Lemma 13.4.1263D0F5AEE0A

Proof. We treat $r\geq 2$ and smooth $V$ by an integration by parts on small balls, then general $V$ by monotone approximation, and reduce $r=1$ to $r=2$ by adding a variable.

Step 1: $r\geq 2$ and $V$ smooth. Here $\| v\| _{L^\infty (L)}=\sup _ L|v|$, as in theorem 7.2.5. If it vanishes, then $v=0$ on $L$, so $\mathrm{dd}^{\mathrm{c}}v=0$ on $\operatorname{Int}L\supset K$, and the left side vanishes. Otherwise both sides are homogeneous of degree one in $v$, and the left side does not change when a constant is added to $v$; replacing $v$ by $v/(2\| v\| _{L^\infty (L)})-3/2$, it therefore suffices to prove that $\int _ K(-V)\, \mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}\leq C\int _ L(-V)T\wedge \beta ^ r$ for plurisubharmonic functions $v$ satisfying $-2\leq v\leq -1$ on $\operatorname{Int}L$. Cover $K$ by finitely many balls $B(a,R_1)$ such that $\overline{B(a,R)}\subset \operatorname{Int}L$ for some $R>R_1$; it suffices to bound the integral over one ball $B(a,R_1)$.

Put $B=B(a,R)$, $\psi =|z-a|^2-R^2$ and $b=\mathrm{dd}^{\mathrm{c}}\psi =\beta /\pi $. Fix $R_1<R_2<R$, and let $\widetilde v=\max \{ v,A\psi \} $, with $A=2/(R^2-R_2^2)$, be the collar modification (7.3.2) of $v$ on $\operatorname{Int}L$, with $N=2$ and with $R^2-R_2^2$ in place of $\delta $. Thus $\widetilde v$ is plurisubharmonic on $\operatorname{Int}L$ and takes values in $[-2,0)$ on $B$; it equals $v$ on $B(a,R_2)$, and $A\psi $ on the collar $\{ R^2-1/A<|z-a|^2<R^2\} $. Choose $0<\delta <1/A$ and put $B_\delta =\{ \psi <-\delta \} $, so that $\widetilde v=A\psi $ on $B\setminus B_\delta $ and $B(a,R_1)\subset B_\delta $. Choose a smooth function $\kappa $ with $0\leq \kappa \leq R^2$, with compact support in $B$ and equal to $-\psi $ on a neighborhood of $\overline{B_\delta }$; this is possible because $0<-\psi \leq R^2$ on $B$. Put $S=\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T$ and $Q=\mathrm{dd}^{\mathrm{c}}V\wedge T$; since $V$ is smooth and plurisubharmonic, $Q$ is a closed positive current of bidimension $(r-1,r-1)$. Since $\mathrm{dd}^{\mathrm{c}}\kappa =-b$ near $\overline{B_\delta }$ and $S=Ab\wedge T$ on $B\setminus B_\delta $,

For the second line we integrate by parts against the smooth compactly supported function $\kappa $: since $V$ is smooth and $\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2}$ is closed, $\mathrm{dd}^{\mathrm{c}}(V\mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2})=\mathrm{dd}^{\mathrm{c}}V\wedge \mathrm{dd}^{\mathrm{c}}\widetilde v\wedge T\wedge b^{r-2}$, and the Bedford–Taylor product is symmetric with respect to the smooth factor.

We bound the two terms on the right. Choose a compact set $L'$ with $\overline B\subset \operatorname{Int}L'$ and $L'\subset \operatorname{Int}L$. Since $0\leq \kappa \leq R^2$, the Chern–Levine–Nirenberg inequality (theorem 7.2.5) applied to $Q$ bounds the first term by a constant times the trace mass of $Q$ on $\overline B$. To bound this mass, test $Q$ against a smooth cutoff $0\leq \chi \leq 1$ equal to one on $\overline B$ and supported in $L'$:

since $V\leq 0$ and the coefficients of $T\wedge \mathrm{dd}^{\mathrm{c}}\chi \wedge \beta ^{r-1}$ are bounded by a fixed multiple of the trace measure. The collar term has the same bound, because $\mathrm{dd}^{\mathrm{c}}\kappa $ is a fixed smooth form. Finally, $S=\mathrm{dd}^{\mathrm{c}}v\wedge T$ on $B(a,R_1)$ and $-V\geq 0$, so the left side of the display bounds $\pi ^{1-r}\int _{B(a,R_1)}(-V)\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$. Summing over the finitely many balls proves the estimate, with a constant depending only on the chosen radii.

Step 2: $r\geq 2$ and $V$ arbitrary. Choose a compact set $L_1$ with $K\subset \operatorname{Int}L_1$ and $L_1\subset \operatorname{Int}L$, depending only on $K$ and $L$. The mollifications $V_ l=V*\rho _{\varepsilon _ l}$, with $\varepsilon _ l\downarrow 0$, are defined, smooth and plurisubharmonic on a neighborhood of $L_1$ for large $l$, and they decrease to $V$ (proposition 5.9.2(2)); in particular $V\leq V_ l\leq 0$, so that $0\leq -V_ l\leq -V$. Step 1 only involves the weight on a neighborhood of the larger compact set, so it applies to $V_ l$ and the pair $(K,L_1)$ and gives

As $l\to \infty $, $-V_ l$ increases to $-V$, and monotone convergence against the positive measure $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta ^{r-1}$ proves the estimate.

Step 3: $r=1$. Let $t$ be the coordinate of $\mathbb C$, put $\beta _ t=\frac{\mathrm i}{2}\mathrm dt\wedge \mathrm d\bar t$, let $D'\Subset D''\Subset D$ be the discs about $0$ of radii $1,2,3$, and let $P\colon U\times D\to U$ be the projection. By proposition 5.6.2, $P^*T$ is a closed positive current of bidimension $(2,2)$ on $U\times D$. Moreover, by corollary 5.6.4(1), the pullback of a current $R=\sum R_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ of order zero on $U$ is $P^*R=\sum (R_{IJ}\otimes \lambda _ D)\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$, where $\lambda _ D$ is Lebesgue measure on $D$. Since $v$ is a locally bounded Borel function, $P^*(vT)=(v\circ P)P^*T$ by corollary 5.6.4(2), and since $P^*$ commutes with $\partial $ and $\bar\partial $,

The functions $V\circ P\leq 0$ and $v\circ P$ are plurisubharmonic (proposition 5.9.3), and $\| v\circ P\| _{L^\infty (L\times \overline{D''})}=\| v\| _{L^\infty (L)}$. We apply Step 2 with $r=2$ to these data, to the Euclidean form $\beta +\beta _ t$ and to the compact sets $K\times \overline{D'}\subset \operatorname{Int}L\times D''=\operatorname{Int}(L\times \overline{D''})$. Since $T\wedge \beta ^2=0$ and $\mathrm{dd}^{\mathrm{c}}v\wedge T\wedge \beta =0$ for bidegree reasons, the trace measures in Step 2 are

that is, the products of the measures $2T\wedge \beta $ and $\mathrm{dd}^{\mathrm{c}}v\wedge T$ with Lebesgue measure on $D$. By Fubini's theorem (proposition 2.2.7), the inequality of Step 2 therefore reads

where $C$ is the constant of Step 2 for $K\times \overline{D'}$ and $L\times \overline{D''}$ in dimension $n+1$, which depends only on $K$, $L$ and $n$. This is the case $r=1$, with $C_{K,L}=8C$. $\square$