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316 6. Introduction to Pseudodifferential Operators

Theorem 6.1.11. (H¨ormander [129, p. 103], Taylor [302, Chap. II, Theorem
m
3.8]) Let A ∈L (Ω) be properly supported. Then
a(x, ξ)= e −ix·ξ A(e iξ·• )(x) (6.1.27)
n
m
n
m
with a ∈ S S S (Ω × IR ) and A = A(x, −iD) ∈ OPS (Ω × IR ). Furthermore,
n
m
if A has an amplitude a(x, y, ξ) with a ∈ S S S (Ω × Ω × IR ) then a(x, ξ) has
the asymptotic expansion
α α
1 ∂ ∂
a(x, ξ) ∼ − i a(x, y, ξ) . (6.1.28)
α! ∂ξ ∂y
α≥0 | y=x
Here we omit the proof.
m
Theorems 6.1.9 and 6.1.11 imply that if A ∈L (Ω) then A can always
be decomposed in the form
A = A 0 + R
n
m
with A 0 ∈ OPS (Ω ×IR ) having a properly supported Schwartz kernel and
a smoothing operator R. Clearly, A 0 is not unique. Hence, to any A and A 0
we can associate with A asymbol
a(x, ξ):= e −ix·ξ A 0 (e iy·ξ )(x) (6.1.29)
and the corresponding asymptotic expansion (6.1.28). Now we define
:= the equivalence class of all the symbols associated (6.1.30)
σ A
n
n
m
:
with A defined by (6.1.29) in S S S (Ω × IR ) S S S −∞ (Ω × IR ) .
m
This equivalence class is called the complete symbol class of A ∈L (Ω).
Clearly, the mapping
n
m
m
n
:
L (Ω)
A → σ A ∈ S S S (Ω × IR ) S S S −∞ (Ω × IR )
induces an isomorphism
m : −∞ m n : −∞ n
L (Ω) L → S S S (Ω × IR ) S S S (Ω × IR ) .
The equivalence class defined by
σ mA := the equivalence class of all the symbols associated
m
n
n
:
with A defined by (6.1.29) in S S S (Ω × IR ) S S S m−1 (Ω × IR )
(6.1.31)
is called the principal symbol class of A which induces an isomorphism
m : m−1 m n : m−1 n
L (Ω) L → S S S (Ω × IR ) S S S (Ω × IR ) .
As for equivalence classes in general, one often uses just one representative
(such as the asymptotic expansion (6.1.28)) of the class σ A or σ mA , respec-
n
m
tively, to identify the whole class in S S S (Ω × IR ).
In view of Theorem 6.1.10, we now collect the mapping properties of
m
the pseudodifferential operators in L (Ω) in the following theorem without
proof.