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328 6. Introduction to Pseudodifferential Operators

(−1) (−1)
Q ◦ Q 0 ◦ A ◦ Q 0 − I = Q ◦ (I + R 2 ) ◦ Q 0 − I
0 0
(−1) (−1)
= Q ◦ Q 0 − I + Q
0 0 ◦ R 2 ◦ Q 0
(−1)
= R 1 + Q ◦ R 2 ◦ Q 0 .
0
(−1) (−1)
Since both, Q 0 and Q are properly supported, Q ◦ R 2 ◦ Q 0 again is a
0 0
(−1)
smoothing operator due to Theorem 6.1.14. Hence, Q ◦Q 0 ◦A◦Q 0 −I =:
0
−∞
S 1 ∈L (Ω).
On the other hand,
(−1) (−1)
A ◦ Q 0 − I = Q ◦ Q 0 ◦ A ◦ Q 0 +(I − Q ◦ Q 0 ) ◦ A ◦ Q 0 − I
0 0
= S 1 − R 1 ◦ A ◦ Q 0
and R 1 ◦ A ◦ Q 0 is a smoothing operator since Q 0 and R 1 are both properly
−∞
supported and R 1 ∈L (Ω). This shows with the previous steps that
−∞
A ◦ Q 0 − I ∈L (Ω). Hence, Q 0 satisfies (6.2.6) and is a parametrix.
(ii) If Q is a parametrix satisfying (6.2.6), then with (6.1.37) we find
1 ∂ α ∂ α
σ A◦Q − aq ∼ σ A (x, ξ) − i q(x, ξ) .
α! ∂ξ ∂x
|α|≥1
n
Hence, σ A◦Q −aq ∈ S S S −1 (Ω×IR ), i.e. (6.2.1) with b = q.Moreover, σ A◦Q −1 ∈
n
S S S −∞ (Ω × IR ) because of (6.2.6).

6.2.1 Systems of Pseudodifferential Operators
The previous approach for a scalar elliptic operator can be extended to gen-
eral elliptic systems. Let us consider the p × p system of pseudodifferential
operators
A =((A jk )) p×p
n
jk
with symbols a (x; ξ) ∈ S S S s j +t k (Ω × IR ) where we assume that there exist
two p–tuples of numbers s j ,t k ∈ IR ; j, k =1,...,p. As a special example we
consider the Agmon–Douglis–Nirenberg elliptic system of partial differential
equations
p s j +t k
jk β
a (x)D u k (x)= f j (x)for j =1,...,p. (6.2.10)
β
k=1 |β|=0
Without loss of generality, assume s j ≤ 0. The symbol matrix associated
with (6.2.10) is given by
s j +t k
jk |β| β jk
a (x)i ξ = a (x; ξ) (6.2.11)
β
|β|=0