Pome

5.29.1 Capacity estimates1C5C06D2C5D9

In this subsection we abbreviate $\operatorname {Cap}=\operatorname {Cap}_\omega $, and, as before, $\omega _ v=\omega +\mathrm{dd}^{\mathrm{c}}v$.

Lemma 5.29.2.5CEF121C5391 There are constants $a,C>0$ such that

87E51BAEC1C0 5.29.3
\begin{equation} \operatorname {Vol}(E)\leq C\exp \bigl (-a\operatorname {Cap}(E)^{-1/n}\bigr ) \end{equation}

for every Borel set $E\subset X$. Moreover, for every $u\in \mathrm{PSH}(X,\omega )$ with $\sup _ Xu=0$ and every $t>1$,

3EF8F920CDD3 5.29.4
\begin{equation} \operatorname {Cap}(\{ u<-t\} )\leq C/t. \end{equation}

Proof. Cover $X$ by finitely many coordinate balls $B'_ j$, with larger concentric balls $B_ j$, $B'_ j\Subset B_ j$. A local capacity test $0\leq v\leq 1$ on $B_ j$ can be turned into a global test as follows. Choose a smooth function $\chi _ j$ with $-2\leq \chi _ j\leq 0$, equal to zero near $\overline{B'_ j}$ and to $-2$ near $\partial B_ j$. Choose $\delta >0$ so small that

\[ \omega +\delta \mathrm{dd}^{\mathrm{c}}\chi _ j\geq 0,\qquad 3\delta /2\leq 1. \]

The function

\[ V=\max \{ \delta (v+\chi _ j),-\delta /2\} +\delta /2 \quad \hbox{on }B_ j, \qquad V=0\quad \hbox{outside }B_ j, \]

is $\omega $-plurisubharmonic on $X$, because near $\partial B_ j$ the constant branch is the larger one, and it takes values between zero and one. On $B'_ j$ we have $\omega _ V\geq \delta \mathrm{dd}^{\mathrm{c}}v$. By the positivity of bounded products,

\[ \operatorname {cap}(E\cap B'_ j,B_ j) \leq \delta ^{-n}\operatorname {Cap}(E). \]

The local volume–capacity estimate, summed over the finite cover, proves (5.29.3).

For the second estimate, choose on each $B_ j$ a smooth negative quadratic function $r_ j$ with $\mathrm{dd}^{\mathrm{c}}r_ j\geq \omega $. For every global test $0\leq v\leq 1$, the functions $u+r_ j\leq 0$ and $v+r_ j$ are plurisubharmonic on $B_ j$. Thus

\[ \begin{split} \int _{\{ u<-t\} }\omega _ v^ n & \leq t^{-1}\int _ X(-u)\omega _ v^ n\\ & \leq t^{-1}\sum _ j\int _{B'_ j}(-u-r_ j) \bigl (\mathrm{dd}^{\mathrm{c}}(v+r_ j)\bigr )^ n\\ & \leq \frac Ct\left(1+\int _ X|u|\, \mathrm{d}V\right). \end{split} \]

Here the second inequality uses $\omega _ v\leq \mathrm{dd}^{\mathrm{c}}(v+r_ j)$, and the third is the weighted Chern–Levine–Nirenberg inequality, applied to local test functions with a common bound. The $L^1$ bound for normalized potentials in theorem 5.25.2 completes the proof after taking the supremum over $v$. $\square$

Corollary 5.29.5.EACB00126847 Fix $p>1$. There is a constant $C_ p$ such that every nonnegative $F\in L^ p(X,\mathrm{d}V)$ satisfies

\[ \int _ E F\, \mathrm{d}V\leq C_ p\| F\| _ p\operatorname {Cap}(E)^2 \]

for every Borel set $E\subset X$.

Proof. By Hölder's inequality and (5.29.3), the integral is at most

\[ C\| F\| _ p\exp (-a\operatorname {Cap}(E)^{-1/n}/p'), \qquad p'=p/(p-1). \]

The function $x\mapsto x^{-2}\exp (-ax^{-1/n}/p')$ is bounded on $0<x\leq \operatorname {Cap}(X)<\infty $, and sets of capacity zero have zero volume. $\square$

The same proof bounds the integral of an $L^ p$ density by any fixed positive power of the capacity. The exponent two will be convenient for the iteration over sublevel sets in lemma 14.19.16.

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub