A Hölder modulus for a bounded potential gives a modulus for integration against its Monge–Ampère measure. The proof uses a smooth approximation of the reference, with integration by parts controlling the change in its measure.
Lemma 5.29.7.D9AF18430F97 Let $B'\Subset B$ be concentric balls in $\mathbb C^ n$, and let $0<\alpha \leq 1$. Suppose $p\in \mathrm{PSH}(B)\cap C^{0,\alpha }(B)$ has $C^{0,\alpha }$ norm at most $N$. For every $M\geq 1$ there is a constant $C_ M$ such that bounded $a,b\in \mathrm{PSH}(B)$ with $|a|,|b|\leq M$ satisfy
\[ \int _{B'}|a-b|(\mathrm{dd}^{\mathrm{c}}p)^ n \leq C_ M\| a-b\| _{L^1(B)}^{\alpha /(\alpha +2n)}. \]
The constants are uniform for references with the stated bound $N$.
Proof.
Choose intermediate concentric balls and a smooth cutoff $0\leq \chi \leq 1$ supported in the innermost of them and equal to one on $B'$. All integrations below take place away from $\partial B$. Let $p_\varepsilon $ be a nonnegative radial convolution of $p$ on a larger intermediate ball. It is plurisubharmonic, and, for small $\varepsilon >0$,
\[ \| p-p_\varepsilon \| _\infty \leq C\varepsilon ^\alpha , \qquad 0\leq \mathrm{dd}^{\mathrm{c}}p_\varepsilon \leq C\varepsilon ^{-2}\mathrm{dd}^{\mathrm{c}}|z|^2. \]
Both references have a common bounded range. Set $f=|a-b|=2\max (a,b)-a-b$, and put
\[ T=\sum _{j=0}^{n-1}(\mathrm{dd}^{\mathrm{c}}p)^ j\wedge (\mathrm{dd}^{\mathrm{c}}p_\varepsilon )^{n-1-j}. \]
This is a closed positive current. Factorization and integration by parts give
\[ \int \chi f\bigl ((\mathrm{dd}^{\mathrm{c}}p)^ n-(\mathrm{dd}^{\mathrm{c}}p_\varepsilon )^ n\bigr ) =\int (p-p_\varepsilon )\mathrm{dd}^{\mathrm{c}}(\chi f)\wedge T. \]
We check that the last signed measure has variation at most $C(1+M)^2$, uniformly in $\varepsilon $. The term $\chi \mathrm{dd}^{\mathrm{c}}f\wedge T$ is a linear combination of the positive measures formed with $\max (a,b),a,b$, whose masses are at most $CM$ by local Chern–Levine–Nirenberg estimates. The term $f\mathrm{dd}^{\mathrm{c}}\chi \wedge T$ has variation at most $CM$.
For the two terms involving $\mathrm df$, use a cutoff $\eta $ equal to one near $\operatorname {supp}\chi $. If $q$ is any of $a,b,\max (a,b)$, translate it so that $0\leq q\leq 2M$. The bounded chain rule and closedness of $T$ give
\[ \int \eta \, \mathrm dq\wedge \mathrm d^{\mathrm c}q\wedge T =\frac12\int q^2\mathrm{dd}^{\mathrm{c}}\eta \wedge T -\int \eta q\mathrm{dd}^{\mathrm{c}}q\wedge T\leq CM^2. \]
The last term is nonnegative before subtraction, and the first is bounded using the local mass of $T$. Since $f=2\max (a,b)-a-b$, its gradient energy is also bounded by $CM^2$. Cauchy–Schwarz against the positive current $T$ now bounds the variation of $\mathrm d\chi \wedge \mathrm d^{\mathrm c}f\wedge T$ and $\mathrm d^{\mathrm c}\chi \wedge \mathrm df\wedge T$ by $CM$. These are all the terms of $\mathrm{dd}^{\mathrm{c}}(\chi f)\wedge T$.
The bounded formulas follow first for local smooth regularizations. To pass to bounded potentials, regularize $a,b,p$ decreasingly on the intermediate balls, write $f$ as the displayed linear combination, and apply lemma 5.27.2 to the expanded identities. The energy bounds are uniform; thus the integration by parts and its estimate survive this limit.
Writing $L=\| a-b\| _{L^1(B)}$, we have proved
16B648CFC04A
5.29.8
\begin{equation} \int \chi f(\mathrm{dd}^{\mathrm{c}}p)^ n \leq C(1+M)^2\varepsilon ^\alpha +C\varepsilon ^{-2n}L. \end{equation}
For small positive $L$, take $\varepsilon =L^{1/(\alpha +2n)}$. A fixed admissible radius and the bounded local mass handle the remaining $L$. For $L=0$, let $\varepsilon \downarrow 0$ in the same inequality.
$\square$
Corollary 5.29.9.F701C1999BDC Let $(X,\omega )$ be compact Hermitian, and let $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$, $0<\alpha \leq 1$. If uniformly bounded quasi-plurisubharmonic functions $a,b$ have a fixed common smooth lower curvature bound, then
\[ \int _ X|a-b|\, \omega _ u^ n \leq C\| a-b\| _{L^1(\mathrm dV)}^{\alpha /(\alpha +2n)}. \]
The constant depends on the bounds for $a,b$, their lower curvature bound and the $C^{0,\alpha }$ norm of $u$.
Proof.
On each of finitely many coordinate balls, choose a fixed smooth strictly plurisubharmonic quadratic $Q$ such that $a+Q,b+Q,u+Q$ are plurisubharmonic and $\mathrm{dd}^{\mathrm{c}}Q\geq \omega $. There $0\leq \omega _ u\leq \mathrm{dd}^{\mathrm{c}}(u+Q)$, so positivity of bounded products gives $\omega _ u^ n\leq (\mathrm{dd}^{\mathrm{c}}(u+Q))^ n$. Apply lemma 5.29.7 on smaller balls covering $X$. The quadratic cancels in the difference, and the local volumes are equivalent to $\mathrm dV$. Summing proves the assertion.
$\square$
The estimate before optimization also controls the tails of unbounded potentials. We need the following local form with a compact ambient space, so that the exponential bound for smooth volume is uniform.
Lemma 5.29.10.EF54C7C9A0A8 Let $(Y,\Omega )$ be compact Hermitian, let $B'\Subset B\subset Y$ be coordinate balls, and let $p\in \mathrm{PSH}(B)\cap C^{0,\alpha }(B)$ have norm at most $N$, with $0<\alpha \leq 1$. There are $c,C>0$, uniform for these references, such that
\[ (\mathrm{dd}^{\mathrm{c}}p)^ n\bigl (B'\cap \{ v<-t\} \bigr )\leq C\mathrm e^{-ct} \quad (t\geq 0) \]
for every $v\in \mathrm{PSH}(Y,\Omega )$ with $\sup _ Yv=0$.
Proof.
Choose a smooth quadratic $Q$ with $\mathrm{dd}^{\mathrm{c}}Q\geq \Omega $ on $B$ and a fixed constant $D$ such that $w=v+Q-D\leq 0$. For integers $m\geq 1$, set
\[ a_ m=\max (w,-m),\qquad b_ m=\max (w,-m-1),\qquad f_ m=a_ m-b_ m. \]
These are bounded plurisubharmonic truncations; $0\leq f_ m\leq 1$, it vanishes on $\{ w\geq -m\} $, and it equals one on $\{ w<-m-1\} $. By lemma 5.26.2 on $(Y,\Omega )$ and the bounded correction $Q-D$,
\[ \| f_ m\| _{L^1(B)}\leq C\mathrm e^{-am} \]
for a fixed $a>0$. Use the proof of lemma 5.29.7, with $M=m+1$, keeping its estimate (5.29.8) before optimization. Taking $\varepsilon =\mathrm e^{-am/(\alpha +2n)}$ gives
\[ \int \chi f_ m(\mathrm{dd}^{\mathrm{c}}p)^ n \leq C(1+m)^2\exp \left(-\frac{a\alpha m}{\alpha +2n}\right) \leq C\mathrm e^{-cm} \]
for some $c>0$. Large $m$ ensure that the convolution radius is admissible; the common local mass handles the other integers. The bounded difference between $w$ and $v$, and then choosing an integer just below the given height, prove the estimate for every real $t\geq 0$.
$\square$
Pome
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