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392 7. Pseudodifferential Operators as Integral Operators

a β (x) = 0 for all |β| = m − 1 .

Repeating this process we finally obtain a β (x) = 0 for all |β|≤ m which
assures the representation (7.1.77) of our theorem.

7.1.4 A Summary of the Relations between Kernels and Symbols

m
As we have seen in this section, a pseudodifferential operator A ∈L (Ω)
c
can be expressed in terms of either a given Schwartz kernel or in terms
of its symbol. The relations between these two representations have been
given explicitly so far. However, for ease of reading, we now summarize these
relations in the following.
Kernel to symbol
m
Let the operator A ∈L (Ω) be given in the form (7.1.8) or, more pre-
c
cisely, in the form (7.1.60) where the Schwartz kernel of the finite part integral
operator in (7.1.60) is a given function in the class Ψhk κ (Ω) with κ = −m−n
for fixed m ∈ IR (see Definition 7.1.1). Then this kernel has an asymptotic
expansion in the form (7.1.2), i.e.,

∞

k(x, x − y) ∼ k κ+j (x, x − y) where k κ+j ∈ Ψhf κ+j .
j=0
In the case that m − j ∈ IN 0 , we assume that the corresponding terms in the
asymptotic expansion satisfy the Tricomi conditions (7.1.56), i.e.,

α
Θ k κ+j (x, Θ)dω(Θ) = 0 for all |α| = m − j,
|Θ|=1
(see Theorem 7.1.7). For n = 1, again this formula is interpreted as in (3.2.14).
With a properly supported part A 0 of A, the symbol corresponding to A 0
can be computed according to the formula (6.1.23), i.e.,

a(x, ξ):= e −ix·ξ A 0 (e iξ• )(x) .

m
The symbol a(x, ξ) admits for A ∈L (Ω) a classical asymptotic expansion
c
∞
0
a(x, ξ) ∼ a (x, ξ) (7.1.80)
m−j
j=0
(see (6.1.42)). Each term in the expansion (7.1.80) can be calculated from the
corresponding pseudohomogeneous terms of the kernel expansion explicitly
depending on m − j.
For m − j< 0 we have: