Proposition 6.9 (Weighted Schauder estimate on the neck, the global version) . [0558]

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Proposition 6.9 (Weighted Schauder estimate on the neck, the global version).

For every sufficiently large parameter $T\gg 1$ , let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$ -invariant Kähler metric $\omega_{T}$ constructed in Section 4.1. Then the following estimate hold:

(6.62)

\|u\|_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal{M}_{T})}\leq C\Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}+\Big\|\frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu+1,\mu}^{1,\alpha}}+\|u\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{T})}\Big),

where the constant $C>0$ is independent of $T\gg 1$ .