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6.1 Basic Theory of Pseudodifferential Operators 307

Remark 6.1.2: The representation (6.1.14) will be also expressed in terms
of Hadamard’s finite part integrals later on.

Remark 6.1.3: In the theorem we use the identity (6.1.11) to ensure that
the integral (6.1.12) decays at infinity with sufficiently high order, whereas
for ξ = 0, we employ the regularization given by the Hadamard’s finite part
integrals. In fact, the trick (6.1.11) used in (6.1.12) leads to the definition of
the oscillatory integrals which allows to define pseudodifferential operators
of order m ≥−n (see e.g. Hadamard [117], H¨ormander [131, I p. 238], Wloka
et al [323]).

Proof:
i) n+m< 0: In this case, the integral has a weakly singular kernel, hence,

Au(x)=(2π) −n u(y) a(x, ξ)e i(x−y)·ξ dξdy
IR n IR n
and

k(x, x − y)=(2π) −n a(x, ξ)e i(x−y)·ξ dξ .
IR n
For the regularity, we begin with the identity (6.1.11) setting z = x − y.
n
N
Since (−∆ ξ ) a(x, ξ) ∈ S S S m−2N (Ω × IR ) integration by parts yields

−2N −n N i(x−y)·ξ
k(x, x − y)= |x − y| (2π) (−∆ ξ ) a(x, ξ) e dξ .
IR n
This representation can be differentiated 2N times with respect to x and
y for x = y. Since N ∈ IN is arbitrary, k is C ∞ for x = y and continuous
for x = y.

ii) n + m ≥ 0: Here, with

1 α α
v(y):= u(y) − (x − y) D u(x) ,
α!
|α|≤2N
(6.1.14) has the form
α
Au(x) = c α (x)D u(x)
|α|≤2N

+(2π) −n v(y)ψ(y)e i(x−y)·ξ dya(x, ξ)dξ .
IR n Ω
Note that the latter integral exists since the inner integral defines a C ∞
function which decays faster than any power of ξ due to the Palay–Wiener–
Schwartz theorem 3.1.3. With (6.1.11) we obtain