Proposition 8.3 (The Weighted Schauder Estimate) . [03JQ]

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Proposition 8.3 (The Weighted Schauder Estimate).

Consider $(\mathcal{M},g_{\beta})$ with a sufficiently large gluing parameter $\beta>0$ . Then there exists a uniform constant $C>0$ (independent of $\beta$ ) such that for every $\omega\in\Omega^{1}(\mathcal{M})$ , it holds that

(8.7)

\|\omega\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}\leq C\Big(\|\mathscr{D}_{g}\omega\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}+\|\omega\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M})}\Big).