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310 6. Introduction to Pseudodifferential Operators

k−1
n
a − a m j ∈ S S S m k (Ω × IR ) . (6.1.17)
j=0
For the proof of this theorem one may set
∞
ξ
a(x, ξ):= χ a m j (x, ξ) (6.1.18)
t j
j=0
by using a C ∞ cut–off function χ(η) with χ(η)=1 for |η|≥ 1and χ(η)=0
1
for |η|≤ and by using a sequence of real scaling factors with t j →∞ for
2
j →∞ sufficiently fast. For the details of the proof see H¨ormander [129,
1.1.9], Taylor [302, Chap. II, Theorem 3.1].
With the help of the asymptotic expansions it is useful to identify in
n
n
m
m
S S S (Ω×IR ) the subclass S S S (Ω×IR ) called classical (also polyhomogeneous)
c
symbols.
n
m
Asymbol a ∈ S S S (Ω × IR ) is called a classical symbol if there exist
n
functions a m−j (x, ξ) with a m−j ∈ S S S m−j (Ω × IR ) ,j ∈ IN 0 such that a ∼
∞

j=0 a m−j where each a m−j is of the form (6.1.5) with (6.1.4) and is of
homogeneous degree m j = m − j; i.e., satisfying
a m−j (x, tξ)= t m−j a m−j (x, ξ)for t ≥ 1and |ξ|≥ 1 . (6.1.19)
n
m
The set of all classical symbols of order m will be denoted by S S S (Ω × IR ).
c
n
m
We remark that for a ∈ S S S (Ω × IR ), the homogeneous functions
c
a m−j (x, ξ)for |ξ|≥ 1 are uniquely determined. Moreover, note that the as-
ymptotic expansion means that for all |α|, |β|≥ 0 and every compact K Ω
there exist constants c(K, α, β) such that
N
∂ ∂
α
β

m−N−|α|−1
a(x, ξ) − a m−j (x, ξ) ≤ c(K, α, β) ξ
∂x ∂ξ

j=0
(6.1.20)
holds for every N ∈ IN 0 .
For a given symbol or for a given asymptotic expansion, the associated
operator A is given via the definition (6.1.8) or with Theorem 6.1.3, in ad-
dition. On the other hand, if the operator A is given, an equally important
question is how to find the corresponding symbol since by examining the
symbol one may deduce further properties of A. In order to reduce A to the
form A(x, D) in (6.1.6), we need the concept of properly supported operators
with respect to their Schwartz kernels K A .
Definition 6.1.4. A distribution K A ∈D (Ω × Ω) is called properly

supported if supp K A ∩ (K × Ω) and supp K A ∩ (Ω × K) are both com-
pact in Ω × Ω for every compact subset K Ω (Treves [306, Vol. I p. 25]).
We recall that the support of K A is defined by the relation z ∈ supp K A ⊂
2n
IR if and only if for every neighbourhood U(z) there exists ϕ ∈ C (U(z))
∞
0