ScalingStacks

Proposition 4.3 . [023J]

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Proposition 4.3.

Let ff be a smooth function on the Tian-Yau space satisfying the integral normalization ∫f​ωT​Yn−1=0\int f\omega_{TY}^{n-1}=0 and the fast decay

|∇T​Ykf|=O⁡(e−c​rD1n−1n),∀k≥0,|\nabla_{TY}^{k}f|=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}}),\quad\forall k\geq 0,

then there is a smooth function uu with ΔT​Y​U=f\Delta_{TY}U=f with fast decay

|∇T​YkU|=O⁡(e−c​rD1n−1n),∀k≥0,|\nabla_{TY}^{k}U|=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}}),\quad\forall k\geq 0,

where the decay rate c>0c>0 may be shrinked.

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