Page 405 -

7.1 Pseudohomogeneous Kernels 389

α

∂ ∂
β
0
a(x, ξ) − a
∂ξ ∂x
m−j (x, ξ)
j<L
α β

∂ ∂
0 m−|α|−J
≤ a m−j (x, ξ) + c(α, β, m, J + n +1)|ξ|
∂ξ ∂x
J≤j≤J+n
m−|α|−J
≤ c 1 (α, β, m, J)|ξ| for |ξ|≥ 1 ,
which are the desired estimates (7.1.71).
m
m
Consequently, A m ∈L (Ω)and A ∈L (Ω) as proposed.
c c
7.1.3 Parity Conditions
Definition 7.1.3. (H¨ormander [131, Vol.I, Section 3.2])
The pseudohomogeneous function k q ∈ Ψhf q for q ∈ Z is of parity σ if it
satisfies the condition
σ
k q (x, −z)=(−1) k q (x, z) for z =0 . (7.1.74)
We now state the following crucial result concerning the transformation of
the finite part integral operators.
Lemma 7.1.9. The parity condition (7.1.74) for k q ∈ Ψhf q with q ∈ Z,
is satisfied if and only if the corresponding homogeneous symbol a 0 (x, ξ)
−n−q
satisfies the parity condition
σ 0
a 0 (x, −ξ)=(−1) a (x, ξ) for ξ =0 . (7.1.75)
−n−q −n−q
Proof: i) Let k q ∈ Ψhf q satisfy the parity condition (7.1.74). Then with
Lemma 7.1.4, the symbol

a ψ (x, ξ):= e −ix·ξ k q (x, x − y)ψ(x − y)e iy·ξ dy
IR n
associated with a suitable cut–off function ψ ∈ C 0 ∞ with ψ(z)=1 for |z|≤
ε, ε > 0, and ψ(z)=0 for |z|≥ 2ε and ψ(−z)= ψ(z) satisfies the parity
condition, i.e.

a ψ (x, −ξ) = e iz·ξ k q (x, z)ψ(z)dz
IR n

= e −iz·ξ k q (x, −z)ψ(z)dz
IR n

σ
= e −iz·ξ (−1) k q (x, z)ψ(z)dz
IR n

σ −i·ξ
= (−1) e k q (x, x − y)e iy·ξ ψ(x − y)dy
IR n
σ
= (−1) a ψ (x, ξ) .