By Lemma 2.14 the initial volume form error is

$$\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1,-1}}\leq C.$$

Applying Corollary 2.24 we can solve the Poisson equation with estimate

$$\Delta_{g^{(2)}}u_{1}=-2E^{(2)},\quad\left\lVert du_{1}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},$$

so in particular

$$\left\lVert\partial\bar{\partial}u_{1}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq C,\quad\left\lVert(\partial\bar{\partial}u_{1})^{2}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.$$

Now $(\omega^{(2)^{\prime}})^{3}=(\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}u_{1})^{3}=(\omega^{(2)})^{3}(1+\frac{1}{2}\Delta_{g^{(2)}}u_{1}+O(|\partial\bar{\partial}u_{1}|^{2}))$, so the new volume form error has improved decay:

$$\frac{3}{4}(E^{(2)^{\prime}}+1)\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(2)^{\prime}})^{3},\quad\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2,-2+2\epsilon}}\leq\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.$$

We notice that the modification to $\omega^{(2)}$ is $C^{0}$-small outside a compact region, where the positive definite condition for the Kähler metric is not affected. Inside the compact set we can add on a locally supported semipositive (1,1)-form to guarantee the Kähler condition, as we have done in Section 2.6. We abuse notation to write this Kähler metric after surgery as $\omega^{(2)^{\prime}}$, which inherits all the analytic properties of $\omega^{(2)}$.

Applying Corollary 2.24 again to solve the Poisson equation with background metric $g^{(2)^{\prime}}$,

$$\Delta_{g^{(2)^{\prime}}}u_{2}=-2E^{(2)^{\prime}},\quad\left\lVert du_{2}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},$$

and using $(\omega^{(2)^{\prime}}+\sqrt{-1}\partial\bar{\partial}u_{2})^{3}=(\omega^{(2)^{\prime}})^{3}(1+\frac{1}{2}\Delta_{g^{(2)^{\prime}}}u_{2}+O(|\partial\bar{\partial}u_{2}|^{2})),$ the new volume form error is now bounded in $C^{k,\alpha}_{-4,-4+4\epsilon}$-norm. Another surgery in the compact region ensures the Kähler property. ∎