Theorem 7.11.2.05F71889C324 Let $B\subset \mathbb C^ n$ be a ball, and let $\mu $ be a finite positive Borel measure on $B$ such that $\mu \leq (\mathrm{dd}^{\mathrm{c}}w)^ n$ for some bounded plurisubharmonic function $w$ on $B$. Put $m=\mu (B)$ and $M=\sup _ Bw-\inf _ Bw$, and for real $p\geq 1$ put
\[ \kappa _{n,p}=2^{n+p}(n+1)!M^ n, \qquad A_ p=\Bigl (1+\frac pn\Bigr )\kappa _{n,p}^{p/(n+p)}m^{n/(n+p)}. \]
Then, for every $\varphi \in \mathcal E_0(B)$, every real $p\geq 1$ and every $t>0$,
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7.11.3
\begin{equation} \mu (\{ \varphi <-t\} ) \leq \min \{ m,\kappa _{n,p}e_ p(\varphi )t^{-(n+p)}\} . \end{equation}
Consequently,
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7.11.4
\begin{equation} \int _ B(-\varphi )^ p\, \mathrm d\mu \leq A_ p\, e_ p(\varphi )^{p/(n+p)}. \end{equation}
The constant $A_ p$ depends only on $n$, $p$, $m$ and $M$.
Proof.
We prove the tail bound by applying Błocki's estimate (proposition 7.4.5) to truncations of $\varphi $, and then integrate it. For $s>0$ let
\[ h_ s=\max \{ \varphi ,-s\} ,\qquad d_ s=h_ s-\varphi =(-\varphi -s)_+. \]
By the remark following definition 7.11.1, there is a compact set $K_ s\subset B$ with $\varphi >-s$ on $B\setminus K_ s$, and since $K_ s$ is compact there is a ball $B_ s\Subset B$, concentric with $B$, which contains $K_ s$. Thus $h_ s=\varphi $ on $B_ s\setminus K_ s$, which is a collar of $\partial B_ s$, and $d_ s$ vanishes outside $K_ s$. The functions $\varphi $, $h_ s$ and $w$ are bounded and plurisubharmonic near $\overline{B_ s}$, $\varphi \le h_ s$, and the oscillation of $w$ on $B_ s$ is at most $M$. Błocki's estimate on $B_ s$, together with $\mu \leq (\mathrm{dd}^{\mathrm{c}}w)^ n$, therefore gives
\[ \int _ B d_ s^{n+1}\, \mathrm d\mu \leq \int _{B_ s}d_ s^{n+1}(\mathrm{dd}^{\mathrm{c}}w)^ n \leq (n+1)!M^ n\int _{B_ s}d_ s(\mathrm{dd}^{\mathrm{c}}\varphi )^ n. \]
Now take $s=t/2$. On $\{ \varphi <-t\} $ we have $d_ s>t/2$. On the other hand, $0\le d_ s\le -\varphi $, and $d_ s=0$ where $-\varphi \le t/2$; since $p\geq 1$, we have $-\varphi \le (t/2)^{1-p}(-\varphi )^ p$ where $-\varphi >t/2$, and hence
\[ \int _{B_ s}d_ s(\mathrm{dd}^{\mathrm{c}}\varphi )^ n \leq \int _{\{ -\varphi >t/2\} }(-\varphi )(\mathrm{dd}^{\mathrm{c}}\varphi )^ n \leq (t/2)^{1-p}e_ p(\varphi ). \]
Combining the two estimates and dividing by $(t/2)^{n+1}$, we obtain
\[ \mu (\{ \varphi <-t\} )\le (n+1)!M^ n(t/2)^{-n-p}e_ p(\varphi )=\kappa _{n,p}e_ p(\varphi )t^{-(n+p)}. \]
Since also $\mu (\{ \varphi <-t\} )\le m$, this proves (7.11.3).
If $m=0$, both sides of (7.11.4) vanish. If $\kappa _{n,p}e_ p(\varphi )=0$, then (7.11.3) shows that $\mu (\{ \varphi <-t\} )=0$ for every $t>0$, so the left side of (7.11.4) vanishes. Otherwise put
\[ T=\left(\frac{\kappa _{n,p}e_ p(\varphi )}m\right)^{1/(n+p)}. \]
The first equality below is the layer-cake formula, which follows from Tonelli's theorem applied to the nonnegative function $(z,t)\mapsto pt^{p-1}\mathbf1_{\{ \varphi (z)<-t\} }$ on $B\times (0,\infty )$. Using (7.11.3) with the bound $m$ for $t<T$ and the second bound for $t\ge T$, we obtain
\[ \begin{aligned} \int _ B(-\varphi )^ p\, \mathrm d\mu & =p\int _0^\infty t^{p-1}\mu (\{ \varphi <-t\} )\, \mathrm dt \leq mT^ p +p\kappa _{n,p}e_ p(\varphi )\int _ T^\infty t^{-n-1}\, \mathrm dt\\ & =mT^ p+\frac pn\kappa _{n,p}e_ p(\varphi )T^{-n}. \end{aligned} \]
By the choice of $T$,
\[ mT^ p=\kappa _{n,p}e_ p(\varphi )T^{-n}=\kappa _{n,p}^{p/(n+p)}m^{n/(n+p)}e_ p(\varphi )^{p/(n+p)}, \]
so the right-hand side equals $(1+p/n)mT^ p=A_ pe_ p(\varphi )^{p/(n+p)}$. This proves (7.11.4).
$\square$
Pome
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