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412    7. Pseudodifferential Operators as Integral Operators

                                                    m
                           Lemma 7.2.10. Let A ∈L (Ω) and m ∈ IN 0 ,
                                                    c


                           k(x, x−y) ∼    k κ+j (x, x−y) and k(x ,x −y ) ∼  k κ+j (x ,x −y ).Then


                                       j≥0                             j≥0
                           the Tricomi conditions

                                      α

                                    Θ k κ+j (x, Θ)dω(Θ)=0=       Θ k κ+j (x ,Θ)dω(Θ)   (7.2.38)
                                                                  α
                               |Θ|=1                        |Θ|=1
                           are satisfied for all |α| = m − j and 0 ≤ j ≤ m.
                           We remark that for 0 ≤ j ≤ m, one has κ + j = −n − (m − j) < 0. Hence,
                           from Definition 7.1.1, we conclude that k κ+j and k κ+j are both positively

                           homogeneous of degree κ + j.
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