NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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CALDERÓN <strong>IN</strong>VERSE PROBLEM ON RIEMANN SURFACES 113 for some smooth differential operator P (x,θ; x∂x,∂θ) in the vector fields x∂x,∂θ down to x = 0. Let us define Vb := x −2 V , which is compactly supported, and H 2m b := u ∈ L 2 (X, dvolgb); m gb u ∈ L2 (X, dvolgb) , m ∈ N0. We also define the following spaces for α ∈ R: Fα := ker x α H 2 b (gb + Vb). Since the eigenvalues of S 1 are {j 2 ; j ∈ N0}, the relative index theorem of Melrose [27, Section 6.2] shows that gb + Vb is Fredholm from xαH 2 b to xαH 0 b if α /∈ Z. Moreover, from [27, Section 2.2.4], we have that any solution of (gb + Vb)u = 0 in xαH 2 b has an asymptotic expansion of the form u ∼ ℓj j>α,j∈Z ℓ=0 x j (log x) ℓ uj,ℓ(θ) as x → 0 for some sequence (ℓj)j of nonnegative integers and some smooth function uj,ℓ on S 1 . In particular, it is easy to check that kerL 2 (X,dvolg)(g + V ) = F1+ɛ for ɛ ∈ (0, 1). <strong>THEOREM</strong> 6.2 Let (X, g) be an asymptotically Euclidean surface, let V1,V2 be two compactly supported smooth potentials, and let x be a boundary-defining function. Let ɛ ∈ (0, 1), and assume that, for any j ∈ Z and any function ψ ∈ ker x j−ɛ H 2 b (g + V1), thereisa ϕ ∈ ker x j−ɛ H 2 b (g + V2) such that ψ − ϕ = O(x ∞ ), and conversely. Then V1 = V2. Proof The idea is to reduce the problem to the compact case. First, we notice that by unique continuation, ψ = ϕ where V1 = V2 = 0. Now it remains to prove that, if Rη denotes the restriction of smooth functions on X to {x ≥ η} and if V is a smooth compactly supported potential in {x ≥ η}, then the set ∞ j=0 Rη(F−j−ɛ) is dense in the set NV of H 2 ({x ≥ η}) solutions of (g + V )u = 0. The proof is well known for positive frequency scattering (see, e.g., [28, Lemma 3.2]). Here it is very similar, so we do not give details. The main argument is to show that it converges in the L2 sense and then uses elliptic regularity; the L2 convergence can be shown as follows. Let f ∈ NV such that ∞ fψdvolg = 0, ∀ ψ ∈ F−j−ɛ. x≥η Then we want to show that f = 0. By[27, Proposition 5.64], there exists k ∈ N and a generalized right inverse Gb for Pb = gb + Vb (here, as before, x 2 Vb = V ) j=0
114 GUILLARMOU and TZOU in x−k−ɛ H 2 b such that PbGb = Id. This holds in x−k−ɛ H 2 b for k large enough since the cokernel of Pb on this space becomes zero for k large. Let ω = Gbf so that (gb + Vb)ω = f ; in particular, this function is zero in {x n, while Brown [5] studied the case of Lipschitz domains with a continuous conductivity. Since the result in our setting is not explicitly written down but is certainly known from specialists, we provide a short proof with few details using the approach of [5]. We prove the following.
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NEAR OPTIMAL BOUNDS IN FREIMAN’S
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SUR LA NON-DENSITÉ DES POINTS ENTI
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30 CASIM ABBAS Recall that the Reeb
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32 CASIM ABBAS that Here the energy
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34 CASIM ABBAS • uτ ( ˙S) ∩ u
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36 CASIM ABBAS punctured surfaces w
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38 CASIM ABBAS that is, the central
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40 CASIM ABBAS even have A(r) ≡ 0
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42 CASIM ABBAS are contact forms on
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44 CASIM ABBAS the standard complex
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46 CASIM ABBAS and Jδε2 = xγ1(r)
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48 CASIM ABBAS (λ0,J0) with vanish
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50 CASIM ABBAS defined as H 1 j (S)
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52 CASIM ABBAS and denoting the cor
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54 CASIM ABBAS formula (2.2) for th
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56 CASIM ABBAS Restricting any of t
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58 CASIM ABBAS where fτ (∞) = li
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60 CASIM ABBAS for τ
Page 61 and 62: 62 CASIM ABBAS Recalling our origin
Page 63 and 64: 64 CASIM ABBAS THEOREM 3.7 Let µ,
Page 65 and 66: 66 CASIM ABBAS Since w2 − w1C 1,
Page 67 and 68: 68 CASIM ABBAS be the conformal tra
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Page 71 and 72: 72 CASIM ABBAS and, for every R>0,
Page 73 and 74: 74 CASIM ABBAS W 1,p loc (C) since
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Page 77 and 78: 78 CASIM ABBAS is finite, the image
Page 79 and 80: 80 CASIM ABBAS Converting from coor
Page 81 and 82: 82 CASIM ABBAS [27] D. MCDUFF and D
Page 83 and 84: 84 GUILLARMOU and TZOU where ∂ν
Page 85 and 86: 86 GUILLARMOU and TZOU Carleman est
Page 87 and 88: 88 GUILLARMOU and TZOU where F ⊂
Page 89 and 90: 90 GUILLARMOU and TZOU LEMMA 2.2.4
Page 91 and 92: 92 GUILLARMOU and TZOU THEOREM 2.3.
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Page 95 and 96: 96 GUILLARMOU and TZOU enough, we h
Page 97 and 98: 98 GUILLARMOU and TZOU Using the fa
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Page 103 and 104: 104 GUILLARMOU and TZOU factor, or
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Page 107 and 108: 108 GUILLARMOU and TZOU where I1 =
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Page 117 and 118: 118 GUILLARMOU and TZOU Proof of Pr
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Page 129 and 130: 130 MARGULIS and MOHAMMADI We fix s
Page 131 and 132: 132 MARGULIS and MOHAMMADI Hence, i
Page 133 and 134: 134 MARGULIS and MOHAMMADI Hence th
Page 135 and 136: 136 MARGULIS and MOHAMMADI Indeed,
Page 137 and 138: 138 MARGULIS and MOHAMMADI For simp
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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