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6.1 Basic Theory of Pseudodifferential Operators 311

such that K A ,ϕ = 0. In accordance with Definition 6.1.4, any continuous
∞
linear operator A : C 0 ∞ → C (Ω) is called properly supported if and only
if its Schwartz kernel K A ∈D (Ω × Ω) is a properly supported distribution.

In order to characterize properly supported operators we recall the fol-
lowing lemma.
Lemma 6.1.4. (see Folland [82, Proposition 8.12]) A linear continuous op-
erator A : C (Ω) → C (Ω) is properly supported if and only if the following
∞
∞
0
two conditions hold:
i) Forany compactsubset K y Ω there exists a compact K x Ω such that
supp v ⊂ K y implies supp(Av) ⊂ K x .
ii) For any compact subset K x Ω there exists a compact K y Ω such that
supp v ∩ K y = ∅ implies supp(Av) ∩ K x = ∅.
Proof:
We first show the necessity of i) and ii) for any given properly supported
operator A. To show i),let K y be any fixed compact subset of Ω. Then define

K x := {x ∈ Ω | there exists y ∈ K y with (x, y) ∈ supp K A ∩ (Ω × K y )}
where K A is the Schwartz kernel of A. Clearly, K x is a compact subset of
Ω since supp K A ∩ (Ω × K y ) is compact because K A is properly supported.
In order to show i), i.e. supp(Av) ⊂ K x , we consider any v ∈ C (Ω) with
∞
0
supp v ⊂ K y and ψ ∈ C (Ω \ K x ). Then
∞
0

Av, ψ = K A (x, y)v(y)dyψ(x)dx
Ω Ω

= K A (x, y)ψ(x)v(y)dxdy =0
K y Ω

since for x ∈ supp ψ ⊂ Ω\K x and all y ∈ supp v we have (x, y) ∈
supp K A ∩ (Ω × K y ). Hence, Av, ψ = 0 for all ψ ∈ C (Ω \ K x ) and we find
∞
0
supp(Av) ⊂ K x .
For ii),let K x be any compact fixed subset of Ω. Then define
K y := {y ∈ Ω | there exists x ∈ K x with (x, y) ∈ supp K A ∩ (K x × Ω)} ,
∞ ∞
which is a compact subset of Ω.For v ∈ C (Ω\K y )and ψ ∈ C (supp(Av)∩
0
0
K x )wehave

Av, ψ = K A (x, y)v(y)ψ(x)dydx =0
Ω Ω
since for every y ∈ supp v we have y ∈ K y and (x, y) ∈ supp K A ∩ (K x × Ω)
for all x ∈ supp ψ. Hence, Av, ψ = 0 for all ψ ∈ C (supp(Av) ∩ K x )which
∞
0
implies supp(Av) ∩ K x = ∅.