Proposition 4.31 . [046J]

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Proposition 4.31.

(Periodic Euclidean region) Let $-3<\delta<0$ . Let $f$ be a function compactly supported in $\mathcal{B}^{-}_{\nu}\cap\{R>A^{-1/2}\}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ (respectively $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ ). Then $\Delta_{a}^{-1}f$ satisfies the $g_{a}$ -Hessian bound on $\mathcal{B}^{-}_{\nu}\cap\{R\gtrsim A^{-1/2}\}$ ,

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu,\quad\text{resp. }\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta}}\leq C.