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6.1 Basic Theory of Pseudodifferential Operators 315

φ j (x)k(x, x − y)φ (y)for (x, y) ∈ B ε (x 0 ) × B ε (y 0 )
I 2
for ε> 0 chosen appropriately small, has only finitely many nontrivial terms.
Hence,
n
R := φ j Aφ ∈ OPS −∞ (Ω × IR ) .
I 2

The operator Q := φ j Aφ , however, is properly supported since it has
I 1

the Schwartz kernel φ j (x)k(x, x − y)φ (y). Each term has compact sup-
I 1
port for fixed x with respect to y and with fixed y with respect to x.
Now, if x ∈ K Ω then, due to the compactness of K, we have supp φ j ∩
K = ∅ only for finitely many indices j whose collection we call J (K). For
every j ∈J (K), there are only finitely many with supp φ j ∩ supp φ = ∅;
we denote the collection of these by J (K, j). Hence, if x traces K, then
φ j (x)k(x, x − y)φ (y) = 0 only for finitely many indices (j, ) with j ∈J (K)
and ∈J (K, j); and we see that

supp φ j (x)K(x, x − y)φ (y) ∩ (K × Ω)
j∈J (K)
∈J (K,j)
is compact since there are only finitely many terms in the sum. If y ∈ K Ω,
we find with the same arguments that

(K ∩ Ω) ∩ supp φ j (x)k(x, x − y)φ (y)
∈J (K)
j∈J (K, )
is compact. Hence, Q is properly supported.
n
m
m
m
We emphasize that L (Ω) = OPS (Ω × IR ) ⊂L (Ω). The operators
−∞
R ∈L are called smoothing operators. (In the book by Taira [301] the
smoothing operators are called regularizers.)
∞

Definition 6.1.5. A continuous linear operator A : C (Ω) →D (Ω) is
0
called a smoothing operator if it extends to a continuous linear operator
from E (Ω) into C (Ω).

∞
The following theorem characterizes the smoothing operators in terms of
our operator classes.
Theorem 6.1.10. (Taira [301, Theorem 6.5.1, Theorem 4.5.2])
The following four conditions are equivalent:
(i) A is a smoothing operator,
(
−∞ m
(ii) A ∈L (Ω)= L (Ω),
m∈IR
n
(iii) A is of the form (6.1.26) with some a ∈ S S S −∞ (Ω × Ω × IR ),
∞
(iv) A has a C (Ω × Ω) Schwartz kernel.