Corollary 2.24 . [0415]

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Corollary 2.24.

(Solution to the Poisson equation) Let $(\delta,\tau)$ fall within the good range of weight exponents. Then given $f\in C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$ , there exists a function $u$ solving $\Delta_{g^{(2)}}u=f$ with gradient bound

\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1,\tau}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4}\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.