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388 7. Pseudodifferential Operators as Integral Operators

α

∂ ∂
β
0 m−|α|−J
a(x, ξ)− a (x, ξ) ≤ c(α, β, J)|ξ| (7.1.71)
∂ξ ∂x
m−j
j<J
for |ξ|≥ 1 and any multiindices α and β. For showing (7.1.71), let L ∈ IN
and the multiindex γ be chosen arbitrarily and consider the estimate
α
∂ ∂
β

γ 0
ξ a(x, ξ) − a m−j (x, ξ) (7.1.72)
∂ξ ∂x
j<L
β β
γ
∂ ∂ k(x, z) ∂ k κ+j (x, z)
−iz·ξ α
≤ e (−iz) − ψ(z) dz
∂z ∂x β ∂x β
j<L
|z|≤1
γ
β
∂ ∂ k κ+j (x, z)
+ e −iz·ξ − i (−iz) α 1 − ψ(z) dz
∂z ∂x β
j<L 1
|z|≥
2
where (7.1.69) is employed. From (7.1.68) we conclude that the integrand of
q
n
the first integral on the right–hand side in (7.1.72) is in C (Ω × IR ) with
q = κ + L − 1+ |α|−|γ|. Hence, this integral is bounded for q = 0 which
corresponds to the choice of γ so that |γ| = L − 1+ |α|− m − n. For the
remaining terms we notice that, for any integer k we have the estimates
γ

∂ β
e −i (−iz) (x, z) 1 − ψ(z)
2k −iz·ξ α ∂ k κ+j
|ξ| β dz
∂z ∂x
1
|z|≥
2
γ

∂
= e −iz·ξ (∆ z ) k −i
∂z
{···}dz
1
|z|≥
2

≤ c |z| −2k−|γ|+κ+j+|α| dz < ∞
1
|z|≥ 2
provided −2k −|γ|+κ+j +|α| < −n.So,wechoose2k ≥−m−|γ|+L+|α|.
Then (7.1.72) implies the estimates
α

∂ ∂
β
0
a(x, ξ) − a
∂ξ ∂x
m−j (x, ξ)
j<L
−|γ|
≤ c(α, β, m, L)|ξ| (7.1.73)
m−|α|−L+n+1
≤ c(α, β, m, L)|ξ| for |ξ|≥ 1 .
To obtain the desired estimates (7.1.71) we now choose L = J + n +1
β

∂ 0
and exploit the homogeneity of a m−j (x, ξ) of degree m − j. Then it
∂x
follows with (7.1.73) and the triangle inequality that