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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
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