Proposition 3.22 . [043E]

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Proposition 3.22.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ . Then in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

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