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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 333

1 i(x−y)ξ
Q(x, −iD)f(x)= e q(x, ξ)f(y)dydξ (6.2.23)
(2π) n
IR n Ω
−2
is a (perhaps not properly supported) parametrix Q ∈L (Ω).
With Q = Q 0 + R, the equations (6.2.6) yield
P ◦ Q = P ◦ (Q 0 + R)= I + C 1 + P ◦ R =: I + R 1 ,
Q ◦ P = (Q 0 + R) ◦ P = I + C 2 + R ◦ P =: I + R 2 . (6.2.24)

Since P is properly supported, P ◦R and R◦P are still smoothing operators;
so are R 1 and R 2 . The equations (6.2.24) read for the differential equation:

P ◦ (Qf)(x) = f(x)+ r 1 (x, y)f(y)dy (6.2.25)
Ω
and

Q(x, −iD) ◦ Pu(x)= u(x)+ r 2 (x, y)u(y)dy (6.2.26)
Ω
n
∞
where r 1 (x, y)and r 2 (x, y)are the C (IR )–Schwartz kernels of the smooth-
ing operators R 1 and R 2 , respectively. We note that both smoothing operators
R 1 and R 2 can be constructed since in (6.2.24) the left-hand sides are known.
∞
∞
Theorem 6.2.4. If R 2 extends to a continuous operator C (Ω) → C (Ω)
∞
and ker(Q ◦ P)= {0} in C (Ω) then the Newton potential operator N is
given by
u(x)=(Nf):=(I + R 2 ) −1 ◦ Q(x, −iD)f(x) . (6.2.27)
∞
where f ∈ C (Ω). Moreover, in this case, the fundamental solution exists
0
and is given by
−1
E(x, y)=(Nδ y )(x)= (I + R 2 ) ◦ Q(x, −iD)δ y (x) (6.2.28)
where δ y (·):= δ(·− y).
Our procedure for constructing the fundamental solution, in principle, can be
extended to general elliptic systems. Similar to the second order case we will
arrive at an expression as (6.2.28). The difficulty is to show the invertibility
as well as the continuous extendibility of I +R 2 . We comment that this is, in
general, not necessarily possible. Hence, for general elliptic linear differential
operators with variable coefficients it may not always be possible to find a
fundamental solution. We shall provide a list of references concerning the
construction of fundamental solutions at the end of the section. In any case,
even if the fundamental solution does not exist, for elliptic operators one can
always construct a parametrix.
From the practical point of view, however, it is more desirable to construct
only a finite number of terms in (6.2.9) rather than the complete symbol. This
leads us to the idea of Levi functions.