Lemma 14.3.1.E2F31793530D There is a constant $C$ depending on $\omega $ such that, for every positive $(1,1)$-form $\eta $, every smooth real $f$, every $\delta >0$, and $0\leq k\leq n-2$,
\[ \begin{split} |\mathrm d^{\mathrm c}f\wedge \mathrm{d}\omega \wedge \eta ^ k\wedge \omega ^{n-k-2}| \leq {}& \frac C\delta \, \mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge \eta ^ k\wedge \omega ^{n-k-1}\\ & +C\delta \, \eta ^ k\wedge \omega ^{n-k}. \end{split} \]
Here the absolute value of a top-degree form is the absolute value of its coefficient relative to the positive volume form $\omega ^ n$.
Proof.
At a given point, choose an $\omega $-unitary frame in which $\eta $ is diagonal, with positive eigenvalues $\lambda _1,\ldots ,\lambda _ n$. Expand $\eta ^ k$ as a sum over $k$-element subsets $I$ of $\{ 1,\ldots ,n\} $, with coefficients that are positive dimensional multiples of $\lambda _ I=\prod _{i\in I}\lambda _ i$. After wedging with $\mathrm d^{\mathrm c}f\wedge \mathrm{d}\omega \wedge \omega ^{n-k-2}$, each surviving term is a bounded coefficient of $\mathrm{d}\omega $ times $\lambda _ I$ times one first derivative $f_ a$ with $a\notin I$. Young's inequality bounds its absolute value by
\[ C\lambda _ I|f_ a|\leq (C/\delta )\lambda _ I|f_ a|^2 +C\delta \lambda _ I. \]
The sum of the first terms is bounded by a fixed multiple of the first form on the right of the assertion, whose diagonal expansion consists of the terms $\lambda _ I|f_ a|^2$ with $a\notin I$. The second form on the right contains all the $\lambda _ I$. Since there are finitely many indices and $\mathrm{d}\omega $ is bounded, the constant $C$ is uniform.
$\square$
Pome
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