Let $\omega$ be a real closed $(1,1)$ form on $X_{U^{*}}$, with

$$\omega={i\over 2}(\alpha\vartheta_{\rm v}\wedge\bar{\vartheta}_{\rm v}+\beta\vartheta_{\rm h}\wedge\bar{\vartheta}_{\rm v}+\bar{\beta}\vartheta_{\rm v}\wedge\bar{\vartheta}_{\rm h}+\gamma\vartheta_{\rm h}\wedge\bar{\vartheta}_{\rm h}).$$

There exists a function $\varphi$ on $X_{U^{*}}$ such that $\omega=i\partial\bar{\partial}\varphi$ if and only if $\omega$ represents the zero cohomology class on $X_{U^{*}}$ and

$$\int_{X_{b}}\beta dx_{1}\wedge dx_{2}=0$$

for all $b\in U^{*}$. Furthermore, for $0<r_{1}<r_{2}$, let $U_{r_{1},r_{2}}=\{y\in U\,|\,r_{1}<|y|<r_{2}\}$. If $r_{1}<r_{1}^{\prime}<r_{2}^{\prime}<r_{2}$ and $\overline{U}_{r_{1},r_{2}}\subseteq U^{*}$, then there exists a constant $C$ depending only on $r_{1},r_{2},r_{1}^{\prime},r_{2}^{\prime}$ and the periods of $f$ over $U_{r_{1},r_{2}}$ such that $\varphi$ can be chosen with

$$\|\varphi\|^{\prime}_{C^{k+2,\alpha}}\leq C(\|\alpha\|_{C^{k,\alpha}}+\|\beta\|_{C^{k,\alpha}}+\|\gamma\|_{C^{k,\alpha}}).$$

Here, we compute the $C^{k,\alpha}$ norm of a function on $f^{-1}(U_{r_{1},r_{2}})$ by thinking of them as functions on ${\cal T}^{*}_{U_{r_{1},r_{2}}}$, which we embed in ${\bf C}^{2}$ by the coordinates $x$ and $y$. We can then use the standard $C^{k,\alpha}$ norms on a bounded open set of ${\cal T}_{U_{r_{1},r_{2}}}^{*}$ which contains a fundamental domain of each fibre. The norm $\|\cdot\|^{\prime}_{C^{k,\alpha}}$ denotes the similar norm of a function over $U_{r_{1}^{\prime},r_{2}^{\prime}}$.