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306 6. Introduction to Pseudodifferential Operators

−2N −n N i(x−y)·ξ
K A (x, y)= k(x, x − y)= |y − x| (2π) (−∆ ξ ) a(x, ξ) e dξ
IR n
for x = y (6.1.12)

m+n
!
where N ≥ +1 for m + n ≥ 0 and N =0 for m + n< 0. In the latter
2
case, K A is continuous in Ω × Ω and for u ∈ C (Ω) and x ∈ Ω we have the
∞
0
representation

Au(x)= k(x, x − y)u(y)dy . (6.1.13)
Ω
If n + m ≥ 0 let ψ ∈ C (Ω) be a cut–off function with ψ(x)=1 for all
∞
0
x ∈ supp u.Then
α
Au(x) = c α (x)D u(x) (6.1.14)
|α|≤2N

1 α α
+ k(x, x − y) u(y) − (y − x) D u(x) ψ(y)dy
α!
|α|≤2N
Ω
where
1 α i(x−y)·ξ

−n
c α (x)=(2π) (y − x) ψ(y)e dya(x, ξ)dξ .
α!
IR n Ω
Alternatively,
Au(x)=(A 1 u)(x)+(A 2 u)(x);
here

(A 1 u)(x)= k 1 (x, x − y)u(y)dy (6.1.15)
Ω
with

k 1 (x, x − y)=(2π) −n 1 − χ(ξ) a(x, ξ)e i(x−y)·ξ dξ ,
IR n
and the integro–differential operator

N
(A 2 u)(x)= k 2 (x, x − y)(−∆ y ) u(y)dy (6.1.16)
Ω
where

k 2 (x, x − y)=(2π) −n |ξ| −2N χ(ξ)a(x, ξ)e i(x−y)·ξ dξ
IR n
and χ(ξ) is the cut–off function with χ(ξ)=0 for |ξ|≤ ε and χ(ξ)=1 for
|ξ|≥ R.