Page 335 -

6.1 Basic Theory of Pseudodifferential Operators 319

To conclude this section, we now return to a subclass, the class of classi-
cal pseudodifferential operators, which is very important in connection with
elliptic boundary value problems and boundary integral equations.
m
Definition 6.1.6. A pseudodifferential operator A ∈L (Ω) is said to
be classical if its complete symbol σ A has a representative in the class
n
n
m
m
S S S (Ω × IR ),cf. (6.1.19). We denote by L (Ω × IR ) the set of all clas-
c
c
sical pseudodifferential operators of order m.
n
m
We remark that for L (Ω × IR ), the mapping
c
m
n
m
n
:
L (Ω)
A → σ A ∈ S S S (Ω × IR ) S S S −∞ (Ω × IR )
c c
induces the isomorphism
m : −∞ m n : −∞ n
L (Ω) L (Ω) → S S S (Ω × IR ) S S S (Ω × IR ) . (6.1.39)
c c
Moreover, we have
(
−∞ m
L (Ω)= L (Ω) . (6.1.40)
c
m∈IR
m
In this case, for A ∈L (Ω), the principal symbol σ mA has a representative
c
ξ

0
m
a (x, ξ):= |ξ| a m x, (6.1.41)
m
|ξ|
n
which belongs to C (Ω×(IR \{0})) and is positively homogeneous of degree
∞
0
m with respect to ξ. The function a (x, ξ) in (6.1.41) is called the homo-
m
m
geneous principal symbol of A ∈L (Ω). Note, in contrast to the principal
c
symbol of A defined in (6.1.31), the homogeneous principal symbol is only
a single function which represents the whole equivalence class in (6.1.31).
Correspondingly, if we denote by

ξ
0
m−j
a m−j (x, ξ):= |ξ| a m−j x, (6.1.42)
|ξ|
the homogeneous parts of the asymptotic expansion of the classical symbol
σ mA , which have the properties
a 0 (x, ξ) = a m−j (x, ξ) for |ξ| ≥ 1 and
m−j
a 0 (x, tξ) = t m−j 0 (x, ξ) for all t > 0 and ξ =0 ,
a
m−j m−j
∞
0
then σ mA may be represented asymptotically by the formal sum a m−j (x, ξ).
j=0