The local Dolbeault lemma on nested polydiscs

If $\eta =0$, take $\psi =0$. Otherwise, by the unique type expansion [F1], some barred coordinate occurs. Let $k$ be the largest index appearing in any barred multi-index of $\eta$. Group the terms uniquely as $\eta =d{\bar{z}}_k\wedge \tau +\theta$, absorbing the permutation signs into $\tau$, so that neither $\tau$ nor $\theta$ contains a barred differential with index at least $k$. Here $\tau$ has type $(p,q-1)$ and $\theta$ has type $(p,q)$.

The product rule [F2] gives $0=\bar{\mathrm{\partial }}\eta =-d{\bar{z}}_k\wedge \bar{\mathrm{\partial }}\tau +\bar{\mathrm{\partial }}\theta$. For each $\mathrm{\ell }>k$, the coefficient terms containing both $d{\bar{z}}_k$ and $d{\bar{z}}_{\mathrm{\ell }}$ can only come from $-d{\bar{z}}_k\wedge \bar{\mathrm{\partial }}\tau$: the form $\theta$ has no barred factor with index at least $k$, so $\bar{\mathrm{\partial }}\theta$ has no term containing both indices. Uniqueness of the wedge expansion [F1] therefore gives ${\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}\tau =0$ for every $\mathrm{\ell }>k$, coefficientwise.

Suppose the current residual form is smooth and closed on a working polydisc $Q=\prod E_j$ containing $P^{\mathrm{\prime }}$, and its largest barred index is $k$. In the descending process, the $k$th factor has not yet been shrunk, so $E_k=D_k$ and $\overline{D_k^{\mathrm{\prime }}}⋐E_k$. Apply the same local operator $T_k$ from [F4] to each of the finitely many coefficient functions of $\tau$ in the unique expansion [F1]. It produces a smooth form ${\psi }_k=T_k\tau$ on a working polydisc $Q^{\mathrm{\prime }}$ that still contains $P^{\mathrm{\prime }}$, with ${\mathrm{\partial }}_{{\bar{z}}_k}{\psi }_k=\tau$. Since every coefficient of $\tau$ has zero ${\bar{z}}_{\mathrm{\ell }}$ derivative for $\mathrm{\ell }>k$ by step 2.1, [F4] preserves those equations for ${\psi }_k$. The full AC premise needed for [F4] is part of the given hypothesis by [F5].

By the coefficient formula [F1], $\bar{\mathrm{\partial }}{\psi }_k=d{\bar{z}}_k\wedge \tau +{\delta }_k$, where every barred index in ${\delta }_k$ is less than $k$: the $k$th derivative supplies the displayed first term, derivatives with index greater than $k$ vanish by step 3.1, and the remaining derivatives have index less than $k$. Set ${\eta }_{k-1}:=\eta -\bar{\mathrm{\partial }}{\psi }_k=\theta -{\delta }_k$. It has type $(p,q)$ and contains only barred indices less than $k$. It remains closed, since $\bar{\mathrm{\partial }}{\eta }_{k-1}=\bar{\mathrm{\partial }}\eta -{\bar{\mathrm{\partial }}}^2{\psi }_k=0$ by [F3].

Repeat steps 1.1–3.1 on each nonzero residual, in descending order of the largest barred index. At each stage the local transform shrinks only the coordinate currently being processed and its output domain still contains $P^{\mathrm{\prime }}$, so all later residuals and the previously constructed primitives restrict to a common neighborhood of $P^{\mathrm{\prime }}$. If no barred factor occurs at index $k$, skip that transform and retain the current domain. After index $1$ is removed, the residual has barred degree $q>0$ but no barred basis factor; the unique expansion [F1] forces it to be zero. There are at most $n$ transforms.

Let $\psi$ be the sum of the finitely many forms ${\psi }_k$ constructed by the repeated step 3.1 procedure in step 5.1, restricted to $P^{\mathrm{\prime }}$. The successive residual identities of step 4.1 telescope, and the final residual is zero by step 5.1; hence $\bar{\mathrm{\partial }}\psi =\eta {\mathrm{\mid }}_{P^{\mathrm{\prime }}}$. Each summand is smooth on a neighborhood of $P^{\mathrm{\prime }}$, so $\psi$ is smooth there and has type $(p,q-1)$.