The following lemma will allow us to define pseudo-differential operators and
their principal symbols in a coordinate invariant way. First we need to localize
the notion of pseudo-differential operator as follows.
A pseudo-differential operator P ∈ Ψd (U ) is said to be compactly supported
if there exists a compact subset A ⊂ U such that P = 0 on Cc∞ (U \ A). Assume
this condition to be fulfilled, and let χ ∈ Cc∞ (U ) be such that χ = 1 on an open
neighborhood of A. Then it is readily seen that P = P ◦ χ on Cc∞ (U ). Conversely,
if P = P ◦ χ for some χ ∈ Cc∞ (U ) then it is readily seen that P is supported by any
compact neighborhood of supp χ. The space of compactly supported operators in
Ψd (U ) will be denoted by Ψdc (U ). According to the following lemma it is sufficient
to consider compactly supported pseudo-differential operators if one is interested
in caluculations modulo smoothing operators.
Lemma 5.19 Let P ∈ Ψd (U ). Then there exists a compactly supported pseudodifferential operator P0 ∈ Ψdc (U ) such that
P − P0 ∈ Ψ−∞ (U ).
Proof By definition we have P = Ψp for some p ∈ S d (U ). Since pr1 supp p
has compact closure A in U, there exists a χ ∈ Cc∞ (U ) with χ = 1 on an open
neighborhood of A. It now follows from Corollary 5.11 that P0 = P ◦ χ has the
required property.
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Let h : U → U 0 be a diffeomorphism of open subsets of Rn . Then the pullback map f 7→ f ◦ h defines a linear isomorphism h∗ : Cc∞ (U 0 ) → Cc∞ (U ). Given
P ∈ Ψd (U ) we define the linear endomorphism
h∗ (P ) := (h∗ )−1 ◦ P ◦ h∗ ∈ End(Cc∞ (U 0 )).
Given p ∈ S d (U ) we define h∗ (p) : U 0 × Rn → C by
h∗ (p)(x0 , ξ 0 ) = p(h−1 x0 , dh−1 (x0 )ξ 0 ).
Proposition 5.20 The map P 7→ h∗ (P ) is bijective from Ψdc (U ) onto Ψdc (U 0 ).
The map p 7→ h∗ (p) is bijective from S d (U ) onto S d (U 0 ) and induces a bijection between the associated quotient spaces S d (U )/S d−1 (U ) and S d (U 0 )/S d−1 (U 0 ).
Moreover, for each P ∈ Ψdc (U ) the principal symbol of h∗ (P ) is given by
σd (h∗ (P )) = h∗ (σd (P )).
Proof
This follows from Gilkey, Lemma 1.3.2, page 24.
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Let E, F be finite dimensional complex linear spaces. Then all definitions
given so far extend trivially to pseudo-differential operators Ψ ∈ Hom(E, F ) ⊗
Ψd (U ), mapping Cc∞ (U, E) ' E ⊗Cc∞ (U ) to Cc∞ (U, F ) ' F ⊗Cc∞ (U ). The principal symbol of order d of such an operator belongs to Hom(E, F )⊗Sd (U )/Hom(E, F )⊗
Sd−1 (U ) ' Hom(E, F ) ⊗ [Sd (U )/Sd−1 (U )].
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Now suppose that E and F are equipped with Hermitian inner products.
With respect to these inner products we define the adjoint A∗ of a linear map A ∈
Hom(E, F ) to be the linear map F → E determined by hA∗ w, viE = hw, AviF
for all v ∈ E and w ∈ F.
We equip Cc∞ (U, E) with the Hermitian inner product given by
Z
hf, gi := hf (x), g(x)iE dx.
U
The inner product on F induces an inner product on Cc∞ (U, F ) in a similar
fashion. A linear map P : Cc∞ (U, E) → Cc∞ (U, F ) is said to have an adjoint
if there exists a linear map P ∗ : Cc∞ (U, F ) → Cc∞ (U, E) such that hP f, gi =
hf, P ∗ gi for all f ∈ Cc∞ (U, E) and g ∈ Cc∞ (U, F ). If it exists the map P ∗ is
uniquely determined and called the adjoint of P.
The following result asserts that a compactly supported pseudo-differential
operator has an adjoint that is pseudo-differential again. Moreover, the symbol
of the adjoint is determined by a suitable asymptotic expansion.
Lemma 5.21 Let P ∈ Hom(E, F )⊗Ψdc (U ). Then P has an adjoint P ∗ contained
in Hom(F, E) ⊗ Ψdc (U ). If p is the symbol of P, determined modulo S −∞ (U ), then
the symbol of P ∗ is determined by the expansion
σ(Pχ∗ ) ∼
X 1
∂ξα Dxα p∗ .
α!
α∈Nn
In particular, the principal symbol is given by σd (Pχ∗ )(x, ξ) = σd (P )(x, ξ)∗ .
Proof See Gilkey, proof of Lemma 1.2.3. The principal symbol is the term of
the series corresponding with α = 0.
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Let now E1 , E2 and E3 be finite dimensional complex linear spaces. If p ∈
Hom(E1 , E2 ) ⊗ S k (U ) and q ∈ Hom(E2 , , E3 ) ⊗ S l (U ), then we define qp : Rn ×
Rn → Hom(E1 , E3 ) by
qp(x, ξ) = q(x, ξ) ◦ p(x, ξ).
It follows by an easy application of the Leibniz rule that qp is a symbol in
Hom(E1 , E3 ) ⊗ S k+l (U ). The bilinear map (p, q) 7→ qp induces a bilinear map
Hom(E1 , E2 ) ⊗ S k (U )/S k−1 (U ) × Hom(E2 , E3 ) ⊗ S l (U )/S l−1 (U )
→ Hom(E1 , E3 ) ⊗ S k+l (U )/S k+l−1 (U )
denoted (σ, τ ) 7→ τ σ. With this notation, we have the following rule for the
calculation of the (principal) symbol of the composition of two pseudo-differential
operators.
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Lemma 5.22 Let E1 , E2 , E3 be finite dimensional complex linear spaces and let
P ∈ Hom(E1 , E2 ) ⊗ Ψk (U ) and Q ∈ Hom(E2 , E3 ) ⊗ Ψl (U ). Then the composition
Q ◦ P belongs to Ψk+l (U ) ⊗ Hom(E1 , E3 ). Let p, q be the symbols of P and Q
respectively. Then the symbol of Q ◦ P is determined by
X 1
∂ξα p Dxα q.
σ(Q ◦ P ) ∼
α!
α∈Nn
In particular, the k + l order principal symbol is given by
σk+l (Q ◦ P ) = σl (Q)σk (P ).
Proof This is proved in Gilkey, Lemma 1.2.3, in the case that P and Q are
compactly supported (although this is not mentioned explicitly). We will show
that the general result can be reduced to this case. Let P, Q be as assumed in the
lemma. We may write P = Ψp and Q = Ψq for suitable symbols with pr1 (supp p)
and pr1 (supp q) having compact closures A and B contained in U. Fix χ ∈ Cc∞ (U )
with χ = 1 on an open neighborhood of A ∪ B. Then Q ◦ P = Q ◦ χ ◦ P. Put
P0 = P ◦ χ and Q0 = Q ◦ χ. Then Q ◦ P = Q0 ◦ P0 + Q0 ◦ P ◦ (1 − χ). It follows
from Corollary 5.11 that P1 := P ◦ (1 − χ) ∈ Ψ−∞ (U ). Now use the lemma below
to conclude that Q0 ◦ P1 ∈ Ψ−∞ (U ). It follows that Q ◦ P − Q0 ◦ P0 ∈ Ψ−∞ . This
completes the reduction.
The assertion about the principal symbol follows from the fact that the principal symbol is the term of the series parametrized by α = 0.
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Lemma 5.23 Let P ∈ Hom(E1 , E2 ) ⊗ Ψ−∞ (U ) and Q ∈ Hom(E2 , E3 ) ⊗ Ψd (U ).
Then Q ◦ P ∈ Hom(E1 , E3 ) ⊗ Ψ−∞ (U ).
Proof It is easy to see that it suffices to prove the statement in case E1 = E2 =
E3 = C.
Let q ∈ S d (U ) be a symbol such that Q = Ψq . The operator P is of the form
b S(Rn ), pr1 (supp K) contained in a compact subset of U.
TK , with K ∈ Cb∞ (Rn ) ⊗
It readily follows that the Fourier transform F1 K of K with respect to the first
n coordinates belongs to S(Rn × Rn ). This implies that the function r : R3n → C
defined by
r(x, ξ, y) = q(x, ξ)F1 K(ξ, y)
belongs to S3d , for every d ∈ R. If f ∈ Cc∞ (U ), then
Z
F(P f )(ξ) =
e−iξx K(x, y) f (y) dy dx.
Rn
The integrand is absolutely integrable. Hence, by interchanging the order of
integration, we obtain
Z
F(P f )(ξ) =
F1 K(ξ, y) f (y) dy.
Rn
From this it follows that Q ◦ P f = Ψr f. By Proposition 5.13 we infer that Q ◦ P ∈
Ψd (U ) for all d ∈ R.
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