ScalingStacks

Lemma 4.7 . [023Q]

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Lemma 4.7.

In the region {x2≤x1,and ​x1≫1}\{x_{2}\leq x_{1},\text{and }x_{1}\gg 1\},

‖d​x1‖k,α,l​o​c=O⁡(x1n−22​n),‖d​x2‖k,α,l​o​c=O⁡(x1−n−22​n​(n−1)​x2n−22​(n−1)),\left\lVert dx_{1}\right\rVert_{k,\alpha,loc}=O(x_{1}^{\frac{n-2}{2n}}),\quad\left\lVert dx_{2}\right\rVert_{k,\alpha,loc}=O(x_{1}^{-\frac{n-2}{2n(n-1)}}x_{2}^{\frac{n-2}{2(n-1)}}),
d​dc​(x1−d1d2​x2)=0,‖d​dc​x2‖k,α,l​o​c=O⁡(x2−1n−1​x1−n−2n⁡(n−1)).dd^{c}(x_{1}-\frac{d_{1}}{d_{2}}x_{2})=0,\quad\left\lVert dd^{c}x_{2}\right\rVert_{k,\alpha,loc}=O(x_{2}^{-\frac{1}{n-1}}x_{1}^{-\frac{n-2}{n(n-1)}}).

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