Page 341 -

6.1 Basic Theory of Pseudodifferential Operators 325

which is (6.1.52) for 1 ≤|α|≤ 2. For |α| > 2, we observe that every
differentiation yields terms which contain at least one factor of the form
∂Φ ∂Φ

(z) − (x) · ξ which vanishes for z = x, except the last term which
∂z ∂x
is of the form
∂
α

i Φ(x) · ξ ,
∂x
which proves (6.1.52).
We comment that in this proof we did not use the explicit representation of
A Φ,0 in the form

∂Φ

(A Φ,0 v)(x)=(2π) −n a x, ξ e ih+i(x −y )·ξ ×
∂x
n
IR Ω
∂Φ ∂Φ
−1

×v(y) det det dy dξ
∂y ∂x
where
∂Φ −1 −1

h(x ,y ,ξ )= (x − y ) · ξ with x = Φ (x )and y = Φ (y ) .

∂x
Now it follows from Theorem 6.1.11 that
α
1 ∂ ∂ α

a Φ (x ,ξ ) ∼ − i a Φ (x ,y ,ξ )

α! ∂ξ ∂y
|α|≥0 | y =x
where
∂Φ ir ∂Φ ∂Φ
−1
a Φ (x ,y ,ξ ) = a x, ξ e det det .

∂x ∂y ∂x
∂Φ ∂Φ

Since det a x, ξ is independent of y , interchanging differentia-

∂x ∂x
tion yields the formula
α
1 ∂Φ ∂ ∂Φ

a Φ (x ,ξ ) ∼ det a x, ξ ×
α! ∂x ∂ξ ∂x
0≤|α|
(6.1.53)
∂ ih(x ,y ,ξ ) ∂Φ
α
−1

× − i e det (y ) .

∂y ∂y | y =x
Needless to say that both formulae (6.1.50) and (6.1.53) are equivalent, how-
∂Φ
−1
ever, in (6.1.53) one needs to differentiate the Jacobian det which
∂y
makes (6.1.53) less desirable than (6.1.50) from the practical point of view.