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308 6. Introduction to Pseudodifferential Operators

I := (2π) −n v(y)ψ(y)e i(x−y)·ξ dy a(x, ξ)dξ
n
IR supp(ψ)

= (2π) −n v(y)ψ(y)|y − x| −2 (−∆ ξ )e i(x−y)·ξ dy a(x, ξ)dξ
n
IR supp(ψ)
and with integration by parts

#
I = lim (2π) −n v(y)ψ(y)|y − x| −2 e i(x−y)·ξ dy
R→∞
|ξ|≤R supp(ψ)

(−∆ ξ )a(x, ξ) dξ

−(2π) −n v(y)ψ(y)|y − x| −2 i(x − y)e i(x−y)·ξ dy · ξ a(x, ξ)dS ξ
|ξ|
|ξ|=R supp(ψ)

$

+(2π) −n v(y)ψ(y)|y − x| −2 e i(x−y)·ξ dy ∇ ξ a(x, ξ) · ξ dS ξ
|ξ|
|ξ|=R supp(ψ)
= lim [I 1 (R)+ I 2 (R)+ I 3 (R)]
R→∞
Now we examine each of the Integrals I j for j =1, 2, 3 separately.
Behaviour of I 1 (R): The inner integral of I 1 reads

−2 i(x−y)·ξ
Φ 1 (x, ξ):= v(y)ψ(y)|y − x| e dy
supp(ψ)⊂IR n
∞
and is a C –function of ξ which decays faster than any power of |ξ|;in
particular of an order higher than n+m−2, as can be shown with integration
by parts and employing (6.1.11) with z and ξ exchanged. Hence, the limit

e
lim I 1 (R)=(2π) −n v(y)ψ(y)|y − x| −2 i(x−y)·ξ dy (−∆ ξ )a(x, ξ) dξ
R→∞
IR n Ω
exists.
Behaviour of I 2 (R): Similarly, the inner integral of I 2 ,

−2 i(x−y)·ξ
Φ 2 (x, ξ):= i v(y)ψ(y)|y − x| (x − y)e dy ,
Ω
is a C –function of ξ which decays faster than any power of |ξ| = R;in
∞
particular of an order higher than m + n − 2. Therefore,
I 2 (R)= O(R m+n−1−ν ) → 0for R →∞ if ν> m + n − 1 .