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7.2 Coordinate Changes and Pseudohomogeneous Kernels 411

(x ,x − y ) ∼ k κ+j (x ,x − y )

k A Φ
j≥0
be the pseudohomogeneous kernel expansion of the transformed Schwartz ker-
nel to A Φ in terms of the new coordinates. Then the parity conditions in
(7.2.35) are invariant under the change of coordinates, i.e.

k κ+j (x ,x − y )=(−1) m−j+σ 0
k κ+j (x ,x − y ) for 0 ≤ j ≤ L. (7.2.36)

Proof: For the proof we employ Lemma 7.1.9. Then (7.2.35) is equivalent
to
a 0 (x, −ξ)=(−1) m−k−σ 0 0 (x, ξ)for ξ =0 and 0 ≤ j ≤ L,
a
m−j m−j
(7.2.37)
where a 0 are the homogeneous symbols of the complete symbol expansion
m−j
of A. Since the symbol a Φ (x ,ξ ) is given by (6.1.45) with (6.1.46), by rear-

ranging terms, the corresponding asymptotic homogeneous expansion of the
symbol reads
0

a Φ,m−j (x ,ξ )
j≥0
j
α
x − y
1 ∂ ∂Φ −1 −1

= − i det Ξ (x ,y ) Ψ ×

α! ∂y ∂y ε
j≥0 =0 |α|=
∂ 0
α −1

× a m−j+ x, Ξ (x ,y ξ | y =x .
∂ξ
We observe that (7.2.37) implies
∂ 0
α −1
a m−j+ x, − Ξ (x ,y ξ

∂ξ
∂ 0
α −1
=(−1) m−j+σ 0 a m−j+ x, Ξ (x ,y ξ .

∂ξ
Then it follows from the above representation that
a

a 0 (x , −ξ )=(−1) m−j+σ 0 0 (x , −ξ )

Φ,m−j Φ,m−j
which implies (7.2.37) by the application of Lemma 7.1.9 again.
To conclude this section, we observe for the Tricomi conditions, that as a
m
m

consequence of Theorem 6.1.13 and A ∈L (Ω), we also have A ∈L (Ω ).

c c
Hence, one may apply Theorem 7.1.7 to the kernels k κ+j and k κ+j of A

and A, respectively. This yields the following lemma which shows that the

Tricomi conditions are invariant under the change of coordinates.