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332 6. Introduction to Pseudodifferential Operators

n
where the cut–off function χ ∈ C (IR ) satisfies
∞
1
χ(ξ)=1 for |ξ|≥ 1and χ(ξ)=0 for |ξ|≤ . (6.2.19)
2
We define the parametrix by
−1
Qf := F (q −2 (ξ)F y →ξ f)
ξ →x
−2 n
with Q ∈L (IR ); and we have
u(x)= Qf(x)+ Rf(x)

with the remainder
1
−1
Rf(x)= F 1 − χ(ξ) f(ξ) . (6.2.20)
)
ξ →x a(ξ)
Since f ∈ C (Ω)wehave f ∈ C ∞ from the Paley–Wiener–Schwartz theorem
∞
0
)
1
3.1.3 (iii). Hence, with 1 − χ(ξ) =0 for |ξ|≥ 1and having a singu-
a(ξ)
larity only at ξ = 0 and having quadratic growth, due to Lemma 3.2.1,

f(ξ) defines a distribution in E . Hence, Rf ∈ C
1 − χ(ξ) 1 ) ∞ and R ∈
a(ξ)
n
−∞ −∞
L (Ω) by applying Theorem 3.1.3 (ii). (Note that R ∈ OPS (Ω ×IR )).
−2
Thus, Q + R ∈L (Ω) defining again N := Q + R, the Newton potential
operator, which can now still be written as
1

−1
Nf(x)= F F y →ξ f(y) (x) . (6.2.21)
ξ →x
a(ξ)
As in the previous case, we still can define the fundamental solution
E(x, y)= Nδ(x − y) (6.2.22)
where δ(x − y) is the Dirac functional with singularity at y. Note that δ ∈
n
n
n
E (IR )and N : E (IR ) →D (IR ) is well defined. E(x, y) will satisfy (3.6.4).

We now return to the more general scalar elliptic equation (5.1.1) with
∞
C –coefficients (but p = 1). Here, the symbol a(x, ξ) in (6.2.4) depends on
both, x and ξ. The construction of the symbols (6.2.8) and (6.2.9) leads to
an asymptotic expansion and via (6.1.18), i.e., by
∞
ξ
q(x, ξ):= Ξ q −2−j (x, ξ)
t j
j=0
we define the symbol q where Ξ ∈ C ∞ is a cut–off function satisfying Ξ(y)=0
1 −2 n
for |y|≤ and Ξ(y)=1 for |y|≥ 1. Then q ∈ S S S (Ω × IR )and
2