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7.1 Pseudohomogeneous Kernels 393

a 0 (x, ξ) = lim k κ+j x, z ψ z e −iξ·z dz for x ∈ Ω (7.1.81)
m−j t
t→+∞
IR n
1
where the cut–off function ψ(z)=1 for |z|≤ and ψ(z)=0 for |z| > 1(see
2
Lemma 7.1.4, equations (7.1.28) and (7.1.29)).
For m ∈ IN 0 and m − j> 0 we have

a 0 m−j (x, ξ)= p.f. k κ+j (x, z)e −iξ·z dz (7.1.82)
IR n
(see Theorem 7.1.6, formula (7.1.53)).

For m ∈ IN 0 and m − j ≥ 0 and if the Tricomi conditions (7.1.56) are
satisfied, then we have

α

a 0 (x, ξ)= c α (x)(iξ) +p.f. k κ+j (x, z)e −iξ·z dz (7.1.83)
m−j
|α|=m−j IR n
(see Theorem 7.1.7, formula (7.1.57)).
Symbol to kernel
m
For the operator A ∈L (Ω), let the classical symbol a(x, ξ) be given
c
by its homogeneous classical asymptotic symbol expansion (7.1.80). Then
the corresponding pseudohomogeneous kernel expansion can be calculated
explicitly as follows.
For m − j< 0 we have

a
k κ+j (x, z)=(2π) −n p.f. e iz·ξ 0 (x, ξ)dξ for x ∈ Ω and z ∈ IR n
m−j
IR n
(7.1.84)
(see in the proof of Theorem 7.1.1, formulae (7.1.48), (7.1.51)).
For m − j ≥ 0 we have

a
k κ+j (x, z)=(2π) −n e iξ·z 0 (x, ξ)ψ(ξ)dξ
m−j
IR n
(7.1.85)

−2 −n iξ·z 0
+|z| (2π) e (−∆ ξ ) a m−j (x, ξ) 1 − ψ(ξ) dξ
IR n
where ∈ IN satisfying 2 >m + n − j (see in the proof of Theorem 7.1.8 i),
formula (7.1.64)). This formula is valid for arbitrary m ∈ IR.
If m ∈ IN 0 then the Schwartz kernel is given by (7.1.85) for j =0,...,m
whereas the operator a 0 m−j (x, −iD) contains a differential operator of the