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6.1 Basic Theory of Pseudodifferential Operators 323

Since the Schwartz kernel of A Φ,0 satisfies

x − y

(x ,x − y )= K Φ ∗ A 0 Φ ∗(x ,x − y )ψ

K A Φ,0
ε
and Φ ∗ A 0 Φ is properly supported, so is A Φ,0 . Now Theorem 6.1.11 can be
∗
applied to A Φ,0 .
−1

Since the function v(y ) 1−ψ x −y det ∂φ vanishes in the vicin-

ε ∂y

ity of x = y identically, we can proceed in the same manner as in the proof
of Theorem 6.1.2 and obtain

x − y

Rv(x ) = (2π) −n e i(x−y)·ξ v(y ) 1 − ψ ×

ε
n
IR Ω
∂φ N
−1

× det (−∆ ξ ) a(x, ξ) dy dξ .
∂y
Here, for N> m + n + 1 we may interchange the order of integration and
∞
find that R is a smoothing integral operator with C –kernel. Therefore,
n
m
Φ ∗ A 0 Φ ∈ OPS (Ω × IR ) and the asymptotic formula (6.1.45) is a conse-

∗
quence of Theorem 6.1.11. This completes the proof.
The transformed symbol in (6.1.45) is not very practical from the compu-
tational point of view since one needs the explicit form of the matrix Ξ(x ,y )

in (6.1.47) and its inverse. Alternatively, one may use the following asymp-
totic expansion of the symbol.
Lemma 6.1.17. Under the same conditions as in Theorem 6.1.16, the com-
plete symbol class of A Φ also has the asymptotic expansion
1 ∂ α ∂ α
iξ ·r
a Φ (x ,ξ ) ∼ a(x, ξ) − i e (6.1.50)
α! ∂ξ | ∂z | z=x
ξ
0≤α ξ= ∂Φ
∂x
where x = Φ(x) and

∂Φ
r = Φ(z) − Φ(x) − (x)(z − x) . (6.1.51)
∂x
Moreover,

∂ iξ ·r
α 0 for |α| =1 ,

− i e | z=x = α
∂z (−i) |α|−1 ∂ Φ(x) · ξ for |α|≥ 2 .
∂x
(6.1.52)
Proof: In view of Theorem 6.1.16 it suffices to consider properly supported
operators A 0 and A Φ,0 . Therefore the symbol class to A Φ,0 can be found by
using formula (6.1.29), i.e.,