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338 6. Introduction to Pseudodifferential Operators

−j−1
is the Schwartz kernel of a pseudodifferential operator T ∈L and
∼j
T = T ◦ N
∼j ∼ ∼j
2κ−1
where T ∈L is generated by the operation T(y)N j (x, y) on the Schwartz
∼
kernel N j of N .
∼j
Note that the operator T is defined here implicitly and does not coincide
∼
with the differential operator T(y).
Proof: From the definition of N j (x, y) with N we have
∼j
1
N δ(·− y)= N j (x, y)= σ N (x, ξ)e i(x−y)·ξ dξ + N j∞ (x, y)
∼j (2π) n ∼j
IR n
where N j∞ (x, y)isthe C –kernel of a smoothing operator. Here, the symbol
∞
n
σ N ∈ S −j−2κ (Ω × IR ) is a representative of the complete symbol of N .
∼j ∼j
Applying T(y) to the kernel, we first consider the new kernel
∂
N j (x, y)
∂x k
∂σ
1 N i(x−y)·ξ ∂

∼j
= (x, ξ) − iξ k σ N j (x, ξ) e dξ + N j∞ (x, y) .
(2π) n ∂x k ∂x k
IR n
Hence, the differentiated kernel defines a new operator with the new symbol
∂σ N
∼j −j−2κ+1 n
(x, ξ) − iξ k σ N (x, ξ) ∈ S (Ω × IR ) . (6.2.50)
∂x k ∼j
Repeating this argument, we find that the Schwartz kernels

α α
a α (x)D N j (x, y)and D N j (x, y)for |α| =2κ
x x
|α|<2κ
n
define pseudodifferential operators in S S S −j−2κ+|α| (Ω × IR )for |α| < 2κ and
n
S S S −j (Ω × IR ), respectively. Therefore,

α
D N j (x, y) • dy = k αj (x, y) • dy for |α| =2κ
x
Ω Ω
defines a Schwartz kernel k αj (x, y) for a pseudodifferential operator in
−j
L (Ω). If k p (x, y) is the properly supported part of the Schwartz kernel
k αj (x, y) due to Theorems 6.1.9 and 6.1.11 then its symbol can be com-
puted by