Introduction to microlocal analysis by Melrose R.

By Melrose R.

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174) a ∈ S∞m (Rp ; Rn ) . 173) is exactly the restriction of the symbol estimates to z ∈ Rp , |ξ| ≤ 2. On the other hand, in |ξ| ≥ 1, a(z, ξ) is homogeneous so ˆ , |Dzα Dξβ a(z, ξ)| = |ξ|m−|β| |Dzα Dξβ a(z, ξ)| ξ ξˆ = |ξ| from which the symbol estimates follow. 2. 8. 54 2. 175) am−j . 176) since the asymptotic expansion of the product is given by the formal product of the asymtotic expansion. 177) 0 and multiplication by (1 + |ξ|2 )m/2 is an isomorphism of the space Sph (Rp ; Rn ) m onto Sph (Rp ; Rn ).

Proof. 87) since all terms with |α| ≥ 1 are of order m + m − |α| ≤ m + m − 1. 81), namely σR (x, ξ) ∼ α (−i)|α| α α Dξ Dx σL (x, ξ) ∼ e−i σL (x, ξ). α! This gives the double sum (still asymptotically convergent) σL (A ◦ B) ∼ β α i| β| β β i|α| α Dξ σL (A)Dxα D D σL (B) . α! β! x ξ Setting γ = α + β this becomes σL (A ◦ B) ∼ γ i|γ| γ! (−1)|γ−α| α D σL (A) × Dξγ−α Dxγ σL (B) . (γ − α)! ξ 44 2. PSEUDODIFFERENTIAL OPERATORS ON EUCLIDEAN SPACE Then Leibniz’ formula shows that this sum over α can be rewritten as σL (A ◦ B) ∼ γ i|γ| γ D σL (A) · Dxγ σL (B) γ!

219) It is important to contrast the behaviour of this ‘semiclassical symbol’ with the usual symbol – with which it is closely related of course. Namely the semiclassical symbol describes in rather complete detail the leading behaviour of the operator at = 0 and is multiplicative. 11 As with the principal symbol rather fine results can be proved by iteration. 220) n (1) n A ∈ Ψ−∞ , A(1) ∈ Ψ−∞ sl (R ) and σsl (A ) = 0 =⇒ A = A sl (R ). 221) A ∈ (1) ) = 0 and so on, one may finally −∞ n ∞ n Ψ−∞ sl (R ) ⇐⇒ A ∈ C ([0, 1]; Ψsl (R )) and N dk A d k =0 = 0.

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