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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 331

This implies with (6.2.6)

jk0 −1
a (x; ξ)= |ξ| s j +t k ((q (x; Θ))) jk
)
s j +t k t j +s k
where q −1 denotes the inverse of ((q jk0 )). The latter exists since
t j +s k −t j −s k
jk0 jk0
det ((q (x; Θ) ))det ((a (x; Θ))) = 1 .
)
)
−t j −s k 0
Hence, A is elliptic in the sense of Agmon–Douglis–Nirenberg. Because of
(6.2.6) we also have A jk ∈L s j +t k (Ω).
In Fulling and Kennedy [86] one finds the construction of the parametrix
of elliptic differential operators on manifolds.

6.2.2 Parametrix and Fundamental Solution

To illustrate the idea of the parametrix we consider again the linear second
order scalar elliptic differential equation (5.1.1) for p = 1, whose differential
operator can be seen as a classical elliptic pseudodifferential operator of order
m =2.
Let us first assume that P has constant coefficients and, moreover, that
the symbol satisfies

n n
jk k n
a(ξ)= a ξ j ξ k − i b ξ k + c = 0 for all ξ ∈ IR . (6.2.16)
j,k=1 k=1
∞
Then the solution of (5.1.1) for f ∈ C (Ω) can be written as
0
1

−1
u(x)= Nf(x):= F F y →ξ f (x) . (6.2.17)
ξ →x
a(ξ)
−2
The operator N ∈L (Ω) defines the volume potential for the operator P,
the generalized Newton potential or free space Green’s operator. When f(y)
is replaced by δ(y − x) then we obtain the fundamental solution E(x, y)for
P explicitly:
1 i(x−y)·ξ

−n
E(x, y)=(2π) e dξ . (6.2.18)
a(ξ)
IR n
If condition (6.2.16) is replaced by
a(ξ) =0 for ξ =0
(as for c = 0 in (6.2.16)) then, as in (6.2.8), we take

1
q −2 (ξ)= χ(ξ)
a(ξ)