NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 63 where the last estimate holds in view of |ξ(ξ − z)| −p/(p−1) dξ d¯ξ ζ =z−1 ξ = C If q solves q = Ɣ(µq + µ),then C =|z| 2−2p/(p−1) ¯∂(αn − α) = µn ∂(αn − α) − µ∂α+ µn ∂α |z 2 ζ 2 − z 2 ζ | −p/(p−1) |z| 2 dζ d¯ζ |ζ (ζ − 1)| −p/(p−1) dζ d¯ζ . C 2πCp = µn ∂(αn − α) + µn − µ + (µn − µ)Ɣ(µq + µ), that is, the difference αn − α again satisfies an inhomogeneous Beltrami equation. By Theorem 3.4, wehave αn − α = A(µnqn + λn), where λn = µn − µ + (µn − µ)Ɣ(µq + µ) and where qn ∈ L p (C) solves qn = Ɣ(µnqn + λn). Combining this with (3.5)and(3.4), we obtain |αn(z) − α(z)| ≤Cp µnqn + λnL p (C) |z| 1−2/p ≤ (Cp sup µnL n ∞ (C) · c ′ pλnLp (C) + Cp λnLp (C)) |z| 1−2/p . (3.6) Since µn − µLp (C) → 0 and (µn − µ)Ɣ(µq + µ)Lp (C) → 0 by Lebesgue’s theorem we also have λnLp (C) → 0 and therefore αn → α uniformly on compact sets. Since ¯∂(αn − α) = µn ∂(αn − α) + λn and αn − α = A(µnqn + λn), we verify that and ∂(αn − α) = Ɣ(µnqn + λn) = qn ¯∂(αn − α) = µnqn + λn. Invoking (3.4) once again, we see that both ∂(αn − α)L p (C) and ¯∂(αn − α)L p (C) can be bounded from above by a constant times λnL p (C) which converges to zero. We also need some facts concerning the classical case where µ ∈ C k,α (BR(0)), BR(0) ={z ∈ C ||z|
64 CASIM ABBAS <strong>THEOREM</strong> 3.7 Let µ, γ, δ ∈ C k,α (BR ′(0)) with 0
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NEAR OPTIMAL BOUNDS IN FREIMAN’S
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Page 29 and 30: 30 CASIM ABBAS Recall that the Reeb
Page 31 and 32: 32 CASIM ABBAS that Here the energy
Page 33 and 34: 34 CASIM ABBAS • uτ ( ˙S) ∩ u
Page 35 and 36: 36 CASIM ABBAS punctured surfaces w
Page 37 and 38: 38 CASIM ABBAS that is, the central
Page 39 and 40: 40 CASIM ABBAS even have A(r) ≡ 0
Page 41 and 42: 42 CASIM ABBAS are contact forms on
Page 43 and 44: 44 CASIM ABBAS the standard complex
Page 45 and 46: 46 CASIM ABBAS and Jδε2 = xγ1(r)
Page 47 and 48: 48 CASIM ABBAS (λ0,J0) with vanish
Page 49 and 50: 50 CASIM ABBAS defined as H 1 j (S)
Page 51 and 52: 52 CASIM ABBAS and denoting the cor
Page 53 and 54: 54 CASIM ABBAS formula (2.2) for th
Page 55 and 56: 56 CASIM ABBAS Restricting any of t
Page 57 and 58: 58 CASIM ABBAS where fτ (∞) = li
Page 59 and 60: 60 CASIM ABBAS for τ
Page 61: 62 CASIM ABBAS Recalling our origin
Page 65 and 66: 66 CASIM ABBAS Since w2 − w1C 1,
Page 67 and 68: 68 CASIM ABBAS be the conformal tra
Page 69 and 70: 70 CASIM ABBAS so that, for z ∈ B
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
Page 75 and 76: 76 CASIM ABBAS (follows from 0 =
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.
Page 93 and 94: 94 GUILLARMOU and TZOU At a point (
Page 95 and 96: 96 GUILLARMOU and TZOU enough, we h
Page 97 and 98: 98 GUILLARMOU and TZOU Using the fa
Page 99 and 100: 100 GUILLARMOU and TZOU certain exp
Page 101 and 102: 102 GUILLARMOU and TZOU −1 where
Page 103 and 104: 104 GUILLARMOU and TZOU factor, or
Page 105 and 106: 106 GUILLARMOU and TZOU Proof of Pr
Page 107 and 108: 108 GUILLARMOU and TZOU where I1 =
Page 109 and 110: 110 GUILLARMOU and TZOU Proof of Le
Page 111 and 112: 112 GUILLARMOU and TZOU COROLLARY 6
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114 GUILLARMOU and TZOU in x−k−
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116 GUILLARMOU and TZOU Therefore,
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118 GUILLARMOU and TZOU Proof of Pr
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120 GUILLARMOU and TZOU [27] R. B.
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122 MARGULIS and MOHAMMADI multiple
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124 MARGULIS and MOHAMMADI As in [E
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126 MARGULIS and MOHAMMADI planes
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128 MARGULIS and MOHAMMADI α ∈ R
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130 MARGULIS and MOHAMMADI We fix s
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132 MARGULIS and MOHAMMADI Hence, i
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134 MARGULIS and MOHAMMADI Hence th
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136 MARGULIS and MOHAMMADI Indeed,
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138 MARGULIS and MOHAMMADI For simp
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140 MARGULIS and MOHAMMADI Note tha
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142 MARGULIS and MOHAMMADI (ii) The
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144 MARGULIS and MOHAMMADI needs to
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146 MARGULIS and MOHAMMADI Proof Le
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148 MARGULIS and MOHAMMADI all j>s,
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150 MARGULIS and MOHAMMADI 0 c
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152 MARGULIS and MOHAMMADI Some imp
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154 MARGULIS and MOHAMMADI of G, an
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156 MARGULIS and MOHAMMADI (ii) for
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158 MARGULIS and MOHAMMADI PROPOSIT
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160 MARGULIS and MOHAMMADI [EMM2]
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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