Now we introduce the global weighted Hölder norms $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}_{\nu}$ by demanding that up to uniform equivalence the norm is

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  $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}\cap\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}$, as defined in Section 3.4.

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  $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}$ on $\mathbb{C}^{3}$ for $|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}$.

On overlapping regions the definitions are equivalent.