Proposition 5.27.356C1E4E8060FF

Proof. We take for $h$ a function from lemma 5.27.29 whose top-degree measure approximates $c\delta _ a$ and whose lower-degree currents are small; lemma 5.27.34 then controls the mixed terms.

Write $a=(a_1,a')$ and $m=n-1$, and fix $R=2\sup _\Omega |z_1|+2$. For $\delta >0$, put $H_\delta (z')=\log (|z'-a'|^2+\delta ^2)$. By the radial calculation in the proof of proposition 5.27.31, $(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$ is a probability measure tending to $\delta _{a'}$ as $\delta \downarrow 0$. Extend every $f_ i$ by zero outside $\Omega $, and choose $\delta $ such that

Choose compact sets $K,L\subset \Omega $ with $K\subset L^\circ $ and $K$ containing the supports of all $f_ i$, and a nonnegative smooth cutoff function $\psi $ with compact support in $\Omega $, equal to one near $L$. Put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. Expanding the top power of $\mathrm{dd}^{\mathrm{c}}(b+h)$, summing the bounds of lemma 5.27.34 with their binomial coefficients and the norms $\| f_ i\| _\infty $, we find $\gamma >0$ such that the mixed terms contribute less than $\tau /3$ to every test integral whenever $h$ is smooth and plurisubharmonic and

Fix $0<\eta \leq t/2$. Apply lemma 5.27.29 on $\mathbb C\times \mathbb C^ m$ with transverse variable $\zeta =z_1-a_1$ and $v=c^{1/m}\varepsilon ^{-1/m}H_\delta $. For $r<n$, lemma 5.27.33 gives

as $A\to \infty $. For the fixed $\delta $, the integrals of these limits against $\psi \beta ^{n-r}$ are finite, and their exponents of $\varepsilon $ are positive. Choose $0<\varepsilon \leq t/2$ so that each of them is less than $\gamma /2$. With these parameters fixed, choose $A$ so large that the integrals $\int _ L(\mathrm{dd}^{\mathrm{c}}h_ A)^ r\wedge \beta ^{n-r}$ are less than $\gamma $, that $v-A\leq 0$ on the projection of $\Omega $ to the $z'$ variables, and that

The last choice follows directly from (5.27.30), because the circles there shrink uniformly over the projected supports of the tests and $\varepsilon (\mathrm{dd}^{\mathrm{c}}v)^ m=c(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$.

Set $h=h_ A-\eta $. As in the proof of proposition 5.27.31, $h$ is smooth, bounded and nonpositive on $\Omega $, with

The logarithm has $L^ p(\Omega )$ norm bounded uniformly for $a_1$ in the projection of $\Omega $ to the first coordinate: after translation, its one-variable integral is bounded by polar integration on one fixed disk. This uniform bound, the weighted Chern–Levine–Nirenberg estimate and the range estimate (5.27.3) give the asserted bound for the integrals over $B'$. Since $\varepsilon +\eta \leq t$, both estimates hold with constants depending only on $\Omega $, $p$, $B'$ and $B$.

Finally, multilinearity of bounded products gives

The choice of $\delta $, the choice of $A$ for the top-degree term and the estimate of the mixed terms each contribute less than $\tau /3$ to every test integral. This proves the required inequalities, with one choice of $h$ for all finite $p$. $\square$