By Vincent Rivasseau (Chief Editor)
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Extra info for Annales Henri Poincaré - Volume 6
We are also interested in the estimates for symbols of commutators. For PDO’s A, Ψl , l = 1, 2, . . , N , denote ad(A; Ψ1 , Ψ2 , . . , ΨN ) = i ad(A; Ψ1 , Ψ2 , . . , ΨN −1 ), ΨN , ad(A; Ψ) = i[A, Ψ], adN (A; Ψ) = ad(A; Ψ, Ψ, . . , Ψ), ad0 (A; Ψ) = A. ,ψN ) and adN (a,ψ) for the symbols of multiple commutators. 1) that the Fourier coeﬃcients of the symbol ad(b, g) are given by ad(b, g)(χ, ξ) = i d(Γ) θ+φ=χ ˆb(θ, ξ + φ)ˆ g (φ, ξ) − ˆb(θ, ξ)ˆ g (φ, ξ + θ) , χ ∈ Γ† , ξ ∈ Rd . 2) Vol. 4 Let b ∈ Sα and gj ∈ Sβj , j = 1, 2, .
Moreover, if the symbol b(x, ξ) is smooth in ξ, then the family H(k) is smooth in k. Note however that for our purposes we need neither this explicit formula, nor the smoothness property. 2). 11) as deﬁnition. 11) preserves the invariance of D(λ) with respect to the choice of the lattice in the following sense: if the operator H happens to be periodic with respect to the lattices Γ and Λ, then DΓ (λ) = DΛ (λ). To this end we write another formula for D(λ) in terms of the spectral projection E(λ) = χ H; (−∞, λ] for H.
3) j=1 with some p = p(l, N, s, α, βj ), and a constant Cl,s independent of b, gj . 2 Asymptotics in the case d = 1 An important part is played by the result obtained in  for a PDO acting in L2 (R). We state this result in the form convenient for our purposes. Let b ∈ S0 ( ξ , Λ) and A = Op(a0 ) + Op(b) with a0 (ξ) = gξ 2 with a constant g > 0. We assume that Λ is a one-dimensional lattice with period τ > 0, so that d(Λ) = τ . Note that here and everywhere below, in the case d = 1 we use the notation x and ξ instead of the boldface letters x, ξ.
Annales Henri Poincaré - Volume 6 by Vincent Rivasseau (Chief Editor)