Lemma 8.11.4.9AF7CFAD9A0B Let $0\leq q\leq m-1$ and let $S$ be a smooth real $(1,1)$-form in $\overline{\Gamma _ m(\omega )}$ pointwise. Put $T=S^ q\omega ^{n-q-2}$. For smooth real $f$ and nonnegative Borel weights $a$, supported on a fixed compact set,
\[ \left|\int a\, \mathrm df\wedge \mathrm d^{\mathrm c}\omega \wedge T\right| \leq C\left(\int a\, \mathrm df\wedge \mathrm d^{\mathrm c}f\wedge \omega \wedge T\right)^{1/2} \left(\int a\, \omega ^2\wedge T\right)^{1/2}. \]
The corresponding inequality with $\mathrm d^{\mathrm c}f\wedge \mathrm d\omega $ holds as well. For a mixed product of $q$ cone forms in place of $S^ q$, it suffices to sum these bounds over the finitely many subset sums of its factors.
Proof.
For $q\leq m-2$, diagonalize $S$ in an $\omega $-unitary frame. Every diagonal coefficient of $S^ q\omega ^{n-q-2}$ is a positive multiple of $S_{q;ij}(\lambda )$ for two omitted coordinates. This is nonnegative by deleted-coordinate positivity, because $q+2\leq m$. The form is a nonnegative sum of simple positive coordinate forms.
For one such simple form, only the remaining complex two-plane contributes to a complementary wedge. The coefficients of $\mathrm d^{\mathrm c}\omega $ are uniformly bounded in unitary frames on the fixed compact set. Its wedge with $\mathrm df$ is consequently bounded by a constant times the gradient norm on that plane. The wedges with $\omega ^2$ and with $\mathrm df\wedge \mathrm d^{\mathrm c}f\wedge \omega $ give, up to fixed positive constants, one and the square of that norm. Pointwise Cauchy–Schwarz over the simple summands, followed by integral Cauchy–Schwarz with weight $a$, proves the estimate. Conjugation gives the other estimate.
For $q=m-1$, at one point choose an $\omega $-unitary coframe $e_ i$ that diagonalizes $S$. Put $\epsilon _ i=\mathrm i e_ i\wedge \bar e_ i$ and $\Omega =\epsilon _1\cdots \epsilon _ n$, so that $\omega =\sum _ i\epsilon _ i$ and $S=\sum _ i\lambda _ i\epsilon _ i$. Writing $B_{ij}=S_{q;ij}(\lambda )$ and $D_ i=S_{q;i}(\lambda )$, ordinary wedge expansion gives
\[ S^ q\omega ^{n-q-2} =q!(n-q-2)!\sum _{i<j}B_{ij} \prod _{k\notin \{ i,j\} }\epsilon _ k. \]
Only the remaining two-plane contributes to its wedge with $\mathrm df\wedge \mathrm d^{\mathrm c}\omega $. If $\partial f=\sum _ i f_ i e_ i$, its absolute density relative to $\Omega $ is bounded by
\[ C\sum _{i<j}(|f_ i|+|f_ j|)|B_{ij}|. \]
The constant bounds the tensor coefficients of $\mathrm d^{\mathrm c}\omega $ in all unitary frames on the fixed compact set. Diagonalization is only pointwise; no derivative of this moving frame is taken.
8.2.8 and finite Cauchy–Schwarz bound this sum by
\[ C\sum _ i|f_ i|\sqrt{D_ i}\sum _ j\sqrt{D_ j} \leq C\left(\sum _ i|f_ i|^2D_ i\right)^{1/2} \left(\sum _ iD_ i\right)^{1/2}. \]
Here $\sum _ iD_ i=(n-q)S_ q(\lambda )$. Moreover
\[ \begin{aligned} S^ q\omega ^{n-q}& =q!(n-q)!S_ q(\lambda )\Omega ,\\ \mathrm df\wedge \mathrm d^{\mathrm c}f\wedge S^ q\omega ^{n-q-1} & =\kappa q!(n-q-1)!\sum _ i|f_ i|^2D_ i\, \Omega , \end{aligned} \]
where $\kappa =1/(2\pi )$ by the book's $\mathrm d^{\mathrm c}$ normalization. These two densities are nonnegative. Thus the pointwise torsion density is bounded by a constant times their geometric mean. Multiplication by $a$ and integral Cauchy–Schwarz give the assertion, including the cases where either density vanishes. The other wedge is conjugate up to a fixed sign, which does not affect its absolute value. For $q\geq 1$, use the exact polarization identity
\[ q!\, \alpha _1\wedge \cdots \wedge \alpha _ q =\sum _{J\subset \{ 1,\ldots ,q\} } (-1)^{q-|J|}\left(\sum _{i\in J}\alpha _ i\right)^ q. \]
Every nonempty subset sum remains in the cone. Applying the single power estimate to each term and taking absolute values proves the mixed assertion. At $q=0$ use the single term $S^0=1$. No Cauchy–Schwarz assertion for an arbitrary weakly positive $(n-2,n-2)$-form is needed.
$\square$
Pome
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