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6.1 Basic Theory of Pseudodifferential Operators 317

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Theorem 6.1.12. If A ∈L (Ω) then the following mappings are continu-
ous (see Treves [306, Chap. I, Corollary 2.1 and Theorem 2.1], Taira [301,
Theorem 6.5.9]):

∞
∞
A : C (Ω) → C (Ω) ,
0

A : E (Ω) → D (Ω) , (6.1.32)
s−m
s
A : H comp (Ω) → H loc (Ω) .
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If in addition, A ∈L (Ω) is properly supported, then the mappings can be
extended to continuous mappings as follows [306, Chap. I, Proposition 3.2],
[301, Theorem 6.5.9]:
A : C (Ω) → C (Ω) ,
∞
∞
0
0
∞
A : C (Ω) → C (Ω) ,
∞
A : E (Ω) → E (Ω) ,

(6.1.33)

A : D (Ω) → D (Ω) ,
s
s−m
A : H comp (Ω) → H comp (Ω) ,
s−m
s
A : H loc (Ω) → H loc (Ω) .
Based on Theorem 6.1.9 and the concept of the principal symbol, one may
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consider the algebraic properties of pseudodifferential operators in L (Ω).
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The class L (Ω) is closed under the operations of taking the transposed and
the adjoint of these operators.
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Theorem 6.1.13. If A ∈L (Ω) then its transposed A ∈L (Ω) and its
m
adjoint A ∈L (Ω). The corresponding complete symbol classes have the
∗
asymptotic expansions
α

1 ∂ ∂ α
σ A ∼ −i σ A (x, −ξ) (6.1.34)
α! ∂ξ ∂x
α≥0
and
α

1 ∂ ∂ α ∗
σ A ∗ ∼ −i σ A (x, ξ) (6.1.35)
α! ∂ξ ∂x
α≥0
where σ A denotes one of the representatives of the complete symbol class.
For the detailed proof we refer to [306, Chap. I, Theroem 3.1].
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With respect to the composition of operators, the class of all L (Ω)is
not closed. However, the properly supported pseudodifferential operators in
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L (Ω) form an algebra. More precisely, we have the following theorem [131,
18.1], [302, Chap. II, Theorem 4.4.].
m 1 m 2
Theorem 6.1.14. Let A ∈L (Ω) ,B ∈L (Ω) and one of them be prop-
erly supported. Then the composition
m 1 +m 2
A ◦ B ∈L (Ω) (6.1.36)