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336 6. Introduction to Pseudodifferential Operators

We are now in the position to define Levi functions of various orders
recursively as follows.
L 0 (x, y) := A 0 (x, −iD; y)δ(·− y) ,
L j (x, y) := L 0 (x, y)+ A 0 (x, −iD; y)T(y)L j−1 (x, y) .
j

= A 0 (x, −iD; y) T(y)A 0 (·, −iD; y) δ(·− y) ,
=0
= A j (x, −iD; y)δ(·− y) . (6.2.40)
L j (x, y) is the Levi function of order j for the operator P.
By applying P to L j (x, y) we obtain

PL j (x, y)= P 0 (y) − T(y) L j (x, y)
j

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

= T(y)A 0 (·, −iD; y) − T(y)A 0 (·, −iD; y) δ(·− y)
=0 =1
j+1
= δ(x − y) − T(y)A 0 (·, −iD; y) δ(·− y) .
By introducing N 0 (x, y):= L 0 (x, y)and
N j+1 (x, y):= A 0 (x, −iD; y)T(y)N j (·,y) (6.2.41)

we can write
PL j = δ(x − y) − T(y)N j+1 (x, y) . (6.2.42)

In the following, we want to show that every N j (x, y) is a Schwartz kernel of
−j−2κ
a pseudodifferential operator N ∈L (Ω) ,j =0, 1,... and, hence, the
∼j
kernel N j (x, y) has the form

−n+j+2κ
N j (x, y)= |x−y| f j x, |x−y|, Θ + log |x−y| g j (x, |x−y|, Θ) ,
)
)
n
∞
f j ,g j ∈ C (Ω × IR ×{Θ ∈ IR ||Θ| =1}) . (6.2.43)
)
)
(The latter will be shown in Theorem 7.1.8).
We further consider the kernels
T j (x, y):= T(y)N j (x, y) (6.2.44)
and will show that T j (x, y) is also a Schwartz kernel of a pseudodifferential
operator T ∈L −j−1 (Ω). Hence, T j (x, y) also has the form (6.2.43) with
∼j
j +2κ replaced by j +1.