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Extra info for Annales Henri Poincaré - Volume 6
6) We shall call such symbols symmetric. Note that Sα (L) = Sβ (L) for any α, β ∈ R and b (α) l,s = Lβ−α b (β) l,s . 7) In fact, the introduction of diﬀerent notation for the same class is done here to reﬂect the dependence on the parameter L. Throughout the entire paper we adopt the following convention. An estimate (or an assertion) is said to be uniform in a symbol b ∈ Sα (resp. Pα ) if the constants in the estimate (or assertion) at hand depend only on the constants Cl,s (α) (resp. Cl,0 ) in the bounds b l,s ≤ Cl,s .
Vol. 3 Density of States 41 Main result We are now in a position to state the main result of the paper. Let V be a Γ-periodic function, and let H = −∆ + V . In the main Theorem below we assume that d = 2. The multidimensional case will be considered in a subsequent publication. 3 Let d = 2. Assume that V ∈ C∞ (Rd ) is Γ-periodic and that Vˆ (0) = 0. Then there is a number λ0 = λ0 (V ) > 0 such that D(λ; H) = 6 1 λ + Oδ (λ− 5 +δ ), ∀δ > 0, 4π for all λ ≥ λ0 . The constant λ0 and the remainder estimate are uniform in V in (0) the sense that they depend only on the constants in the bounds V l,0 ≤ Cl .
1 Let d = 1 and let b ∈ Sα (L, Γ), α ≤ 0. Then for r ≤ L the symbol b belongs to Sα ( ξ , Γ) and b (α) l,s; ξ ≤ Cl,s b with a constant Cl,s depending only on l, s. V. Sobolev Ann. Henri Poincar´e Proof. 10) for the classes with a constant weight. One uses the fact that |ξ| ≤ L/2 + r/2 ≤ L on the support of ζθ , and thus |∂ξsˆb(θ, ξ)| ≤ b l,s;L θ −l Lα−s ≤ Cl,s b l,s;L −l θ ξ α−s , for all ξ ∈ supp ζθ and θ ∈ Θr . 4) that |∂ξs ϕθ (ξ; L)| + |∂ξs ζθ (ξ; L)| ≤ Cs ξ −s . The above estimates lead to the proclaimed estimate for the norm b (α) l,s; ξ .