7.4 Weighted estimates and Monge–Ampère capacityCD19E1C1FD36

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 $q=n$. $\square$

Proof. For each competitor $v$, the measure $(\mathrm{dd}^{\mathrm{c}}v)^ n$ is increasing and countably subadditive on Borel sets: $\int _{\bigcup _ jE_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sum _ j\int _{E_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sum _ j\operatorname {Cap}_ D(E_ j)$. Taking the supremum over $v$ gives the same properties for $\operatorname {Cap}_ D$. Outer capacity is increasing by definition. For countable subadditivity, given sets $A_ j$ with $\sum _ j\operatorname {Cap}^*_ D(A_ j)<\infty $ and $\varepsilon >0$, choose open sets $G_ j\supset A_ j$ with $\operatorname {Cap}_ D(G_ j)<\operatorname {Cap}^*_ D(A_ j)+\varepsilon 2^{-j}$; then $\operatorname {Cap}^*_ D(\bigcup _ jA_ j)\le \operatorname {Cap}_ D(\bigcup _ jG_ j)\le \sum _ j\operatorname {Cap}^*_ D(A_ j)+\varepsilon $.

Finiteness on a compact set $K$ follows from theorem 7.2.5 with $T=1$, applied to $K$ and a compact neighborhood $L$ of $K$ in $D$, since $\| v\| _{L^\infty (L)}\le 1$ for every competitor. Inner regularity of each Radon measure $(\mathrm{dd}^{\mathrm{c}}v)^ n$ (definition 2.2.1) gives $\int _ E(\mathrm{dd}^{\mathrm{c}}v)^ n=\sup _ K\int _ K(\mathrm{dd}^{\mathrm{c}}v)^ n\le \sup _ K\operatorname {Cap}_ D(K)$ over compact $K\subset E$, and taking the supremum over $v$ gives (1). For (2), $\operatorname {Cap}_ D(E_ j)\le \operatorname {Cap}_ D(E)$ by monotonicity, and for every competitor $v$ monotone convergence gives $\int _ E(\mathrm{dd}^{\mathrm{c}}v)^ n=\lim _ j\int _{E_ j}(\mathrm{dd}^{\mathrm{c}}v)^ n\le \lim _ j\operatorname {Cap}_ D(E_ j)$; taking the supremum over $v$ gives (2). For (3), choose $a\in D$ and $d>0$ with $D\subset B(a,d)$; the function $|z-a|^2/d^2$ is a competitor, and by (7.2.4) and proposition 7.2.2(3) its Monge–Ampère measure is $n!\pi ^{-n}d^{-2n}\, \mathrm dV_{2n}$. Hence (3) holds with $C_ D=\pi ^ nd^{2n}/n!$.

For (4), the first inequality holds because the restriction to $B$ of a competitor for $D$ is a competitor for $B$, with the same Monge–Ampère measure on $B$. For the second, write $B=B(a,r)$ and choose $\lambda >0$ such that $\rho =\lambda (|z-a|^2-r^2)<-1$ on a neighborhood of $K$. Given a competitor $v$ for $B$, let $w$ be the function equal to $\max \{ v-1,\rho \} $ on $B$ and to $\rho $ on $D\setminus B$. Since $v-1\le 0$ on $B$ and $\rho =0$ on $\partial B$, proposition 5.9.5 shows that $w$ is plurisubharmonic on $D$. It equals $v-1$ near $K$, satisfies $w\ge -1$ on $D$, and is bounded above by $\Lambda =\max \{ 0,\sup _ D\rho \} $, which does not depend on $v$. Thus $(w+1)/(\Lambda +1)$ is a competitor for $D$, and by locality its Monge–Ampère measure near $K$ is $(\Lambda +1)^{-n}(\mathrm{dd}^{\mathrm{c}}v)^ n$. This gives the second inequality with $C=(\Lambda +1)^ n$; the first inequality holds for all Borel subsets of $B$.

For the outer capacities in (4), let $K'$ be a compact neighborhood of $K$ in $B$; the constant obtained for $K'$ also serves for $K$, and we use it. If $A\subset K$ and $G$ is an open subset of $D$ or of $B$ containing $A$, then $G\cap \operatorname{Int}K'$ is open in both $B$ and $D$, contains $A$, lies in $K'$, and has no larger capacity than $G$. Applying the inequalities to these sets and taking infima over $G$ gives them for the outer capacities of $A$.

For (5), let $F\colon D\to D'$ be biholomorphic. By proposition 5.9.3, applied to $F$ and to $F^{-1}$, the map $v\mapsto v\circ F$ is a bijection from the competitors for $D'$ onto the competitors for $D$, and by proposition 7.2.2(5), $(\mathrm{dd}^{\mathrm{c}}(v\circ F))^ n=F^*\bigl ((\mathrm{dd}^{\mathrm{c}}v)^ n\bigr )$. We claim that for every positive Radon measure $\mu $ on $D'$, regarded as a current of bidegree $(n,n)$, the pullback $F^*\mu $ is the image of $\mu $ under $F^{-1}$, that is, $\int \psi \, \mathrm d(F^*\mu )=\int \psi \circ F^{-1}\, \mathrm d\mu $ for every test function $\psi $ on $D$.

If $\mu =g\, \mathrm dV_{2n}$ with $g$ smooth, then $F^*\mu =(g\circ F)\, |\det F'|^2\, \mathrm dV_{2n}$, because $F^*(\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n)=\det F'\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$ and similarly for the conjugate forms; since $|\det F'|^2$ is the Jacobian determinant of $F$ as a map of $\mathbb R^{2n}$, the change of variables formula gives the claim. Both sides of the claimed identity are weakly continuous in $\mu $ (proposition 5.6.2), and on relatively compact open subsets of $D'$ every $\mu $ is the weak limit of its smooth regularizations, so the claim holds in general. Consequently $(F^*\mu )(F^{-1}(E'))=\mu (E')$ for Borel sets $E'\subset D'$, since two Radon measures that agree on test functions agree on Borel sets (proposition 2.5.3(4)). Applying this to $\mu =(\mathrm{dd}^{\mathrm{c}}v)^ n$ and $E'=F(E)$ and taking the supremum over $v$ gives $\operatorname {Cap}_ D(E)=\operatorname {Cap}_{D'}(F(E))$. Since $F$ is a homeomorphism, it maps the open sets containing a set $A$ bijectively onto the open sets containing $F(A)$, so the outer capacities agree as well. $\square$

Proof. Choose an open neighborhood $U$ of $K$ with compact closure in $\operatorname{Int}L$. For every competitor $v$, theorem 7.4.1, applied to $\overline U\subset \operatorname{Int}L$ with $q=n$ and $u_1=\cdots =u_ n=v$, gives

Since $V$ is upper semicontinuous, $U\cap \{ V<-s\} $ is an open set containing $K\cap \{ V<-s\} $; taking the supremum over $v$ proves the estimate. Since $V$ is locally integrable (theorem 5.9.1), $\| V\| _{L^1(L)}<\infty $, and letting $s\to \infty $ gives $\operatorname {Cap}^*_ D(K\cap \{ V=-\infty \} )=0$.

Now let $E$ be as in the statement. Let $x\in D$, let $W$ and $V_ W$ be as in the hypothesis, and choose $r>0$ with $\overline{B(x,3r)}\subset W$; put $B_ x=B(x,3r)$, $K_ x=\overline{B(x,r)}$ and $L_ x=\overline{B(x,2r)}$. Then $E\cap B_ x\subset \{ V_ W=-\infty \} $, and the restriction of $V_ W$ to $B_ x$ is plurisubharmonic. The estimate just proved, applied in $B_ x$ to this restriction and to $K_ x\subset \operatorname{Int}L_ x$, gives $\operatorname {Cap}^*_{B_ x}(E\cap K_ x)=0$; since every open set in $B_ x$ is open in $D$, the first inequality of proposition 7.4.3(4) gives $\operatorname {Cap}^*_ D(E\cap K_ x)=0$. A compact set $K\subset D$ is covered by finitely many interiors $\operatorname{Int}K_ x$, and subadditivity gives $\operatorname {Cap}^*_ D(E\cap K)=0$. Finally, by theorem 7.4.1, applied on $B_ x$ to $K_ x\subset \operatorname{Int}L_ x$ with $q=n$, the function $V_ W$ is integrable on $K_ x$ for every measure $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$ as in the statement, so the Borel set $K_ x\cap \{ V_ W=-\infty \} $ has measure zero for all these measures. Since $D$ has a countable basis, countably many sets $\operatorname{Int}K_ x$ cover $D$, and the union of the corresponding sets $K_ x\cap \{ V_ W=-\infty \} $ is a Borel set with the required property. $\square$

Proof. First suppose that $u$, $h$ and $v$ are smooth; then $\mathrm{dd}^{\mathrm{c}}u$, $\mathrm{dd}^{\mathrm{c}}h$ and $\mathrm{dd}^{\mathrm{c}}v$ are positive $(1,1)$-forms, by proposition 5.9.2. Subtracting $\sup _ Bv$ from $v$, we may assume $-M\le v\le 0$ on $B$. Put $w=h-u\ge 0$; it vanishes on a collar of $\partial B$. Let $p\ge 2$, and let $R$ be a smooth closed strongly positive form of bidegree $(n-1,n-1)$; the exterior product of $R$ with a positive $(1,1)$-form is then a nonnegative multiple of the volume form, by proposition 5.3.2, since positive $(1,1)$-forms are strongly positive. Since $\mathrm{d}w\wedge \mathrm{d}^{\mathrm c}w$ is a positive $(1,1)$-form by (7.3.1) and $\mathrm{dd}^{\mathrm{c}}h\wedge R\ge 0$, the chain rule $\mathrm{dd}^{\mathrm{c}}(w^ p)=pw^{p-1}\mathrm{dd}^{\mathrm{c}}w+p(p-1)w^{p-2}\mathrm dw\wedge \mathrm d^{\mathrm c}w$ gives

Since $w^ p$ has compact support in $B$ and $R$ is closed, (7.2.3) gives $\int _ Bw^ p\mathrm{dd}^{\mathrm{c}}v\wedge R=(\mathrm{dd}^{\mathrm{c}}(vR))(w^ p)=(vR)(\mathrm{dd}^{\mathrm{c}}(w^ p))$. As $v\le 0$ and $-v\le M$, the preceding inequality therefore yields

For $k=1,\ldots ,n$, apply this with $p=n+2-k$ and $R=(\mathrm{dd}^{\mathrm{c}}v)^{n-k}\wedge (\mathrm{dd}^{\mathrm{c}}u)^{k-1}$, a product of positive $(1,1)$-forms, hence strongly positive (proposition 5.3.2) and closed. This gives

and chaining these $n$ inequalities from $k=1$ to $k=n$ gives the asserted inequality, since $(n+1)n\cdots 2=(n+1)!$.

In general, write $B=B(a,r)$ and let $V$ be a collar of $\partial B$ on which $u=h$. Since $\partial B$ is compact, there is $r''<r$ with $\{ z:r''\le |z-a|\le r\} \subset V$; choose $r'\in (r'',r)$ and put $B'=B(a,r')$. Then $h-u$ vanishes on $B\setminus \overline{B(a,r'')}$. Convolve $u$, $h$ and $v$ with the same radial kernel $\rho _\varepsilon $, with $\varepsilon <\min \{ (r'-r'')/2,r-r'\} $ so small that the regularizations are defined, smooth and plurisubharmonic on a neighborhood of $\overline{B'}$ (proposition 5.9.2). At points of $\overline{B'}$ they are averages over balls of radius $\varepsilon $ contained in $B$, where $u\le h$; hence $u_\varepsilon \le h_\varepsilon $ on $\overline{B'}$. Moreover $h_\varepsilon -u_\varepsilon $ vanishes on the collar $\{ z:r''+\varepsilon <|z-a|<r-\varepsilon \} $ of $\partial B'$ and outside $\overline{B(a,r''+\varepsilon )}$ in $B'$, hence outside the compact set $\overline{B(a,(r'+r'')/2)}\subset B'$. The values of $v_\varepsilon $ on $B'$ are averages of values of $v$ on $B$, so the oscillation of $v_\varepsilon $ on $B'$ is at most $M$.

Apply the smooth case on $B'$ and let $\varepsilon =\varepsilon _ k\downarrow 0$. The weights $(h_\varepsilon -u_\varepsilon )^{n+1}$ and $h_\varepsilon -u_\varepsilon $ are polynomials in decreasing sequences of bounded plurisubharmonic functions, and they, as well as their limits $(h-u)^{n+1}$ and $h-u$, vanish in $B'$ outside the fixed compact set $\overline{B(a,(r'+r'')/2)}$. Testing against a function $\chi \in C^\infty _ c(B')$ equal to $1$ on this set, corollary 7.3.8 lets both sides pass to the limit. Since $h-u$ vanishes on $B\setminus B'$, the limiting inequality on $B'$ is the asserted inequality on $B$. $\square$