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396 7. Pseudodifferential Operators as Integral Operators

κ+j

π α (x , −Θ )+ |x − y |
|α|−1
π α (x , −Θ )
|α|=1 |α|≥2
κ+j

= 1+ |x − y | q ,κ+j (x ,Θ )

π α (x , −Θ )
|α|=1 ≥1
where the functions q ,κ+j (x ,Θ ) defined on (Ω ×{|Θ | =1}) are obtained

by using a power series representation. Collecting terms, we obtain

f κ+j (x, x − y) ∼ |x − y | f κ+p (x ,Θ ) (7.2.9)
κ+p
j≥0 p≥0
where
κ+j

f κ+p (x ,Θ )= q ,κ+j (x ,Θ )f κ+j Ψ(x ),Θ(Θ ) .

π α (x , −Θ )
j+ =p |α|=1
j, ≥0
If κ ∈ Z then

k(x, x − y) ∼ f κ+j (x, x − y)+ log |x − y|p κ+j (x, x − y)
j≥0 j≥0
κ+j≥0
where p κ+j (x, z) are homogeneous polynomials of degree κ + j. According to
our previous analysis it suffices to consider only the terms

Q(x, x − y):= log |x − y|p κ+j (x, x − y) .
j≥0
κ+j≥0
In view of (7.2.6) we may rewrite

Q(x, x − y) ∼ log |x − y |p κ+j (x, x − y)
j≥0
κ+j≥0

π α (x , −Θ ) p κ+j (x, x − y) (7.2.10)

+ log
|α|=1

+ log 1+ |x − y | q ,1 (x ,Θ ) p κ+j (x, x − y) .

≥1
For the polynomials p κ+j we use (7.2.4) to obtain

p κ+j (x, x − y) ∼ p κ+j Ψ(x ) , π α (x ,x − y )
|α|≥1

∼ p κ+ ,j (x ,x − y ) (7.2.11)
≥j+κ
with the homogeneous polynomials p κ+ of degrees κ + .