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340 6. Introduction to Pseudodifferential Operators

Note that the special choice of T j in (6.2.44) and N j in (6.2.41) leads to the
assumption in Lemma 6.2.6.
Proof: From the definition (6.2.52) of the kernel function N j+1 one has the
representation
1 1 i(x−z)·ξ
N j+1 (x, y)= e T j (z, y)dzdξ .
0
(2π) n a (y, ξ)
2κ
IR n Ω
Hence, this kernel defines the operator
1 e i(x−z)·ξ
(N f)(x)= T j (z, y)f(y)dzdydξ .
0
∼j+1 (2π) n a (y, ξ)
2κ
IR n Ω Ω
−j−1 −j−1
Since T ∈L , we also have T ∈L due to Theorem 6.1.8 and,
∼j ∼j
for the properly supported part T jp we may use (6.1.27) which yields

e i(y−z)·ξ T jp (z, y)dz = σ (y, −ξ) .
T jp
Ω
Hence, performing the integration with respect to z yields

1 e i(x−y)·ξ
(N f)(x)= σ T (y, −ξ)f(y)dydξ
0
∼j+1 (2π) n a (y, ξ) jp
2κ
IR n Ω (6.2.53)
1 e i(x−z)·ξ
+ T j∞ (z, y)f(y)dzdydξ .
0
(2π) n a (y, ξ)
2κ
IR n Ω Ω
The first term on the right–hand side defines an operator of the form (6.1.26)
with the amplitude function
χ(ξ) −(j+2κ+1) n
a(x, y, ξ)= σ T (y, −ξ) 0 ∈ S S S (Ω × Ω × IR ) . (6.2.54)
jp a (y, ξ)
2κ
Here, χ(ξ) is again the cut–off function with χ(ξ)=1 for |ξ|≥ 1and χ(ξ)=0
1
for |ξ|≤ . The remaining (1 − χ(ξ))a(x, y, ξ) is a distribution with compact
2
support, and, hence, defines a smoothing operator.
The second term in (6.2.53) has the Schwartz kernel
e
i(x−z)·ξ
k(x, y):= 0 T j∞ (z, y)dzdξ
a (y, ξ)
2κ
IR n Ω
∞
which is C (Ω × Ω). Hence, it defines a smoothing operator. This implies
N ∈L −(j+2κ+1) (Ω), completing the proof.
∼j+1