Lemma 2.5 . [03ZV]

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Lemma 2.5.

In the above region (2.14) near $\mathfrak{D}_{1}$ ,

\begin{cases}\left\lVert\alpha_{2}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2},\\ \left\lVert\alpha_{3}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2},\\ \left\lVert\alpha_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq CA^{1/2}.\end{cases}

Consequently, if $C_{1}$ is chosen to be large enough, then

|V^{ij}_{(1)}-V^{ij}_{\text{Taub}}|\leq\frac{CA^{1/4}}{|\vec{\mu}|_{a}}\ll a_{ij}\leq V^{ij}_{\text{Taub}},\quad|W_{(1)}-W_{\text{Taub}}|\leq\frac{CA^{3/4}}{|\vec{\mu}|_{a}}\ll A\leq W_{\text{Taub}}.