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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 335

2κ−n
L 0 (x, y)= |x − y| 00 (x, |x − y|, Θ) + log |x − y| 01 (x, |x − y|, Θ) (6.2.34)
)
)
:
∞
with Θ =(x − y) |x − y| where the functions 00 and 01 are C (Ω × IR + ×
)
n
{Θ ∈ IR ||Θ| =1}). The integral operator defined by the kernel L 0 (x, y)
)
)
will be denoted by N ;
∼0
i(x−y)·ξ
1 e
(N v)(x):= L 0 (x, y)v(y)dy = v(y)dydξ . (6.2.35)
0
∼0 (2π) n a (y, ξ)
2κ
n
Ω IR Ω
This is a pseudodifferential operator of the form (6.1.26) with the amplitude
function
0
a(x, y, ξ)= a (y, ξ) −1 ;
2κ
and L 0 (x, y) is the Schwartz kernel of N .
∼0
0
We remark that the only singularity of a (y, ξ) −1 at ξ = 0 can be
2κ
handled, by introducing the cut–off function χ(ξ), in the same manner
n
0
as in Remark 3.2.2. Hence, χ(ξ)a (y, ξ) −1 ∈ S S S −2κ (Ω × Ω × IR )and
2κ
−2κ
N ∈L (Ω).
∼0
To derive higher order Levi functions, we return to (6.2.30). Define the
pencil of pseudodifferential operators by
−1 0 −1
A 0 (x, −iD; y)f (x) = F a (y, ξ) F z →ξ f(z)
ξ →x 2κ

= E 0 (x, z; y)f(z)dz . (6.2.36)
Ω
−2κ
Then A 0 (x, D; y) ∈L (Ω) for every y ∈ Ω.Moreover,
P 0 (y)A 0 (x, −iD; y)= I and A 0 (x, −iD; y)P 0 (y)= I. (6.2.37)
In accordance with the Neumann series, we introduce the recursive se-
quence of operators
j

A j (x, −iD; y):= A 0 (x, −iD; y)T(y) A 0 (x, −iD; y) (6.2.38)
=0
= A 0 (x, −iD; y)+ A j−1 (x, −iD; y)T(y)A 0 (x, −iD; y) ,
j =1, 2,... . We note that
j

PA j (x, −iD; y) = P 0 (y) − T(y) A 0 (·, −iD; y)T(y) A 0 (x, −iD; y)
=0
j+1

= I − T(y)A 0 (·, −iD; y) (6.2.39)
for every y ∈ Ω and every j ∈ IN 0 .