Theorem 14.19.6.492E047184B3 Let $u,v\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$ and $0<\varepsilon <1$. For $s>0$, put
\[ m_\varepsilon =\inf _ X\bigl (u-(1-\varepsilon )v\bigr ),\qquad U_{\varepsilon ,s}=\{ u<(1-\varepsilon )v+m_\varepsilon +s\} . \]
If $0<s<\varepsilon ^3/(32B_0n^2)$, then, with $a=B_0n^2s\varepsilon ^{-3}$,
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14.19.7
\begin{equation} (1-4a)^ n\int _{U_{\varepsilon ,s}}\omega _{(1-\varepsilon )v}^{\, n} \leq \int _{U_{\varepsilon ,s}}\omega _ u^ n. \end{equation}
In particular, the reverse factor $(1-4a)^{-n}$ tends to one as $s\downarrow 0$ for fixed $\varepsilon $.
Proof.
Write $v_\varepsilon =(1-\varepsilon )v$, $U=U_{\varepsilon ,s}$ and
\[ \phi =\max (u,v_\varepsilon +m_\varepsilon +s),\qquad T_ k=\omega _ u^ k\wedge \omega _\phi ^{n-k},\qquad I_ k=\int _ U T_ k\quad (0\leq k\leq n). \]
Set $I_{-1}=I_{-2}=0$. Then $0\leq \phi -u\leq s$, this difference vanishes outside $U$, and on $U$ plurifine locality gives $\omega _\phi =\omega _{v_\varepsilon }\geq \varepsilon \omega $ in all bounded products.
By lemma 5.28.3, for every fixed positive mixed product $W$ of $n-1$ bounded-potential factors,
\[ \mathbf1_{\{ u=\phi \} }\omega _ u\wedge W \leq \mathbf1_{\{ u=\phi \} }\omega _\phi \wedge W. \]
For $0\leq k\leq n-1$, apply this with $W=\omega _ u^ k\wedge \omega _\phi ^{n-k-1}$. Since $u=\phi $ on $X\setminus U$, the difference $T_ k-T_{k+1}=(\omega _\phi -\omega _ u)\wedge W$ is nonnegative there. Consequently
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14.19.8
\begin{equation} I_ k-I_{k+1} \leq \int _ X\mathrm{dd}^{\mathrm{c}}(\phi -u)\wedge W =\int _ X(\phi -u)\mathrm{dd}^{\mathrm{c}}W. \end{equation}
This integration by parts is valid for bounded potentials. To see this, approximate $u$ and $\phi $ by decreasing sequences of smooth, uniformly bounded $\omega $-plurisubharmonic functions, using theorem 5.32.1. For smooth functions the identity is ordinary integration by parts. The form $\mathrm{dd}^{\mathrm{c}}W$ expands as below into bounded Hessian products and smooth forms, and bounded Bedford–Taylor convergence, also against bounded quasi-continuous coefficients converging in capacity, passes each term to the limit.
Set $r=n-k-1$. Since $\mathrm{d}\omega _ u=\mathrm{d}\omega _\phi =\mathrm{d}\omega $, expansion gives
\[ \begin{split} \mathrm{dd}^{\mathrm{c}}W={}& k\mathrm{dd}^{\mathrm{c}}\omega \wedge \omega _ u^{k-1}\wedge \omega _\phi ^ r +r\mathrm{dd}^{\mathrm{c}}\omega \wedge \omega _ u^ k\wedge \omega _\phi ^{r-1}\\ & +k(k-1)\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega \wedge \omega _ u^{k-2}\wedge \omega _\phi ^ r\\ & +2kr\, \mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega \wedge \omega _ u^{k-1}\wedge \omega _\phi ^{r-1}\\ & +r(r-1)\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega \wedge \omega _ u^ k\wedge \omega _\phi ^{r-2}. \end{split} \]
Terms containing a negative power have coefficient zero and are omitted. On $U$ we may replace each $\omega ^ j$ in an upper bound by $\varepsilon ^{-j}\omega _\phi ^ j$, since $\omega _\phi \geq \varepsilon \omega $ there. Using $\phi -u\leq s$ and $\varepsilon ^{-2}\leq \varepsilon ^{-3}$ in (14.19.8), we obtain
\[ I_ k-I_{k+1}\leq B_0s\varepsilon ^{-3} \bigl (k(k-1)I_{k-2}+k(1+2r)I_{k-1}+r^2I_ k\bigr ). \]
Each coefficient is at most $n^2$, and $a=B_0n^2s\varepsilon ^{-3}$. Hence
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14.19.9
\begin{equation} I_ k-I_{k+1}\leq a(I_{k-2}+I_{k-1}+I_ k). \end{equation}
Put $q=1-4a$; the assumption on $s$ gives $a<1/32$, so $q>7/8$. Suppose inductively that $qI_ j\leq I_{j+1}$ for $j<k$. Then $I_{k-1}\leq q^{-1}I_ k$ and $I_{k-2}\leq q^{-2}I_ k$ whenever the indices exist. Since
\[ 1+q^{-1}+q^{-2}\leq 1+8/7+64/49<4, \]
(14.19.9) gives $qI_ k\leq I_{k+1}$. The case $k=0$ follows from the same formula with the two absent terms set to zero. By induction, $q^ n I_0\leq I_ n$. Since $T_0=\omega _{v_\varepsilon }^ n$ on $U$ and $T_ n=\omega _ u^ n$, this is the theorem. In dimension one all torsion terms vanish and the same argument applies.
$\square$
Pome
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