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6.1 Basic Theory of Pseudodifferential Operators 305

The set of all standard pseudodifferential operators A(x, −iD) of order
m
n
m will be denoted by OPS (Ω × IR ) and forms a linear vector space to-
gether with the usual linear operations. Note that the differential operator of
order m,
α

∂
A(x, −iD)= a α (x) −i , (6.1.9)
∂x
|α|≤m
n
m
with C –coefficients on Ω belongs to OPS (Ω × IR ) with the symbol
∞
(6.1.3).
For a standard pseudodifferential operator one has the following mapping
properties.
Theorem 6.1.1. (H¨ormander [131, Theorem 18.1.6], Egorov and Shubin [68,
p. 7 and Theorem 1.4, p.18], Petersen [247, p.169 Theorem 2.4])
n
m
The operator A ∈ OPS (Ω ×IR ) defined by (6.1.6) is a continuous operator
∞
A : C (Ω) → C (Ω) . (6.1.10)
∞
0
s
The operator A can be extended to a continuous linear mapping from H (K)
into H s−m (Ω) foranycompactsubset K Ω. Furthermore, in the framework
loc
of distributions, A can also be extended to a continuous linear operator

A : E (Ω) →D (Ω) .
The proof is available in textbooks as e.g. in Egorov and Shubin [68],
Petersen [247]. The main tool for the proof is the well known Paley–Wiener–
Schwartz theorem, Theorem 3.1.3.
For any linear continuous operator A : C ∞ → C (Ω) there exists a
∞
0
distribution K A ∈D (Ω × Ω) such that

∞
Au(x)= K A (x, y)u(y)dy for u ∈ C (Ω)
0
Ω
where the integration is understood in the distributional sense. Due to the
Schwartz kernel theorem, the Schwartz kernel K A is uniquely determined by
the operator A (see H¨ormander [131], Schwartz [276] or Taira [301, Theorem
4.5.1]).
n
m
For an operator A ∈ OPS (Ω × IR ), the Schwartz kernel K A (x, y)has
the following smoothness property.
Theorem 6.1.2. (Egorov and Shubin [68, p. 7]) The Schwartz kernel K A of
n
m
n
A(x, D) ∈ OPS (Ω × IR ) is in C (Ω × Ω \{(x, x) | x ∈ IR }).
∞
Moreover, by use of the identity
N iz·ξ
e iz·ξ = |z| −2N (−∆ ξ ) e with N ∈ IN , (6.1.11)
K A has the representation in terms of the symbol,