Lemma 3.19 . [0437]

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Lemma 3.19.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ . Then $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is estimated on $\mathcal{B}^{+}_{\nu}$ by

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\nu\begin{cases}A^{\delta/4}\ell^{\delta}\quad&\ell\lesssim A^{1/2},\\ A^{3\delta/4}\quad&\ell\gtrsim A^{1/2}.\end{cases}