Theorem 7.11.33A26D05A19EF5

Proof. Write $B=B(a,R)$ and $\rho (z)=|z-a|^2-R^2$. We say that $v\in \mathcal E_0(B)$ has a quadratic collar if $v=c\rho $ on a neighborhood of $\partial B$ in $B$ for some constant $c>0$. We first prove the inequality with constant one for potentials with quadratic collars (Steps 1 and 2), and then approximate arbitrary elements of $\mathcal E_0(B)$ by such potentials while keeping their energies under control (Steps 3 and 4). For functions $b_0,\ldots ,b_ n$ which either all have quadratic collars, or are smooth real functions on a neighborhood of $\overline B$ with $b_2,\ldots ,b_ n$ plurisubharmonic, put

The integral is finite: in the first case every mixed product of the $b_ l$ is smooth near $\partial B$ and locally finite on the rest of $B$, and in the second the integrand is a smooth form on a neighborhood of $\overline B$.

Step 1. Smooth functions. Let $b_0,\ldots ,b_ n$ be smooth real functions on a neighborhood of $\overline B$, with $b_2,\ldots ,b_ n$ plurisubharmonic and $b_0=0$ on $\partial B$, and put $S=\mathrm{dd}^{\mathrm{c}}b_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}b_ n$, with $S=1$ if $n=1$. With $\mathrm d^{\mathrm c}$ as in (5.2.10), we claim that

Fix a smooth function $\theta \colon \mathbb R\to [0,1]$ with $\theta =1$ on $(-\infty ,-2]$ and $\theta =0$ on $[-1,\infty )$, and put $\chi _ k=\theta (k\rho )$ for $k\geq 1$; then $\chi _ k\in C^\infty _ c(B)$. The $(2n-1)$-form $\chi _ kb_0\, \mathrm d^{\mathrm c}b_1\wedge S$ is smooth with compact support in $B$, and $S$ is closed, so proposition 5.2.6 and the Leibniz rule give

As $k\to \infty $, $\chi _ k\to 1$ pointwise on $B$, and the forms $\mathrm db_0\wedge \mathrm d^{\mathrm c}b_1\wedge S$ and $b_0\, \mathrm{dd}^{\mathrm{c}}b_1\wedge S$ have bounded coefficients on $B$; by dominated convergence, the first and the last integral tend to $\int _ B\mathrm db_0\wedge \mathrm d^{\mathrm c}b_1\wedge S$ and $-I(b_0,\ldots ,b_ n)$. In the middle integral, $\mathrm d\chi _ k=k\theta '(k\rho )\, \mathrm d\rho $ vanishes outside the set where $-2\leq k\rho \leq -1$. Let $\Lambda $ be the maximum of $|\nabla b_0|$ on $\overline B$. If $z\in B$ and $\zeta \in \partial B$ is a nearest boundary point, then $b_0(\zeta )=0$ and $|z-\zeta |=R-|z-a|$, so the mean value theorem on the segment from $z$ to $\zeta $ gives

Hence $|b_0\, k\theta '(k\rho )|\leq (\Lambda /R)|k\rho |\, |\theta '(k\rho )|\leq 2\Lambda \sup |\theta '|/R$ on $B$, uniformly in $k$, and $b_0\, k\theta '(k\rho )\to 0$ pointwise on $B$. Since $\mathrm d\rho \wedge \mathrm d^{\mathrm c}b_1\wedge S$ has bounded coefficients on $B$, dominated convergence shows that the middle integral tends to zero. This proves (7.11.35).

As observed at the beginning of section 7.3, $\mathrm db\wedge \mathrm d^{\mathrm c}c\wedge S=\mathrm dc\wedge \mathrm d^{\mathrm c}b\wedge S$ for smooth real $b,c$. Moreover, $\mathrm db\wedge \mathrm d^{\mathrm c}b=(\mathrm i/2\pi )\partial b\wedge \bar\partial b$ is a positive $(1,1)$-form, and so are the forms $\mathrm{dd}^{\mathrm{c}}b_ l$, $l\geq 2$, since the complex Hessian of a smooth plurisubharmonic function is positive semidefinite (proposition 5.9.2). Positive $(1,1)$-forms are strongly positive (section 5.3), so by proposition 5.3.2 the $(n,n)$-form $\mathrm db\wedge \mathrm d^{\mathrm c}b\wedge S$ is a nonnegative multiple of the volume form. Consequently the real bilinear form

defined for smooth real functions $b,c$ on a neighborhood of $\overline B$ vanishing on $\partial B$, is symmetric and positive semidefinite, and by (7.11.35) we have $Q(b,c)=I(b,c,b_2,\ldots ,b_ n)$. The Cauchy–Schwarz inequality for $Q$, which follows from the nonnegativity of the quadratic polynomial $t\mapsto Q(b+tc,b+tc)$, therefore gives

The symmetry of $Q$ allows us to interchange the first two arguments of $I$, and the symmetry of the products allows us to permute the last $n$ arguments. Hence $I$ is symmetric in all $n+1$ arguments when they are smooth plurisubharmonic functions on a neighborhood of $\overline B$ vanishing on $\partial B$.

Step 2. Quadratic collars. Let $u_0,\ldots ,u_ n$ have quadratic collars, say $u_ l=c_ l\rho $ near $\partial B$. We first show that the assertions of Step 1 persist for them. Extend each $u_ l$ by $c_ l\rho $ to a plurisubharmonic function on a neighborhood of $\overline B$. Let $\vartheta _\varepsilon $ be the radial mollifiers of section 2.5, denoted $\rho _\varepsilon $ there; we change the letter because $\rho $ is now the quadratic function above. Because $\rho $ is a quadratic polynomial and $\vartheta _\varepsilon $ is radial, $\rho *\vartheta _\varepsilon =\rho +\gamma \varepsilon ^2$ with the constant $\gamma =\int |y|^2\vartheta _1(y)\, \mathrm dV(y)$. Hence

equals $c_ l\rho $ on a fixed neighborhood of $\partial B$ for all small $\varepsilon $. By proposition 5.9.2, the convolutions $u_ l*\vartheta _\varepsilon $ are smooth and plurisubharmonic on a neighborhood of $\overline B$ for small $\varepsilon $, and they decrease to $u_ l$ as $\varepsilon \downarrow 0$. Hence each $u_{l,\varepsilon }$, which differs from $u_ l*\vartheta _\varepsilon $ by a constant, is smooth and plurisubharmonic on a neighborhood of $\overline B$ and vanishes on $\partial B$, although the $u_{l,\varepsilon }$ need not decrease. By theorem 7.3.7,

weakly, and the constant $c_0\gamma \varepsilon ^2$ changes the integral against a fixed test function only by an amount tending to zero, by the local mass bounds of theorem 7.2.5. Near $\partial B$, all the $u_{l,\varepsilon }$ and $u_ l$ coincide with $c_ l\rho $. Splitting the integrals over $B$ by a cutoff in $C_ c(B)$ which equals one outside this neighborhood of $\partial B$, we conclude that $I(u_{0,\varepsilon },\ldots ,u_{n,\varepsilon })\to I(u_0,\ldots ,u_ n)$, and likewise for every rearrangement of the potentials. Passing to the limit in the smooth case proves that $I$ is symmetric and satisfies (7.11.36) for potentials with quadratic collars.

We now prove the inequality with constant one for these potentials. Since $u_ l=c_ l\rho $ near $\partial B$ with $c_ l>0$, the energy $e_1(u_ l)$ is positive: already its integral over a neighborhood of $\partial B$ is positive. Since $I$ is positively homogeneous of degree one in each argument and $e_1(tv)=t^{n+1}e_1(v)$ for $t>0$, both sides of the inequality are multiplied by the same factor when each $u_ l$ is replaced by $e_1(u_ l)^{-1/(n+1)}u_ l$. We may therefore assume that $e_1(u_ l)=1$ for every $l$. Consider the finitely many ordered $(n+1)$-tuples whose entries are taken from the functions $u_0,\ldots ,u_ n$, and let $F$ be the largest value of $I$ on these tuples. Since $I(u_ l,\ldots ,u_ l)=e_1(u_ l)=1$, we have $F\geq 1$. Among the tuples on which $I$ equals $F$, choose one for which the sum of the squares of the multiplicities of its distinct entries is largest.

Suppose that this tuple contains two distinct functions $b$ and $c$, with multiplicities $r$ and $s$. By symmetry we may place them in the first two slots. In (7.11.36), the left side is $F^2$, while the two factors on the right are nonnegative values of $I$ on tuples of our family, hence at most $F$. Therefore both factors equal $F$. Replacing one entry $c$ by $b$ changes the sum of the squares of the multiplicities by $(r+1)^2+(s-1)^2-r^2-s^2=2(r-s)+2$, and replacing one entry $b$ by $c$ changes it by $2(s-r)+2$. One of these numbers is positive, which contradicts the choice of the tuple. Hence the chosen tuple consists of a single function repeated $n+1$ times, and $F=1$. In particular $I(u_0,\ldots ,u_ n)\leq 1$, which is the inequality with constant one.

Step 3. Approximation with controlled energy. Let $u\in \mathcal E_0(B)$, put $m_\rho =\int _ B(\mathrm{dd}^{\mathrm{c}}\rho )^ n$, and for $j\geq 1$ let

Since $u-\delta _ j\leq -\delta _ j$ while $j\rho >-\delta _ j$ near $\partial B$, the function $v_ j$ equals $j\rho $ on a neighborhood of $\partial B$ in $B$. Since $v_ j$ and $j\rho $ coincide outside a compact subset of $B$, lemma 7.11.18 gives $\int _ B(\mathrm{dd}^{\mathrm{c}}v_ j)^ n=j^ nm_\rho $, and hence $v_ j\in \mathcal E_0(B)$. Moreover, $v_ j\to u$ uniformly on $B$. Indeed, let $s>0$, and choose a neighborhood $V$ of $\partial B$ with $u>-s$ on $V\cap B$. Since $u-\delta _ j\leq v_ j\leq 0$, we have $|v_ j-u|\leq s+\delta _ j$ on $V\cap B$. On the set $B\setminus V$, whose closure is a compact subset of $B$, the function $\rho $ is bounded away from zero and $u$ is bounded, so $j\rho <u-\delta _ j$, and hence $v_ j=u-\delta _ j$, there for all large $j$.

Next we estimate $e_1(v_ j)$. Let $t>4\delta _ j$. Since $u\leq v_ j+\delta _ j$ and $v_ j\leq 0$,

for the first inclusion note that on $\{ v_ j<-t\} $ we have $v_ j/2<-t/2<-t/4-\delta _ j$, hence $u\leq v_ j+\delta _ j<v_ j/2-t/4$. As $u$ tends to $0$ and $v_ j/2-t/4$ tends to $-t/4$ at $\partial B$, the comparison theorem (theorem 7.3.11) applies to $u$ and $v_ j/2-t/4$ and gives

For $0<t\leq 4\delta _ j$ we use the total mass $j^ nm_\rho $ instead. Integrating over $t$ by the layer-cake formula $e_1(v)=\int _0^\infty (\mathrm{dd}^{\mathrm{c}}v)^ n(\{ v<-t\} )\, \mathrm dt$, we obtain

Step 4. The general case. Apply the construction of Step 3 to each $u_ l$, obtaining functions $v_{l,j}\in \mathcal E_0(B)$ which equal $j\rho $ near $\partial B$. They are uniformly bounded, since $u_ l-1\leq v_{l,j}\leq 0$, and they converge uniformly, hence locally in capacity, to $u_ l$. Let $\chi \in C_ c(B)$ with $0\leq \chi \leq 1$. The weight $-u_0$ is bounded and quasi-continuous (theorem 7.5.2), so the weighted form of theorem 7.5.8 and Step 2 give

By (7.11.37), the right side is at most

Letting $\chi $ increase to one proves (7.11.34). Vanishing energies need no separate treatment, since the energies of the $v_{l,j}$ are positive. $\square$