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330 6. Introduction to Pseudodifferential Operators

and all other coefficients are zero. Then the symbol matrix (6.2.12) for (2.3.1)
reads
⎛ ⎞
2
µ|ξ| 0 0 −iξ 1
0 2 0
⎜ µ|ξ| −iξ 2 ⎟
a(x, ξ)= ⎜ ⎟
⎝ 0 0 µ|ξ| 2 −iξ 3 ⎠
0
−iξ 1 −iξ 2 −iξ 3
For n = 2, the third column and third row are to be discarded. The charac-
teristic determinant becames
2 2n
H(x, ξ) = det a(x, ξ)= µ |ξ| for n =2, 3 .
Hence, the Stokes system is elliptic in the sense of Agmon–Douglis–Nirenberg.

Theorem 6.2.3. (see also Chazarain and Piriou [39, Chap.4, Theorem 7.7])
A =((A jk )) p×p is elliptic in the sense of Agmon–Douglis–Nirenberg if and
only if there exists a properly supported parametrix Q 0 =((Q jk )) p×p with
−t j −s k
Q jk ∈L (Ω).

Proof: Assume that A is elliptic. By the definition of ellipticity, the homo-
geneous principal symbol is then of the form

jk0 r
a (x; ξ)=((|ξ| δ j ))((a (x; Θ)))((|ξ| δ rk ))
s j
t r
)
s j +t k s j +t k
where Θ = ξ and Einstein’s rule of summation is used, and where
)
|ξ|
jk n
H(x; Θ) = det ((a (x; Θ) )) for every x ∈ Ω and Θ ∈ IR , |Θ| =1 .
)
)
)
)
s j +t k
This implies that for every fixed x ∈ Ω there exists R(x) > 0 such that
jk jk
det((a (x; ξ))) = 0 for all |ξ|≥ R(x). By setting a(x; ξ):=((a (x; ξ))) p×p
n
and using an appropriate cut–off function χ ∈ C (Ω × IR ) with χ(x, ξ)=1
∞
for all x ∈ Ω and |ξ|≥ 2R(x) and with χ(x, ξ) = 0 in some neighbourhood
of the zeros of det a(x, ξ), we define q −m (x, ξ) by (6.2.8) as a matrix–valued
function. Recursively, then q −m−j (x, ξ) can be obtained by (6.2.9) providing
us with asymptotic symbol expansions of all the matrix elements of the op-
erator Q which we can find due to Theorem 6.1.3. The properly supported
parts of Q define the desired Q 0 satisfying the second equation in (6.2.6),
i.e., Q 0 is a left parametrix. To show that Q 0 is also a right parametrix, i.e.
to satisfy the first equation in (6.2.6), the arguments are the same as in the
scalar case in the proof of Theorem 6.2.2.
−t j −s k
Conversely, if Q is a given parametrix for A with Q jk ∈L (Ω) being
properly supported then its homogeneous principal symbol has the form
jk0 −t j −s k jk0
((q (x; ξ))) = ((|ξ| q (x; Θ)))
)
−t j −s k −t j −s k
−t j r0 −s k
=((|ξ| δ j ))q (x; Θ)))((|ξ| δ rk )) .
)
−t j −s k