 By Mahmoudi F., Malchiodi A.

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35 C εk H d (Kε ,N Kε ) and the last ε−δ 2d d 2 j=0 (1 + ε |µj | )βj . 8 Let u2 = u ˆ2 + u ˜2 = Cε−k βj Ψj (εy, ζ) + j=ε−δ +1 j=0 δ∈ k 2,k m ∈ H2 . Then, choosing βj ψjm (εy)ˆ vj,ε (|ζ|) ζ|ζ| in (89), one has  (96) u2 2 HΣε 2δ 1 (1 + O(ε1−γ + ε2− k ))  βj2 ∂1 w0 εk j=0 =  Cε−k ε−δ H 1 (Rn+1 ) + βj2  . + j=ε−δ +1 Proof. We first claim that the following formula holds u ˆ2 2HSε (97) 1 = k ε ε−δ 2δ βj2 1 + O(ε2−2γ + ε2− k ) 2 . H 1 (Rn+1 ) + ∂1 w0 j=0 Proof of (97). We write ε−δ ε−δ βj ψjm (εy)∂m w0 (ζ)χε (|ζ|) u ˆ2 = u ˆ2,1 + u ˆ2,2 := βj Ψj (εy, ζ).

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