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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 339

σ K (x, ξ)= k p (x, y)e i(y−x)·ξ dξ . (6.2.51)
Ω
The factors (a α (y) − a α (x)) in (6.2.31) induce new kernels of the form

α
a α (y) − a α (x) D N j (x, y)= a α (y) − a α (x) k αj (x, y)
x
and can be written as asymptotic sums

β
c αjβ (x)(x − y) k αj (x, y)
|β|≥1
by using the Taylor expansion for a α . The new Schwartz kernels

α
α
(x − y) k p (x, y)+(x − y) k ∞ (x, y)
generate properly supported pseudodifferential operators with symbols

β
σ(x, ξ)= (x − y) k p (x, y)e i(y−x)·ξ dy .
Ω
The latter can be rewritten as
β
∂ i(y−x)·ξ

σ(x, ξ) = −i k p (x, y)e dy
∂ξ
Ω
β

∂ n
= −i σ K (x, ξ) ∈ S S S −j−|β| (Ω × IR ) with |β|≥ 1 .
∂ξ
Consequently, T(y)N j (x, y)= T j (x, y) defines the Schwartz kernel of a
pseudodifferential operator T ∈L −j−1 (Ω) and, also, defines the operator
∼j
2κ−1
T with T ∈L (Ω).
∼ ∼
We now justify the assumption for N made in Lemma 6.2.5. For N given
∼j ∼j
−2κ
by the Schwartz kernel N j in (6.2.43), we have already shown N ∈L (Ω).
∼0
j−2κ
For j ∈ IN, the following lemma justifies the desired property of N ∈L .
∼j
Lemma 6.2.6. Let T j (x, y) be the Schwartz kernel of a pseudodifferential
−j−1
operator T ∈L with j ∈ IN 0 .Then
∼j
A 0 (x, D; y)T j (·,y)=: N j+1 (x, y) (6.2.52)
defines the Schwartz kernel of a pseudodifferential operator
N ∈L −(j+2κ+1) (Ω).
∼j+1