By Melrose R.
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Sample text
Recall that this map is accomplished in two steps, first taking the Taylor series at y = x, integrating by parts and taking an asymptotic sum. This constructs m n b ∈ S∞ (Rn ; Rn ) so that qL (b) − I(a) ∈ Ψ−∞ ∞ (R ). Then the case m = −∞ is m done directly by estimation. Given a convergent sequence in S∞ (R2n ; Rn ), each of the terms in the Taylor series converges and it follows that the asymptotic sums m can be arranged to converge, that is if an → a in S∞ (R2n ; Rn ) then there exists m n n n bn → b ∈ S (R ; R ) such that qL (bn ) − I(an ) → qL (b) − I(a) ∈ Ψ−∞ ∞ (R ).
Then smoothness is just iterative regularity in this sense. Essentially by definition this means that A ∈ n ∞ m n n C ∞ ([0, 1] ; Ψm ∞ (R )) is the left-reduced symbol a = σL (A( )) ∈ C ([0, 1]; S∞ (R ; R )). 17. Linear invariance n It is rather straightforward to see that the algebra Ψ∞ ∞ (R ) is invariant under n affine transformations of R . In particular if Ta x = x + a, for a ∈ Rn , is translation by a and Ta∗ f (x) = f (x + a), Ta∗ : S(Rn ) −→ S(Rn ) is the isomorphism on functions then a new operator is defined by n m n Ta∗ Aa f = ATa∗ f and A ∈ Ψm ∞ (R ) =⇒ Aa ∈ Ψ∞ (R ).
Proof. The uniqueness part is easy. 50). 52) m−N aj ) ∈ S∞ (Rp ; Rn ). aj − a − a−a = a− j=0 j=0 −∞ −N Since S∞ (Rp ; Rn ) is just the intersection of the S∞ (Rp ; Rn ) −∞ p n that a − a ∈ S∞ (R ; R ), proving the uniqueness. over N it follows So to the existence of an asymptotic sum. To construct this (by Borel’s method) we cut off each term ‘near infinity in ξ’. Thus fix φ ∈ C ∞ (Rn ) with φ(ξ) = 0 in |ξ| ≤ 1, φ(ξ) = 1 in |ξ| ≥ 2, 0 ≤ φ(ξ) ≤ 1. 53) 0 > 1 > ··· > j ↓ 0. 54) a(z, ξ) = φ( j ξ)aj (z, ξ).