Corollary 4.6 . [023N]

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Corollary 4.6.

Let $f$ be a smooth function on the Tian-Yau space satisfying the fast decay

|\nabla_{TY}^{k}f|=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}}),\quad\forall k\geq 0,

then there is a smooth function $U$ with $\Delta_{TY}U=f$ in the form

U=\text{const}\cdot u_{0}+\tilde{U},\quad|\nabla_{TY}^{k}U|=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}}),\quad\forall k\geq 0,

for some possibly shrinked $c>0$ .