ScalingStacks

Lemma 4.30 . [046G]

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Lemma 4.30.

The volume form error E(1)E^{(1)} satisfies the estimate on Mν−M^{-}_{\nu}:

‖E(1)‖C−1,01,α≤CA−3/4ν2.\left\lVert E^{(1)}\right\rVert_{C^{1,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

In the subregion {R≳A1/4}⊂Mν−\{R\gtrsim A^{1/4}\}\subset M^{-}_{\nu}, and any fixed large kk,

‖E(1)‖C−1,0k,α≤CA−3/4ν2.\left\lVert E^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

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