By definition $\phi$ is smooth on $Q_{T}\setminus H\times(-\infty,0]$. Using (4.17) it is easy to see that $\phi$ extends to a continuous function on $Q_{T}$. Hence for all fixed $z$, the following equation holds in the sense of currents on $D$

(4.150)

$$T\omega_{D}+d_{D}d_{D}^{c}\phi(z)=\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z).$$

Elliptic regularity then implies that $\phi$ is smooth on each slice $\{z\}\times D$ for $z\neq 0$. Now for $z\leq 0$ we can write

(4.151)

$$\phi(z)=\int_{T_{-}}^{z}uhdu+\phi(T_{-}).$$

We then see that $\phi$ is indeed smooth on $Q_{T}\setminus P$. Over the $S^{1}$ fibration $\mathcal{M}$, we know $\phi$ is globally continuous, and it is smooth and satisfies the equation (4.140) on ${\mathcal{M}^{*}}$. Now again by standard theory on pluri-subharmonic functions we conclude the current equation holds on $\mathcal{M}$. Since we know $\omega$ is $C^{2,\alpha}$ in local holomorphic coordinates on $\mathcal{M}$, elliptic regularity gives that $\phi$ is in $C^{4,\alpha}$ in local holomorphic coordinates. This implies that $\phi$ is $C^{3,\alpha}$ in the smooth topology we defined, since we know the holomorphic coordinate functions are $C^{3,\alpha}$. ∎