ScalingStacks

Lemma 3.25 . [043M]

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Lemma 3.25.

Over the region {|μ→|a≤13A1/2}\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\},

|Ψ1−1∗αi−α~i|≤CA−3/4|μ→|a,|∇gak(Ψ1−1∗αi−α~i)|ga≤CA−1/4−k/2,i=1,2,3.|\Psi_{1}^{-1*}\alpha_{i}-\tilde{\alpha}_{i}|\leq CA^{-3/4}|\vec{\mu}|_{a},\quad|\nabla_{g_{a}}^{k}(\Psi_{1}^{-1*}\alpha_{i}-\tilde{\alpha}_{i})|_{g_{a}}\leq CA^{-1/4-k/2},\quad i=1,2,3.

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