators and the main problem is just to
make sense of this statement.
As noted above, we could work with the symbol spaces ‘with bounds’ S M (Rn )
and it is conventional to do so for several, good, reasons. Some of these disappear
with 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 the
full symbol spaces involves little more than change of notation. An example of a
PSEUDODIFFERENTIAL OPERATORS
9
classical symbol is precisely a in (2.1). It differs from a general symbol in the sense
of (2.2) by ‘having an asymptotic expansion in smooth homogeneous functions at
infinity’. Rather than discuss the usual formulation of this concept let us look at a
in (2.1) in terms of the radial compactification of Rn .
Recall that we discussed this compactification earlier, by identifying Rn with
the hyperplane ηn+1 = 1 in Rn+1 and then taking the stereographic projection to
the half-sphere (not the full sphere minus a point although we did that two). This
means considering the bijection
1
(ξ, 1)
∈ Sn ∩ {ηn+1 > 0}, hξi = (1 + |ξ|2 ) 2 .
(2.6)
F : Rn 3 ξ 7−→ η =
hξi
Since it is topologically, and differentially, a ball let me write the closure of the
image 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 sure
that 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 )∗ hξi−1 = ηn+1 |Bn is smooth right up to the
boundary where it vanishes simply – it is a defining function for the boundary of
Bn . The point of all this discussion is that with a in (2.1),
m
(F −1 )∗ a = ηn+1
b, b ∈ C ∞ (Bn ) = C ∞ (Rn+1 )
Bn
.
That is, transferred to the ball, the upper half-sphere, it becomes a smooth function
which vanishes to precisely order m at the boundary (in this case m is an integer
but we do not assume this in general).
Definition 1. The space of classical symbols of real order s is defined to be
−s
s
(2.7)
Scl
(Rn ) = F ∗ ηn+1
C ∞ (Bn ) ⊂ S s (Rn ).
The sign reversal in the order here is just the usual historical baggage. In particular
(2.8)
s+k
s
Scl
(Rn ) ⊂ Scl
(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
s
s
s
(2.2) hold with M = s for all a ∈ Scl
(Rn ). If a ∈ Scl
(Rn ) then a = F ∗ (ηn+1
b),
∞
n
s
b ∈ C (B ), where b is certainly bounded, so |a| ≤ Chξi which is the top estimate
in (2.2). The (infinitely many) other estimates reduce to the same bound after
application of any number (repeated arbitrarily) of the vector fields ξi ∂ξj .
Lemma 2. There are vector fields on Bn (smooth first order differential operators
without constant terms) Vij such that ξi ∂ξj (F ∗ a) = F ∗ (Vij a) and these vector fields
are tangent to the boundary of Bn , meaning that Vij ηn+1 = ηn+1 eij , 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 are
things to check to feel confident about it. To define C ∞ (Bn ) there are two obvious
choices, we can say u ∈ C ∞ (Bn ) if u = ũ Bn for some ũ ∈ C ∞ (Rn ) or we can
just demand that u is smooth in the interior of Bn (which remember is the closed
10
PSEUDODIFFERENTIAL OPERATORS
ball) with all derivatives uniformly bounded. By integration the second definition
means all derivatives are continuous up to the boundary. If we take the second,
apparently weaker definition, then the topology is given by the supremum norms
on the derivatives and there is a nice theorem of Seeley that says there exists a
continuous linear extension map
(2.9)
E : C ∞ (Bn ) −→ {ũ ∈ Cc∞ (Rn ), supp ũ ⊂ K}, Eu
Bn
=u
where we can take K to be the ball of radius 2 for instance. So these two definitions
are the same. If you use Borel’s Lemma you can see that the restriction map is
surjective but Seeley gives quite a bit more.
So, it is perhaps a relief to know that the space of pseudodifferential operators
n
Ψm
cl,S (R ), as defined below, is actually identified linearly, and topologically, with
the space
hξim F ∗ C ∞ (Bn ; S(Rn )),
(2.10)
so ultimately with
(2.11)
C ∞ (Bn ; S(Rn )).
The idea of ‘quantization’ is to turn a function, such as one of these symbols, into
an operator.
Recall that a space like (2.11) is straightforward to define starting from the
definition of C ∞ (Bn ). We can start with the continuous maps from the ball to
the Fréchet space S(Rn ) forming C(Bn ; S(Rn )). Then consider once-differentiable
elements, 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 which
are required to be in C(Bn ; S(Rn )). This definition can then be iterated to define
infinite differentiability and the space (2.11)
Exercise 3. Show that the space C ∞ (Bn ; S(Rn )) is naturally identified with the
subspace of C ∞ functions on the interior of Bnη × Rny for which
(2.12)
suphyiN |Dηα Dyβ (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 the
space of smooth functions on Rn with support in the (closed) ball Bn . These can
also be identified with the subspace of C ∞ (Bn ) with all derivatives vanishing at the
boundary – the existence of an extension map from this subspace is then clear! We
can 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 complement
(2.13)
For a ∈ C ∞ (Bnη ; S(Rn )y ) conesupp(a) = {(η, x) ∈ Sn−1 × Rn
ˆ x)}{
∃ ã ∈ C˙∞ (Bn ; S(Rn )) s.t. a = ã near (ξ,
We can think of the cone support as represnting a cone in (Rnξ \ {0}) × Rny , conic
in the first variable, such that on the complementary open cone F ∗ a is rapidly
PSEUDODIFFERENTIAL OPERATORS
11
decaying. Since the ‘comparison’ functions C˙∞ (Bn ; S(Rn )) form an ideal, and as
for the usual support of smooth functions
conesupp(ab) ⊂ conesupp(a) ∩ conesupp(b),
(2.14)
conesupp(a) = ∅ ⇐⇒ a ∈ C˙∞ (Bn ; S(Rn ))
although this last result is not quite obvious (you can see it by checking that the
ˆ x) from conesupp(a) means that all derivatives of a restricted
absence of a point ξ,
n−1
n
ˆ x) so if conesupp(a) is empty these boundary values
to S
× R vanish near (ξ,
are all zero, from here it is not so hard).
Exercise 4. One can refine the definition of the cone support to be ‘uniform near
infinity’ in the second variable. Define the uniform cone support uconesupp by
working 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 are
C˙∞ (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 Rn
in that variable the definitions are the same. For classical symbols with Schwartz
coefficients, 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)
Z
X
C ∞ (Bn ; S(Rn )) 3 a(η, y) =
aγ (η)eγ (y), aγ = a(·, y)eγ (y)dy ∈ C ∞ (Bn )
γ
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 constant
CN such that the coefficients in (2.16) satisfy
(2.17)
kaγ k ≤ CN (1 + |γ|)−N .
Proof. This follows directly from Proposition 2. Namely, as norms fixing the topology of C ∞ (Bn ) we can take the supremum over Bn of the derivatives up to multiorder 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
s (Rn ) for any s.
‘coefficients’ aγ ∈ S(Rn ) are multipliers on H\
m
∞
n
n
Proposition 3. For any a ∈ Scl,S
(Rn ; Rn ) = F ∗ x−m
n+1 C (B ; S(R ), the series
(2.16) converges in B(H s (Rn ); H s−m (Rn ) for each s to define a bounded operator
which we write as
X
(2.18)
a(Dy , y)u =
aγ (D)(eγ (y)u(y)).
γ
Somewhat perversely the ‘coefficients’ are written on the right here. We will see
below that these operators restrict to continuous linear maps S(Rn ) −→ S(Rn ),
so the operators on the Sobolev spaces are all determined by continuous extension
and it is not necessary to distinguish then by the space on which they act.
12
PSEUDODIFFERENTIAL OPERATORS
Exercise 5. Show the boundedness of the operators or order m ≤ 0 on Lp (Rn )
for 1 < p < ∞ and on the Hölder spaces C k,γ (Rn ) for 0 < t < 1, k ∈ N0 . The
hard work, really the Marcinkiewicz multiplier theorem in the first case and less
hard in the second (see [5, §7.9]) is to show the boundedness in this sense of the
convolution operators aγ (D) in terms of a continuous seminorm on S 0 (Rn ). For
classical symbols boundedness can be reduced to the homogenous case and the
estimates in [5, §4.5] can be used instead.
n
Definition 2. We define the space Ψm
cl,S (R ) as consisting of the operators on
Sobolev spaces given by the sum (2.18) for a classical symbol a, so as the image of
a ‘right quantization’ map
(2.19)
m
n
qR : Scl,S
(Rn ; S(Rn )) −→ Ψm
cl,S (R ).
The right and ‘R’ refers to fact that the coefficients are put on the right. Note
that 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
X
X
(2.20)
B(x, y) =
Bγ (x, y) =
Aγ (x − y)eγ (y) ∈ S 0 (R2n ),
γ
γ
In fact if we write z = x − y and set
X
(2.21) A(z, y) = B(z + y, y) =
Aγ (z)eγ (y) converges in
γ
S(Rny ; H −m−s (Rn )), s > n/2,
and (1 − χ(z))A(z, y) ∈ S(R2n ).
Here χ ∈ Cc∞ (Rn ) is identically equal to 1 near 0.
We define a support set for these operator in terms of the symbol:(2.22)
WF0 (qR (a)) = conesupp(a) ⊂ Sn−1 × Rn .
This is also called the ‘operator wavefront set’ and has the property
(2.23)
WF0 (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)
WF0S (qR (a)) = uconesupp(a) ⊂ Sn−1 × Bn .
n
This definition of Ψm
cl,S (R (=) 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’
X
(2.25)
a(x, D)u =
eγ (x)(aγ (D)u).
γ
In general a(x, D) 6= a(D, y). As we show below, the space of operators obtained
this way is indeed the same as (2.18). That is (2.25) also defines a linear bijection
(2.26)
m
n
qL : Scl,S
(Rn ; Rn ) −→ Ψm
cl,S (R )
i.e. it has the same range.
PSEUDODIFFERENTIAL OPERATORS
13
There are other quantizations (especially ‘Weyl quantization’, see below). More
generally we can consider
−s
m
(2.27)
Scl,S
(Rn ; R2n ) = F ∗ ηn+1
C ∞ (Bn ; S(R2n
(x,y) )
where we allow the ‘symbol’ to depend on both x and y and also ξ. Then the
Hermite expansion for functions on R2n gives
X
m
(2.28)
Scl,S
(Rn ; R2n ) 3 b(x, y, ξ) =
eγ (x)bγ,γ 0 (ξ)eγ 0 (y)
γ,γ 0
where the convergence is strong enough that the operator
X
eγ (x)(bγ,γ 0 (D)(eγ 0 (·)u(·)))
(2.29)
b(x, D, y) =
γ,γ 0
converges in B(H s (Rn ); H s−m (Rn )). We can write this ‘general’ quantization map
as
(2.30)
m
n
q : Scl,S
(Rn ; R2n ) −→ Ψm
cl,S (R )
because it again has the same range.
The Schwartz kernel of q(a) is given the series
X
(2.31)
B(x, y) =
eγ 0 Aγ 0 ,γ (x − y)eγ (y) ∈ S 0 (R2n )
γ,γ 0
since this is true for the indivdual terms in (2.29) and the resulting series converges
in S 0 (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.
m
Exercise 8. If a ∈ Scl
(Rn ) and e ∈ S(Rn ) show that the kernel
x+y
)
(2.33)
A(x − y)e(
2
defines a bounded operator H s (Rn ) −→ H s−m (Rn ) for any s ∈ R – for instance
divide A(z) into a part A1 with compact support in |z| < 1 and a Schwarz part A2
and write
(2.34)
x+y
x+y
x+y
) = A1 (x − y)φ(x − y)e(
) + A2 (x − y)(1 − φ(x − y))e(
)
A(x − y)e(
2
1
2
for an appropriate cutoff φ ∈ Cc∞ (Rn ); then apply q to the first part and observe that
the second part is in S(R2n ). Show that this operator is independent of the choice of
m
cut-off and estimate the norm in terms of a seminorm on a(ξ)e(X) ∈ Scl,S
(Rn ; Rn ).
Conclude that the Hermite expansion of symbols leads to a Weyl quantization map
qW with values in the bounded operators just as for qR and qL . Use the discussion
in §5 to show that this is another bijection
(2.35)
m
n
qW : Scl,S
(Rn ; Rn ) −→ Ψm
cl,S (R )
which has the useful property (among others) that (qW (a))∗ = qW (ā).
14
PSEUDODIFFERENTIAL OPERATORS
3 Algebra of pseudodifferential operators
Let me collect a substantial, although not exhaustive, list of properties of the
n
space Ψm
cl,S (R ), 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 the
S is there because the coefficients are, by assumption, in S(Rn ). This space has
certain defects, in particular Id ∈
/ Ψ0cl,S (Rn ) because of the assumed rapid decay
of the coefficients. There are many variants of these spaces but we have to start
somewhere.
n
Theorem 1. The spaces Ψm
cl,S (R ) have the following properties:
n
(P1) (Boundedness) The elements of Ψm
cl,S (R ) define bounded operators
(3.1)
A : H s (Rn ) −→ H s−m (Rn ) ∀ s ∈ R.
n
m
n
For each j and A ∈ Ψm
cl,S (R ), xj A ∈ Ψcl,S (R ), the commutator [xj , A] ∈
n
Ψm−1
cl,S (R ) and in consequence
(3.2)
A : hait H s (Rn ) −→ hxi−t H s−m (Rn ) ∀ t, s
A : S(Rn ) −→ S(Rn ), A : S 0 (Rn ) −→ S 0 (Rn ).
(P2) (Left-right reduction) The maps qL and qR are injective and qR , qL and q
have the same range for each m ∈ R, Futhermore
(3.3)
qR (a) = qL (b), a, b ∈ C ∞ (Bn ; S(Rn )) =⇒ conesupp(a) = conesupp(b).
n
m
n
(P3) (Symbol map) For any m ∈ R, Ψm−1
cl,S (R ) ⊂ Ψcl,S (R ) and there is a short
exact sequence
(3.4)
σ
m
n
m
n
m ∞ n−1
Ψm−1
; S(Rn ))
cl,S (R ) −→ Ψcl,S (R ) −→ |ξ| C (S
where |ξ|m C ∞ (Sn−1 ; S(Rn )) should be interpreted as the space of those elements of C ∞ (Rn \ {0}; S(Rn )) which are homogeneous of degree m in the
n
first n variables. So σm is surjective and has null space exactly Ψm−1
cl,S (R ).
0
(P4) (Multiplicativity) For any m and m , in terms of operator composition
0
0
(3.5)
m+m
n
m
n
n
Ψm
cl,S (R ) ◦ Ψcl,S (R ) ⊂ Ψcl,S (R ),
σm+m0 (AB) = σm (A)σm0 (B), WF0 (AB) ⊂ WF0 (A) ∩ WF0 (B).
(P5) (Adjoints) Taking adjoints with respect to Lebesgue measure,
(3.6)
n
∗
m
n
Ψm
cl,S (R ) 3 A 7−→ A ∈ Ψcl,cS (R )
is an isomorphism and σm (A∗ ) = σm (A), WF0 (A∗ ) = WF0 (A).
m−k
(P6) (Asymptotic completeness) For any sequence Ak ∈ Ψcl,S
(Rn ), k ∈ N0 ,
m
n
there exists A ∈ Ψcl,S (R ) such that
(3.7)
A−
N
X
−1
Ak ∈ Ψm−N
(Rn ) ∀ N ∈ N0 .
cl,S
k=0
This relationship is written
(3.8)
A∼
N
X
k=0
Ak =⇒ WF0 (A) ⊂
[
k
WF0 (Ak ).
PSEUDODIFFERENTIAL OPERATORS
15
(P7) (Pseudo/Microlocality) For any A ∈ Ψ∗cl,Rn (Rn ),
singsupp(Au) ⊂ singsupp(u), ∀ u ∈ S 0 (Rn ),
(3.9)
WF(Au) ⊂ WF(u) ∩ WF0 (A), ∀ u ∈ S 0 (Rn )
where the second of these implies the first.
(P8) (Residual Algebra)
\
0
−∞
n
n
2n
m
n
Ψm−k
(3.10)
cl,S (R ) = ΨS (R ) ≡ S(R ) = {A ∈ Ψcl,S (R ); WF (A) = ∅}
k∈N0
is the space of integral operators with Schwartz kernels
Z
Bu(x) =
B(x, y)u(y)dy, B ∈ S(R2n ) and B, B ∗ : S 0 (Rn ) −→ S(Rn ).
Rn
4 Elliptic regularity
Rather than proceeding directly to check all these properties, let’s see what we
can achieve by using them. Suppose that
X
(4.1)
P (x, D) =
pα (x)Dα , pα ∈ C ∞ (Ω),
|α|≤m
is a differential operator with smooth coefficients in some open set Ω ⊂ Rn . This is
not in our algebra unless the coefficients happen to be Schwartz functions on the
whole 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
X
(4.3)
Pm (x, ξ) =
pα (x)ξ α 6= 0 on Ω × Rn \ {0}.
|α|=m
Theorem 2. [Elliptic Regularity] If P as in (4.1) is elliptic in Ω in the sense of
(4.3) then
singsupp(u) = singsupp(P (x, D)u),
(4.4)
WF(u) = WF(P (x, D)u), ∀ u ∈ C −∞ (Ω).
Proof. The inclusions
singsupp(u) ⊃ singsupp(P (x, D)u),
(4.5)
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 local
statements in the sense that we just need to show
(4.6)
x̄ ∈
/ singsupp(P (x, D)u) =⇒ x̄ ∈
/ singsupp(u)
and similarly for WF . So we can look near a particular point x̄ ∈ 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)
n
If φ ∈ Cc∞ (Ω) and ψ ∈ Cc∞ (Ω), supp(φ) ⊂ {ψ = 1} then ∃ A ∈ Ψ−m
cl,S (R ),
n
WF0 (A) ⊂ Sn−1 × supp(φ) s.t. A ◦ P (x, D)ψ = φ Id −E, E ∈ Ψ−∞
S (R ).
16
PSEUDODIFFERENTIAL OPERATORS
n
Notice that P (x, D)ψ ∈ Ψm
cl,S (R ) since now the coefficients are compactly supported 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 x̄. We can choose another ψ 0 ∈ C ∞ (Ω) (this is being a bit pedantic) so that
ψψ 0 = ψ and then we see that
(4.8)
P (x, D)ψu = P (x, D)ψ(ψ 0 u) = φf + g, f = P (x, D)u,
supp(g) ⊂ {φ 6= 1} =⇒ x̄ ∈
/ supp(g),
and where everything is now in Cc−∞ (Ω) ⊂ S 0 (Rn ). So we can apply A on the left
and see that
(4.9)
AP (x, D)ψ(ψ 0 u) = Aφf + Ag = φu − E(ψ 0 u) =⇒
φu = A(φf ) + Ag + E(ψ 0 u).
The last term is in S(Rn ) by (P8). Since x̄ ∈
/ supp(g), and in particular x̄ ∈
/
singsupp(g), pseudolocality shows that x̄ ∈
/ singsupp(Ag). Thus indeed we get (4.6)
since x̄ ∈
/ singsupp(P (x, D)u) implies x̄ ∈
/ singsupp(φf ), f = P (x, D)u, so x̄ ∈
/
singsupp(A(φf )) and then x̄ ∈
/ singsupp φu and so finally x̄ ∈
/ singsupp u since
φ(x̄) 6= 0.
The proof for WF is the same, using microlocality instead of pseudolocality. This shows the efficacy of the calculus of pseudodifferential operators, although
you might rightly object that the proof is a little clumsy because of the need to
insert all these cutoffs. This is indeed the case and we can get rid of that annoyance
by working with pseudodifferential operators on the open set Ω; this just requires
a 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),
n
P (x, D)ψ ∈ Ψm
cl,S (R ), σm (P (x, D)ψ) = Pm (x, ξ)ψ(x) =
X
pα (x)ψ(x)ξ α .
|α|=m
Now we are looking for A ∈
(4.11)
n
Ψ−m
cl,S (R )
and for any choice
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)
∈ |ξ|−m C ∞ (Sn−1 ; S(Rn ))
Pm (x, ξ)ψ(x)
(with the usual caveat that the function is taken to be zero outside the support of
the 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 homogeneous
of degree −m as claimed. Set
A0 = qR (a0 (1 − µ(ξ))
where µ(ξ) ∈ Cc∞ (Rn ) is equal to 1 in a neighbourhood of 0 – here we are using
the surjectivity of the symbol map in (3.4) in a slightly stronger sense since we also
PSEUDODIFFERENTIAL OPERATORS
17
have WF0 (A0 ) ⊂ Sn−1 × K where K = supp(φ). With this initial choice
(4.13) E1 = A0 P (x, D)ψ − φ Id ∈ Ψ0cl,S (Rn ), σ0 (E1 ) = a0 Pm ψ − φ = 0 =⇒
0
n
n−1
E1 ∈ Ψ−1
× K.
cl,S (R ) and WF (E1 ) ⊂ S
Then we can continue by induction to construct a succession of operators Aj j = 0,
. . . N so that
(4.14) Aj ∈ Ψ−m−j
(Rn ), WF0 (Aj ) ⊂ Sn−1 × K s.t.
cl,S


N
X

Aj  P (x, D)ψ = φ Id +EN +1 ,
j=0
−1
EN +1 ∈ Ψ−N
(Rn ), WF0 (EN −1 ) ⊂ Sn−1 × K.
cl,S
Then we can choose AN +1 = qR (aN +1 )
(4.15) aN +1 =
σ−N −1 (EN +1 )
∈ |ξ|−m−N −1 C ∞ (Sn−1 ; S(Rn ))
Pm (x, ξ)
=⇒ WF0 (AN +1 ) ⊂ Sn−1 × K.
This makes sense because of the ellipticity and the last condition in (4.14) which
ensures that WF0 (EN +1 ) ⊂ Sn−1 × K.
Now we use the asymptotic completeness result to show that we can choose one
X
n
(4.16)
A∼
A−j , A ∈ Ψ−m
cl,S (R ).
j
The inductive arrangement means that
(4.17)
n
AP (x, D)ψ − φ Id ∈ Ψ−N
cl,S (R ) ∀ N
which implies, by the last property above, that it is a smoothing operator, with
kernel in S(R2n ).
Note that we do not really need all the properties listed above to prove elliptic
regularity. For instance, it is enough to stop at a sufficiently large N in the construction 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 to
dispense 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
ˆ 6= 0 for some x̄ ∈ Ω and ξˆ ∈ Sn−1 show that there exists A ∈ Ψ−m (Rn ),
Pm (x̄, ξ)
cl,S
ψ, φ ∈ Cc∞ (Ω) such that
(4.18)
ˆ ∈
AP (x, D)ψ = φ Id −E, (x̄, ξ)
/ WF(E), φ(x̄) 6= 0.
Denoting the set of such ‘elliptic points’ of P as Ell(P ) ⊂ Sn−1 × Ω conclude that
for any u ∈ C −∞ (Ω),
(4.19)
WF(u) ∩ Ell(P ) ⊂ WF(P (x, D)u) ⊂ WF(u).
18
PSEUDODIFFERENTIAL OPERATORS
5 Proof of properties of pseudodifferential operators
We start with (P1). Since yj is a multiplier on S(Rn ) it follows immediately that
n
qR (a)yj = qR (ayj ) ∈ Ψm
cl,S (R ); this applies to any such multiplier, i.e. any smooth
function of slow growth so for instance
n
a(D, y)hyiM ∈ Ψm
cl,S (R ) ∀ M.
(5.1)
m
For a convolution operator a(D) where a ∈ Scl
(Rn ) we know that
[xj , A] = xj a(D) − a(D)yj = b(D), b(ξ) = i∂ξj a(ξ).
m
Scl,S
(Rn ; Rn )
Since ∂ξj :
(2.18) shows that
(5.2)
m−1
−→ Scl,S
(Rn ; Rn ), the convergence of the operator series
[xj , a(D, y)] = b(D, y), b(ξ, y) = i∂ξj a(ξ, y).
n
α
Thus xj a(D, y) ∈ Ψm
cl,S (R ) as well and, iterating, it follows that x a(D, y) ∈
m
n
Ψcl,S (R ). Thus
(5.3)
xα a(D, y)hyiM : H s (Rn ) −→ H s−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 ) = hxi−m H m (Rn ), S(Rn ) = hxi−m H m (Rn ).
m
m
The injectivity of qR in (P2) is a weak form of the Schwartz kernel theorem, here
just amounting to the fact that a can be recovered from the kernel of the operator
qR (a). The kernel of qR (a) is given by B in (2.20). Using (2.21) we can recover the
expansion of the symbol by taking the Hermite expansion of A(z, y) in y and then
the Fourier transform of the resulting distribution.
From this the exactness of the symbol sequence in (3.4) follows with σm (qR (a)) =
∞
n
n
|ξ|m am , am = xm a Sn−1 ×Rn being the restriction of a ∈ x−m
n+1 C (B ; S(R )) to the
−m
boundary. That is to say, after removal of a factor of ηn+1 , (3.4) just arises from
the exact ‘restriction sequence’
ηn+1 C ∞ (Bn ; S(Rn ) −→ C ∞ (Bn ; S(Rn ) −→ C ∞ (Sn−1 ; S(Rn ).
Exercise 10. (Maybe on a second reading ...) Show that for the more general
operators
(5.5)
n
m
n
n
Ψm
S (R ) = qR (SS (R ; R )
defined by ‘non-classical symbols’ the discussion above carries over except that the
principal symbol sequence has to be interpreted in terms of a quotient
(5.6)
σ
m
m−1
n
m
Ψm−1
(Rn ) −→ Ψm
)(Rn ; Rn )
S (R ) −→ (SS /SS
S
Go through the rest of the proofs checking what happens in the non-classical case.
Next consider the ‘residual algebra’ defined to be
\
\
−∞
n
n
n
n
Ψ−∞
Ψm−k
ΨM
cl,S (R )
S (R ) =
cl,S (R ), m ∈ R =⇒ ΨS (R ) =
M
k
(5.7)
−∞
n
2n
B : ΨS (R ) ←→ S(R )
where B is the map to the Schwartz kernel. Here the first part is the definition
and the claim is that the result is independent of m hence the second part holds.
n
So, fix m and consider qR (a) ∈ Ψ−∞
S (R ). The symbol map, applied iteratively,
PSEUDODIFFERENTIAL OPERATORS
19
shows that all terms in the Taylor series of a at the boundary of Bn vanish. Thus
a ∈ 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, taking
a 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,
n
m
k
∞
n
n
If Ak = qR (ak ) ∈ Ψm−k
cl,S (R ) then xn+1 ak ∈ xn+1 C (B ; S(R )). An appropriate
form of Borel’s Lemma shows that there exists b ∈ C ∞ (Bn ; S(Rn )) such that b −
N
P
N +1 ∞
−m
n
n
xm
n+1 ak ∈ xn+1 C (B ; S(R )). Then A = qR (xn+1 b) satisfies (3.7). From the
k=0
discussion of the residual algebra above, the difference of two such ‘asymptotic
−∞
n
n
sums’ is in Ψm−N
cl,S (R ) for all N and hence in ΨS (R ).
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 q
n
2n
turns a symbol F ∗ a, a ∈ x−m
n+1 CI(B ; S(R ) into an operator q(a) by expanding
2n
it in Hermite series on R
X
ˆ γ 0 (y).
(5.8)
a(x, y, ξ) =
eγ (x)aγ,γ 0 (ξ)e
γ,γ 0
Then each term becomes the operator
(5.9)
u 7−→ q(eγ aγ,γ 0 eγ 0 )u = eγ (x)(Aγ,γ 0 ∗)(eγ 0 u).
Then we use the convergence of (5.8) to sum (5.9) over γ, γ 0 in B(H s , H s−m ) for
any s. The continuity of this operator can be expressed by
(5.10)
kq(a)kB(H s ,H s−m ) ≤ kaks,m
∞
n
2n
for some continuous seminorm k · ks,m on x−m
n+1 C (B ; S(R ). The Schwartz kernel
is given by the corresponding series (2.31).
m
Given a ∈ Scl,S
(Rn ; R2n ) we ‘prepare’ it by writing
(5.11) a = a1 + a2 , F ∗ ai (x, ·, y) = Â(x, ·, y), F ∗ a(x, ·, y) = Â(x, ·, y),
A1 (x, z, y) = φ(z)A(x, z, y), φ ∈ Cc∞ (Rn ), φ(z) = 1 in |z| < 1.
That is, treating x and y as parameters we divide a into a part for which F ∗ a1
has Fourier transform compactly supported near 0 in the dual variable z. Now, the
properties of symbols imply that A2 ∈ S(R3n ) is Schwartz in all variables – being a
Schwartz function of z with values in the Schwartz functions in the other variables.
From
The Hermite expansion respects this deivision so
(5.12)
n
q(a) = q(a1 ) + q(a2 ), q(a2 ) ∈ Ψ−∞
S (R )
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 as
in (5.8):
X
ˆ y) =
ˆ γ 0 (y)
(5.13)
a(x, ξ,
lγ (x)aγ,γ 0 (ξ)r
γ,γ 0
20
PSEUDODIFFERENTIAL OPERATORS
∞
n
2n
provided it converges in x−m
n+1 C (B ; S(R ) and has the correct limit. In particular
this shows that
(5.14)
∞
n
2n
[xi , q(a)] = q((xi − yi )a) = q(−i∂ξj a), ∂ξj a ∈ x−m+1
n+a C (B ; S(R )
comes from a symbol one order lower. Iterating this statement we see that
(5.15)
−m+|α| ∞
q((xi − yi )α a) = q((−i∂ξ )α a), ∂ξα a ∈ xn+a
C (Bn ; S(R2n )
n
We showed that q(a2 ) ∈ Ψ−∞
S (R ) so are reduced to examing q(a1 ) for the first
part a1 in (5.11); let me again denote this a to simplify the notation but now we
know that it is ‘prepared’ in the sense that
(5.16)
q(a(·, x, y)) = q(a(·, x, y)φ(x − y)), φ ∈ Cc∞ (Rn )
since this formula holds for the Schwartz kernels.
m
Consider Taylor’s formula with remainder for g(·, x, y) = ηn+1
b(·, x, y) (just removing the leading factor to make everything smooth) around x = y :
X
X
(5.17)
g(·, x, y) =
(x − y)α gα (·, y) +
(x − y)α hα (·, x, y).
|α|≤N
|α|=N +1
This identity only holds in |x − y| < 2. If we insert the cutoff we see that
(5.18)
X
X
a(·, x, y)φ(x − y) =
(x − y)α aα (·, y)φ(x − y) +
(x − y)α bα (·, x, y)φ(x − y)
|α|≤N
|α|=N +1
−m
−m
where aα = ηn+1
gγ , bγ = ηn+1
hγ have the leading powers restored. Now all terms
m
are in the symbol space Scl,S (Rn ; R2n ) and the identity (5.18) holds globally.
The identities (5.16) and (5.15) show that
X
(5.19) q(a) = q(φ(x − y)a) =
q((−i∂ξ )α aα (·, y)φ(x − y))
|α|≤N
+
X
q((−i∂ξ )α bα (·, x, y)φ(x − y))
|α|=N +1
for any N. As noted above, the Schwartz kernel an operator such as
q((−i∂ξ )α aα (·, y)φ(x − y)
can be computed from any convergent expansion (5.13) for its symbol. Set
(5.20)
m
fα (·, y) = (−i∂ξ )α aα (·, y) ∈ Scl,S
(Rn ; Rn ).
Then taking the expansion of fα (·, y) as usual in a ‘single’ Hermite expansion and
multiplying by φ(x − y)
X
(5.21)
fα (·, y)φ(x − y)) =
fα,γ (·)eγ (y)φ(x − y)
γ
m
converges in Scl,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) with
more indices. Summing the series for the Schwartz kernel over this series first we
find that the kernel of fα (·, y)φ(x − y)) is
X
(5.22)
Bα (x, y) =
Eα,γ (x − y)eγ (y)φ(x − y), Êα,γ = fα,γ .
γ
PSEUDODIFFERENTIAL OPERATORS
21
However, the preparation of the symbol above means that all the Eα,γ (z) are supported 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
X
m−|α|
(5.24)
Aα , Aα = qR (aα ) ∈ Ψcl,S (Rn ).
|α|≤N
So now we may invoke the asymptotic summability result shown above to find one
n
A ∈ Ψm
cl,S (R ) such that
X
(5.25)
A∼
Aα .
|α|
It follows now from (5.19) again that for all N
(5.26) E = q(a) − A = q(aφ(x − y)) − A = A(N )
X
−1
+
q((−i∂ξ )α bα (·, x, y)φ(x − y)), A(N ) ∈ Ψm−N
(Rn ).
cl,S
|α|=N +1
Here there is a single operator, so with a fixed Schwartz kernel, expressed in terms
of symbols all of order m − N − 1. The regularity of the kernels in (2.32), together
with the fact that they are all supported in |x − y| < 1, shows that E has kernel in
S(R2n ) finally we have shown that
(5.27)
q(a) = qR (ã)
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
m
checked by a similar argument. Namely ‘preparing’ the symbol a ∈ Scl,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) + e
where e is residual. Thus it is enough to show that any residual operator is in
the 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 B̃(x, z, y) ∈ S(R3n ) and so
is the kernel of q(b) where b ∈ C˙∞ (Bn ; S(R2n ) is obtained from B̃ by inverse Fourier
transform in z.
Directly from the definitions taking adjoints, with respect to Lebesgue measure
as usual, interchanges qR and qL :
(5.28)
(qR (a))∗ = qL (ā) =⇒ σm (qR )∗ ) = σm (qR (a))
which is (P5). Using this we conclude that the range of qR is contained in that of
qL , so these are equal.
The fundamental property is multiplicativity but this now follows easily. Namely
n
m0
n
if A ∈ Ψm
cl,S (R ), B ∈ Ψcl,S (R ) the composite A ◦ B is certainly well-defined as an
n
operator on S(R ). 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 of
convolution operators in the middle)
0
m+m
(5.29) A ◦ B = q(a(·, x)b(·, y)) ∈ Ψcl,S
(Rn ) =⇒ σm+m0 (A ◦ B) = σm (A)σm0 (B).
This is (P4) with the formula for the principal symbol following from
This, I think, only leaves (P7) which is a good exercise!
22
PSEUDODIFFERENTIAL OPERATORS
6 Diffeomorphism-invariance
For any open set Ω ⊂ Rn we can consider the
m
n
Ψm
cl,c (Ω) ⊂ Ψcl,S (R )
(6.1)
with the defining support property
(6.2)
n
∗
n
∞
n
A ∈ Ψm
cl,S (R ) s.t. A, A : S(R ) −→ Cc (Ω) ⊂ S(R ).
Lemma 5. The defining condition (6.2) that A ∈ Ψm
cl,c (Ω) is equivalent to the
condition that the Schwartz kernel B ∈ S 0 (R2n ) of A have compact support within
the product
supp(B) b Ω × Ω.
(6.3)
Proof.
Proposition 4. If F : Ω −→ Ω0 is a diffeomorphism between open subsets of Rn
with inverse G : Ω0 −→ Ω then
AF u = F ∗ (AG∗ u), u ∈ Cc∞ (Ω) defines a bijection
(6.4)
0
m
Ψm
cl,c (Ω ) 3 A 7−→ AF ∈ Ψcl,c (Ω).
7 Localization
We define the large space
(7.1)
Ψm
cl (Ω)
as consisting of continuous linear operators
X
n
A : Cc∞ (Ω) −→ C ∞ (Ω) s.t. Aφ =
χi Aij χj , Aij ∈ Ψm
cl,S (R )
i,j
for one (and hence any) partition of unity χi ∈ Cc∞ (Ω) for Ω with locally finite
supports (meaning that for any K b Ω there are only finitely many j such that
supp(χj ) ∩ K 6= ∅.
m
m
Ψm
cl,c (Ω) ⊂ Ψcl,P (Ω) ⊂ Ψcl (Ω)
References
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[2]
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, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32
[4]
(1979), 359–443.
[5]
, The analysis of linear partial differential operators, vol. 1, Springer-Verlag, Berlin,
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