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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 341

6.2.4 Levi Functions for Elliptic Systems

In view of the construction of Levi functions for scalar elliptic differential
operators of order 2κ we now are able to extend the approach to systems
(6.2.10), i.e.
p s j +t k
jk β
a (x)D u k (x)= f j (x) ,j =1,...,p
β
k=1 |β|=0
where s j ≤ 0and t k ≥ 0, which are elliptic in the sense of Agmon–Douglis–
Nirenberg, i.e.

jk |β| β
H 2κ (x, ξ) := det a i ξ =0
β
p×p
|β|=s j +t k
n
for ξ ∈ IR \{0} where a jk is set to be zero for terms with order less than
β
s j +t k . We note that the determinant itself defines a scalar elliptic differential
operator

jk β
H 2κ (x; D) := det a (x)D
β
|β|=s j +t k

p p
t
of order 2κ = s j + k=1 k whose homogeneous symbol is H 2κ (x; ξ).

j=1
Let B(x, ξ) denote the cofactor matrix of a 0 (x, ξ)=((a jk (x; ξ))) p×p de-
s j +t k
fined by
a 0 (x, ξ)B(x, ξ)= B(x, ξ)a 0 (x, ξ)= H 2κ (x, ξ)((δ jk ))) p×p . (6.2.55)
It is not difficult to see that the cofactor matrix ((B k (x, ξ))) p×p given implic-
itly by (6.2.55) defines a system of differential operators B k (x, iD) of orders
2κ − t k − s . Then we can define
a −1 (y; D) := B(y, iD)H −1 (y, iD) ,
0 −2κ
where
1 e i(x−z)·ξ
−1
H −2κ (y, iD)f (x):= f(z)dzdξ . (6.2.56)
(2π) n H 2κ (y, ξ)
IR n Ω
The integral in (6.2.56) is defined in the sense of Hadamard’s finite part
(−1)
integral. Hence, the integral operator H (y; iD) is a pseudodifferential op-
−2κ
erator of order −2κ having the homogeneous symbol 1/H 2κ (y, ξ) with y as
(−1)
a frozen parameter. Consequently, the operator a 0 is a right inverse to
a 0 (y; D), the principal part of the differential equations with constant coeffi-
cients frozen at y. The differential operator a(x; D) can be decomposed by

a(x; D)= a 0 (y; D) − T(y; D)