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7.1 Pseudohomogeneous Kernels 387
n
k(x, z) − k κ+j (x, z) ∈ C (Ω × IR )for 0 ≤ <κ + J (7.1.68)
0≤j<J
where k κ+j ∈ ψhf κ+j .If m ∈ IN 0 then, for 0 ≤ j ≤ m we require k κ+j to
satisfy the Tricomi conditions (7.1.56) with κ + j instead of κ.
n
Note that condition (7.1.68) implies k ∈ C (Ω×IR \{0}). With a cut-off
∞
function ψ(z), the integral operator
Au(x):= p.f. k(x, x − y)u(y)dy
Ω
∞
operating on u ∈ C (Ω) can be written as
0
Au = A m u + Ru = p.f. k(x, x − y)ψ(x − y)u(y)dy
Ω
+ 1 − ψ(x − y) k(x, x − y)u(y)dy
Ω
where R is a smoothing operator since it has a C –kernel. For A m we have
∞
the representation
A m u(x) = F −1 a(x, ξ))u(ξ)
ξ →x
where
a(x, ξ) := (2π) n/2 F z →ξ k(x, z)ψ(z) (7.1.69)
n
∞
is in C (Ω × IR ) due to the Paley–Wiener–Schwartz theorem 3.1.3.
For the pseudohomogeneous functions k κ+j we know from Theorem 7.1.7,
that the operators
∞
A m−j u(x)= k κ+j (x, x − y)u(y)dy for u ∈ C (Ω)
0
Ω
m−j
are pseudodifferential operators A m−j ∈L (Ω) where for 0 ≤ j ≤ m with
c
m ∈ IN 0 the Tricomi conditions are assumed to be satisfied. Moreover, their
homogeneous symbols are given by
a 0 (x, ξ)=(2π) n/2 F z →ξ k κ+j (x, z) .
m−j
m
So, in order to show A m ∈L (Ω) it suffices to show
c
0 m−L n
a(x, ξ) − a m−j (x, ξ)χ(ξ) ∈ S S S (Ω × IR ) (7.1.70)
j<L
1
where χ(ξ)is the cut–off function with χ(ξ)=0 for |ξ|≤ and χ(ξ)=1 for
2
|ξ|≥ 1. We note that (7.1.70) is equivalent to the estimate