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6.1 Basic Theory of Pseudodifferential Operators 309

Behaviour of I 3 (R): The inner integral of I 3 is Φ 1 (x, ξ). Now, choose
ν> m + n − 2, then lim I 3 (R)=0.
R→∞
To complete the proof, we repeat the process by applying this technique
N times. Finally, we obtain

A(vψ)=(2π) −n v(y)ψ(y)|y − x| −2N e i(x−y)·ξ dy ×
n
IR supp(ψ)
N
× (−∆ ξ ) a(x, ξ) dξ .
Replacing v(y) by its definition, we find (6.1.14) after interchanging the order
of integration.
In the same manner as for m + n< 0 one finds that

N
(−∆ ξ ) a(x, ξ)e i(x−y)·ξ dσ
IR n
is in C ∞ for x = y. The alternative formulae (6.1.15) and (6.1.16) are direct
N
consequences of the Fourier transform of (−∆ ξ ) a(x, ξ) .
n
m
For the symbol calculus in S S S (Ω × IR ) generated by the algebra of
pseudodifferential operators it is desirable to introduce the notion of asymp-
totic expansions of symbols and use families of symbol classes.
n
m
We note that the symbol classes S S S (Ω × IR ) have the following proper-
ties:
n
n
m
a) For m ≤ m we have the inclusions S S S −∞ (Ω × IR ) ⊂ S S S (Ω × IR ) ⊂

n
n
n
m
S S S m (Ω × IR ) where S S S −∞ (Ω × IR ):= 9 S S S (Ω × IR ) .
m∈IR
α

β ∂ n
∂
n
m

b) If a ∈ S S S (Ω × IR ) then a ∈ S S S m−|α| (Ω × IR ).
∂x ∂ξ
n
n
n
m
c) For a ∈ S S S (Ω×IR )and b ∈ S S S m (Ω×IR )wehave ab ∈ S S S m+m (Ω×IR ).
n
Definition 6.1.3. Given a ∈ S S S m 0 (Ω×IR ) and a sequence of symbols a m j ∈
n
S S S m j (Ω × IR ) with m j ∈ IR and m j >m j+1 →−∞. We call the formal sum
∞
an asymptotic expansion of a if for every k> 0 there holds
a m j
j=0
k−1 ∞
n
a − a m j ∈ S S S m k (Ω × IR ) and we write a ∼ a m j .
j=0 j=0
n
∈ S S S m 0 (Ω × IR ) is called the principal symbol.
The leading term a m 0
is given, the following theorem holds.
In fact, if a sequence a m j
n
∈ S S S m j (Ω × IR ) with m j >m j+1 →−∞ for j →
Theorem 6.1.3. Let a m j
n
∞. Then there exists a symbol a ∈ S S S m 0 (Ω × IR ),uniquemodulo S S S −∞ (Ω ×
n
IR ), such that for all k> 0 we have