Proposition 8.9.3.EB2FA13CDDE0 For $1\leq p<m$, the recursion defines a real closed current of order zero and bidegree $(p,p)$. If $M=\max _ a\| u_ a\| _{L^\infty (X)}$, then
\[ \| D_ p(u_1,\ldots ,u_ p)\| _ K \leq C_{K,U,\omega ,p}(1+M)^ p. \]
For $p=m$, it defines a real closed distributional current. For any smooth admissible sequences $u_ a^{j_ a}\downarrow u_ a$ with a common uniform bound, one has
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8.9.4
\begin{equation} D_ p(u_1,\ldots ,u_ p) =\lim _{j_ p\to \infty }\cdots \lim _{j_1\to \infty } \mathrm{dd}^{\mathrm{c}}u_1^{j_1}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ p^{j_ p} \end{equation}
in distributions. The index $j_1$ tends to infinity first. Below degree $m$, all these limits have locally uniformly bounded coefficient variations.
Proof.
The case $p=1$ is the preceding lemma. Suppose the assertions hold through degree $p-1<m$. Since $u_ p$ is bounded Borel and $D_{p-1}$ has measure coefficients, their product is well defined. Applying $\mathrm{dd}^{\mathrm{c}}$ gives a real current that is both $\mathrm d$- and $\mathrm d^{\mathrm c}$-closed.
For a fixed smooth complementary test form $\eta $, dominated convergence against the fixed signed measures gives
\[ \begin{aligned} \langle D_ p,\eta \rangle & =\langle u_ pD_{p-1},\mathrm{dd}^{\mathrm{c}}\eta \rangle \\ & =\lim _{j_ p\to \infty } \langle u_ p^{j_ p}D_{p-1},\mathrm{dd}^{\mathrm{c}}\eta \rangle . \end{aligned} \]
With $j_ p$ fixed, the inductive iterated limit for $D_{p-1}$ can be paired with the smooth form $u_ p^{j_ p}\mathrm{dd}^{\mathrm{c}}\eta $. Integration by parts for the smooth closed factors identifies the resulting pairing with the smooth product on the right of (8.9.4). This proves that identity also when $p=m$.
For $p<m$, expand the smooth product using $\mathrm{dd}^{\mathrm{c}}u_ a^{j_ a}=\sigma _ a^{j_ a}-\omega $. A term with $q$ admissible factors, $0\leq q\leq p$, has the form
\[ \sigma _{a_1}^{j_{a_1}}\wedge \cdots \wedge \sigma _{a_ q}^{j_{a_ q}}\wedge \omega ^{p-q}. \]
Every such $q$ is strictly less than $m$. Apply 8.2.7 to each coefficient test, and then the smooth mass estimate from 8.7.1 to the potential sum. The term $q=0$ is the fixed smooth form $\omega ^ p$. For $q\geq 1$, the sum of its admissible forms is $q(\omega +\mathrm{dd}^{\mathrm{c}}\bar u)$, where $\bar u$ is their average and $\| \bar u\| _\infty \leq M+1$. The result is a coefficient variation bound $C(1+M)^ p$, after choosing the approximations with norm at most $M+1$ as in the preceding proof. For arbitrary uniformly bounded approximations the same argument uses their common bound. The bounds survive each distributional limit, proving that $D_ p$ has order zero and the stated estimate. This completes the induction. No coefficient bound for the unwedged degree-$m$ current is used or asserted.
$\square$
Pome
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