Theorem 7.4.17C68264563C0

Proof. For $q\le n-1$ we integrate $V$ by parts against a cutoff function on a ball, after making the factors coincide near its boundary (Steps 1–4); the case $q=n$ follows by adding one variable (Step 5).

Step 1: reductions. By theorem 5.9.1, $V$ is locally integrable, so $\| V\| _{L^1(L)}<\infty $. For $q=0$ the left-hand side equals $n!\int _ K|V|\, \mathrm{d}V_{2n}\le n!\| V\| _{L^1(L)}$, by (5.2.8). Let $q\ge 1$. If $\| u_ l\| _{L^\infty (L)}=0$ for some $l$, then $u_ l=0$ on $\operatorname{Int}L$, so the product vanishes on $\operatorname{Int}L\supset K$ by locality. Otherwise we replace each $u_ l$ by $u_ l/(2\| u_ l\| _{L^\infty (L)})-3/2$, which takes values in $[-2,-1]$ on $L$; by proposition 7.2.2(3) this multiplies the left-hand side by $\prod _ l(2\| u_ l\| _{L^\infty (L)})^{-1}$. For each $a\in K$ choose $R_ a>0$ with $\overline{B(a,R_ a)}\subset \operatorname{Int}L$; finitely many of the balls $B(a,R_ a/2)$ cover $K$, and this choice depends only on $K$ and $L$. Hence it suffices to prove the following: for concentric balls $B'=B(a,R')$ and $B=B(a,R)$ with $R'<R$ and $\overline B\subset \operatorname{Int}L$, there is a constant $C$, depending only on $B'$, $B$, $L$, $n$ and $q$, such that

whenever $u_1,\ldots ,u_ q$ are plurisubharmonic on $\Omega $ with values in $[-2,-1]$ on $L$.

Step 2: a common collar. Put $\psi =|z-a|^2-R^2$ and $b=\mathrm{dd}^{\mathrm{c}}\psi =\beta /\pi $, and choose $\delta >0$ with $\overline{B'}\subset \{ \psi <-\delta \} $. Replace each $u_ l$ by $\max \{ u_ l,A\psi \} $, where $A=2/\delta $; this function is plurisubharmonic on $\Omega $ (proposition 5.9.5). By (7.3.2), with $N=2$ and $U=\operatorname{Int}L$, it is unchanged on $\{ \psi <-\delta \} $, it equals $A\psi $ on the collar $V_0=\{ z\in \operatorname{Int}L:\psi (z)>-\delta /2\} $ of $\partial B$, and it takes values in $[-2,0)$ on $B$. Let $S=\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ be the product of the modified functions. Put $\tau =\delta /4$ and $B_\tau =\{ \psi <-\tau \} $. Then $\overline{B'}\subset B_\tau $, and on the neighborhood $V_0$ of $\overline B\setminus B_\tau $ we have $S=(A\, \mathrm{dd}^{\mathrm{c}}\psi )^ q=A^ qb^ q$ by proposition 7.2.2(1),(2). By lemma 2.5.1(4) there is $\theta \in C^\infty _ c(B)$ with $0\le \theta \le 1$ and $\theta =1$ on a neighborhood of $\overline{B_\tau }$; the function $\chi =-\psi \theta $ is smooth, nonnegative, has compact support in $B$, and equals $-\psi $ on a neighborhood of $\overline{B_\tau }$. All these choices depend only on $B'$, $B$ and $L$.

Step 3: smooth $V$ and $q\le n-1$. Suppose that $q\le n-1$ and that $V$ is smooth and plurisubharmonic on a neighborhood of $\overline B$ with $V\le 0$ on $B_\tau $. The signed measure $S\wedge b^{n-q-1}\wedge \mathrm{dd}^{\mathrm{c}}\chi $ has compact support in $B$. Splitting its integral against $V$ over $B_\tau $, where $\mathrm{dd}^{\mathrm{c}}\chi =-b$, and over $B\setminus B_\tau \subset V_0$, where $S=A^ qb^ q$, we obtain

The second equality holds because the first integral is $(VS\wedge b^{n-q-1})(\mathrm{dd}^{\mathrm{c}}\chi )=\bigl (\mathrm{dd}^{\mathrm{c}}(VS\wedge b^{n-q-1})\bigr )(\chi )$, and $\mathrm{dd}^{\mathrm{c}}(VS\wedge b^{n-q-1})=\mathrm{dd}^{\mathrm{c}}V\wedge S\wedge b^{n-q-1}$ by (7.2.3), the current $S\wedge b^{n-q-1}$ being closed. The smooth form $\mathrm{dd}^{\mathrm{c}}V$ is closed, and it is strongly positive, since its coefficient matrix is positive semidefinite (proposition 5.9.2) and weak and strong positivity agree in bidegree $(1,1)$ (section 5.3); so by proposition 7.2.2(4), with $T=1$ and $\gamma =\mathrm{dd}^{\mathrm{c}}V$, the current $\mathrm{dd}^{\mathrm{c}}V\wedge S$ is the product $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \mathrm{dd}^{\mathrm{c}}V$ with the closed positive base current $\mathrm{dd}^{\mathrm{c}}V$ of bidimension $(n-1,n-1)$.

Choose a compact set $L'\subset B$ whose interior contains $\operatorname{Supp}\chi $. The first term is at most $\pi ^{q+1-n}\sup \chi $ times the mass of the trace measure of $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \mathrm{dd}^{\mathrm{c}}V$ on $\operatorname{Supp}\chi $. Since the modified $u_ l$ take values in $[-2,0)$ on $B$, theorem 7.2.5, applied to $\operatorname{Supp}\chi \subset \operatorname{Int}L'$, shows that the first term is at most $C\| \mathrm{dd}^{\mathrm{c}}V\| _{L'}$, with $C$ depending only on $\chi $, $L'$, $n$ and $q$. Let $\chi '\in C^\infty _ c(B)$ satisfy $0\le \chi '\le 1$ and $\chi '=1$ on $L'$. Since $\mathrm{dd}^{\mathrm{c}}V\wedge \beta ^{n-1}\ge 0$,

where $C$ bounds the coefficient of the smooth $(n,n)$-form $\mathrm{dd}^{\mathrm{c}}\chi '\wedge \beta ^{n-1}/(n-1)!$. The collar term is bounded by the same quantity, because $b^{n-1}\wedge \mathrm{dd}^{\mathrm{c}}\chi $ has bounded smooth coefficients. Hence

Step 4: general $V$ and $q\le n-1$. Let $V\in \mathrm{PSH}(\Omega )$, and choose $r_0>0$ with $2r_0<\operatorname {dist}(\overline B,\mathbb C^ n\setminus \operatorname{Int}L)$. The function $V$ is subharmonic (theorem 5.9.1). If $x$ lies within distance $r_0$ of $\overline B$, then $\overline{B(x,r_0)}\subset \operatorname{Int}L$, and the ball submean inequality of theorem 3.4.5(2) gives

put $c=c_0\| V\| _{L^1(L)}$. For $0<\varepsilon <r_0$ the regularization $V_\varepsilon =V*\rho _\varepsilon $ is smooth and plurisubharmonic on the neighborhood $\{ z:\operatorname {dist}(z,\overline B)<r_0\} $ of $\overline B$ (proposition 5.9.2), since this neighborhood lies in $\Omega _\varepsilon $. It satisfies $V_\varepsilon \le c$ on $\overline B$, since $V_\varepsilon (x)$ is an average of values of $V$ at points within distance $\varepsilon $ of $x$, and $\int _ B|V_\varepsilon |\, \mathrm{d}V_{2n}\le \int \rho _\varepsilon (y)\int _ B|V(x-y)|\, \mathrm dV_{2n}(x)\, \mathrm dV_{2n}(y)\le \| V\| _{L^1(L)}$ by Tonelli's theorem, since $B-y\subset L$ for $|y|\le \varepsilon $. Applying Step 3 to $V_\varepsilon -c$ gives

As $\varepsilon \downarrow 0$, the nonnegative functions $c-V_\varepsilon $ increase to $c-V$, by (3.4.13), so monotone convergence gives the same bound for $\int _{B_\tau }(c-V)\, S\wedge b^{n-q}$. Since $V\le c$ on $B_\tau $, we have $|V|\le (c-V)+c$ there, and the mass of $S\wedge b^{n-q}$ on $B_\tau $ is bounded by a constant depending only on $B'$, $B$, $n$ and $q$, by theorem 7.2.5 applied to $\overline{B_\tau }\subset B=\operatorname{Int}\overline B$, since the modified $u_ l$ take values in $[-2,0]$ on $\overline B$. Since $S$ coincides with the original product on $B'\subset B_\tau $, by locality, this proves the estimate of Step 1 for $q\le n-1$. By the finite cover and the normalizations of Step 1, the theorem holds for $q\le n-1$, in every dimension.

Step 5: the case $q=n$. Consider $\mathbb C^{n+1}$ with coordinates $(z,w)$, the unit disc $\Delta \subset \mathbb C$, and the projection $p(z,w)=z$; let $\beta _ z$, $\beta _ w$ be the forms (5.2.8) in the variables $z$ and $w$, so that $\beta _ z+\beta _ w$ is the form $\beta $ of $\mathbb C^{n+1}$, and $\beta _ w=\mathrm dV_2(w)$. By proposition 5.9.3, the functions $u_ l\circ p$ are locally bounded and plurisubharmonic on $\Omega \times \Delta $, and $V\circ p$ is plurisubharmonic there, since it is not identically $-\infty $ on any connected component $\Omega _0\times \Delta $ of $\Omega \times \Delta $. Let $\mu =\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$, a positive measure on $\Omega $. We claim that

If the $u_ l$ are smooth, then $\mathrm{dd}^{\mathrm{c}}(u_ l\circ p)=p^*\mathrm{dd}^{\mathrm{c}}u_ l$, so the product of these $n$ factors is the pullback $p^*\mu $ of the $(n,n)$-form $\mu =g\, \mathrm dV_{2n}$ (proposition 7.2.2(1)). Its exterior product with $\beta _ z$ is $p^*(\mu \wedge \beta )=0$, since $\mu \wedge \beta $ has bidegree $(n+1,n+1)$ on $\mathbb C^ n$, and its exterior product with $\beta _ w$ is $g(z)\, \mathrm dV_{2n}(z)\wedge \mathrm dV_2(w)$, the density of $\mu \otimes \mathrm dV_2$.

In general, let $u_{l,\varepsilon }=u_ l*\rho _\varepsilon $, which are smooth plurisubharmonic functions on $\Omega _{\varepsilon _0}$ for $\varepsilon <\varepsilon _0$, decreasing to $u_ l$ as $\varepsilon \downarrow 0$ (proposition 5.9.2). By theorem 7.3.7, applied on $\Omega _{\varepsilon _0}\times \Delta $ to the functions $u_{l,\varepsilon }\circ p$, which decrease to $u_ l\circ p$, the left-hand sides for $u_{l,\varepsilon }$ converge weakly to the left-hand side for $u_ l$. By the same theorem on $\Omega _{\varepsilon _0}$, the measures $\mu _\varepsilon =\mathrm{dd}^{\mathrm{c}}u_{1,\varepsilon }\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{n,\varepsilon }$ converge weakly to $\mu $; hence $\mu _\varepsilon \otimes \mathrm dV_2\to \mu \otimes \mathrm dV_2$, because for a test function $\varphi $ on $\Omega _{\varepsilon _0}\times \Delta $ we have $\int \varphi \, \mathrm d(\mu _\varepsilon \otimes \mathrm dV_2)=\int \Phi \, \mathrm d\mu _\varepsilon $ with $\Phi (z)=\int _\Delta \varphi (z,w)\, \mathrm dV_2(w)$, a test function on $\Omega _{\varepsilon _0}$ (differentiate under the integral sign). Since $\varepsilon _0$ is arbitrary, the claim follows from the smooth case.

Now apply the case already proved in dimension $n+1$, with $q=n\le (n+1)-1$, to the compact sets $\widetilde K=K\times \{ |w|\le 1/2\} $ and $\widetilde L=L\times \{ |w|\le 3/4\} $ in $\Omega \times \Delta $; note that $\widetilde K\subset \operatorname{Int}\widetilde L$. By the claim and Fubini's theorem, the left-hand side of the estimate for $V\circ p$ and the $u_ l\circ p$ is $\frac\pi 4\int _ K|V|\, \mathrm d\mu $, while $\| V\circ p\| _{L^1(\widetilde L)}=\frac{9\pi }{16}\| V\| _{L^1(L)}$ and $\| u_ l\circ p\| _{L^\infty (\widetilde L)}=\| u_ l\| _{L^\infty (L)}$. Hence the estimate holds in $\mathbb C^ n$ with the constant $\frac94C_{\widetilde K,\widetilde L,n}$, which depends only on $K$, $L$ and $n$. This proves the case $q=n$. $\square$