Lemma 2.19 . [040T]

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Lemma 2.19.

In the situation of Lemma 2.18,

\left\lVert\Delta_{g^{(2)}}\Delta_{a}^{-1}f-f\right\rVert_{C^{k,\alpha}_{\delta-1,\tau}(\mathbb{C}^{3})}\leq C.

In particular for a large enough constant $C_{2}$ ,

\left\lVert\Delta_{g^{(2)}}\Delta_{a}^{-1}f-f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3}\cap\{\text{dist}(\cdot,\mathfrak{D})>C_{2}/2\})}\leq\frac{C}{C_{2}}\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}\ll\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.