4.3 Local potentials: plurisubharmonicity [00TP]

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4.3 Local potentials: plurisubharmonicity

The following lemma is a special case of the principle that for a subharmonic function, the standard mean value inequality has interesting strengthenings if there is more information about microscopic averages.

Lemma 4.6.

Let $\Phi$ be a subharmonic function on $B_{2}^{n}\times T^{k}=B_{2}\times\mathbb{R}^{k}/\epsilon\mathbb{Z}^{k}$ equipped with the Euclidean metric $g=\sum_{1}^{n}dx_{i}^{2}+\sum_{1}^{k}dy_{j}^{2}$ , where $0<\epsilon\ll 1$ . Let $v$ be the averaging function of $\Phi$ over the $T^{k}$ fibres. Assume $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int|\Phi|\lesssim 1$ and a Lipschitz bound $\text{Lip}(v)\lesssim 1$ , then on $B_{1}\times T^{k}$ we have $\Phi\leq v+C\epsilon^{1/2}$ .

Proof.

(courtesy of W. Feldman) By passing to the universal cover $B_{2}\times\mathbb{R}^{k}$ , the standard mean value inequality implies

\sup_{B_{3/2}\times T^{k}}\Phi\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int|\Phi|\lesssim 1.

Let $p\in B_{1}\times T^{k}$ , which lifts to a point $p$ in $B_{1}\times\mathbb{R}^{k}$ . Consider the Euclidean ball $B_{g}(p,\epsilon R)\subset B_{3/2}\times\mathbb{R}^{k}$ , where $R\gg 1$ is a parameter to be chosen. Then by the mean value inequality,

\Phi(p)\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B_{g}(p,\epsilon R)}\Phi.

Define the subset $E\subset B_{g}(\epsilon R)$ as the union of all interior lattice cubes, then

B_{g}(p,\epsilon R)\setminus E\subset B_{g}(p,\epsilon R)\setminus B_{g}(p,\epsilon(R-C)),

and by the lattice periodicity of $\Phi$ we have $\int_{E}\Phi=\int_{E}v$ . By partitioning the integral $\int_{B_{g}(p,\epsilon)}\Phi$ into the contributions from $E$ and $B_{g}(p,\epsilon R)\setminus E$ ,

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B_{g}(p,\epsilon R)}\Phi\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B_{g}(p,\epsilon R)}v+CR^{-1}\sup_{B_{g}(p,\epsilon R)}(\Phi-v)\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B_{g}(p,\epsilon R)}v+CR^{-1}.

By the Lipschitz bound of $v$ , the RHS is bounded above by

v(p)+\text{Lip}(v)\epsilon R+CR^{-1}\leq v(p)+C(\epsilon R+R^{-1}).

Choosing $R=\epsilon^{-1/2}$ gives $\Phi(p)\leq v(p)+C\epsilon^{1/2}$ . ∎

Back to the setting of Prop. 4.4,

Corollary 4.7.

(Local potential upper bound) On $U^{s}_{w,\delta}$ , then $\phi-\bar{\phi}\leq Cs^{-1/2}.$

Proof.

The psh property of $\phi$ implies subharmonicity. By the Harnack inequality in Prop. 4.1 the average $L^{1}$ -integral is bounded, and by Prop. 4.4 there is a Lipschitz bound on the local average function $\bar{\phi}$ . ∎

Corollary 4.8.

(Local $L^{1}$ -oscillation bound) Over one log scale inside $U_{w,\delta}^{s}$ ,

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}|\phi-\bar{\phi}|d\mu_{s}\leq Cs^{-1/2}.

Proof.

Recall the local oscillation of $\bar{\phi}$ in one log scale is $O(s^{-1})$ . Since the local sup of $\phi$ differs from the local average of $\phi$ by $O(s^{-1/2})$ , the local $L^{1}$ -oscillation is likewise bounded by $O(s^{-1/2})$ . ∎

Remark 4.9.

The $s$ -dependence is probably not optimal.

We now seek a local $L^{1}$ -oscillation bound on the charts of boundary type $U_{P}$ (cf. Remark 3.10). The idea is that any chart of boundary type overlaps with some chart of toric type in an annulus region, where the $L^{1}$ -oscillation bound is already known. It would be enough to transfer the $L^{1}$ -oscillation bound from the annulus to the deep interior of the chart.

Lemma 4.10.

Let $\Phi$ be a psh function on the $\{|z_{i}|\leq 4,\forall i\}\subset\mathbb{C}^{n}$ . Then

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{B_{1}}|\Phi|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{\{1<|z_{i}|<4,\forall i\}}|\Phi|.

Proof.

We induct on dimension. For $n=1$ , the unit ball is already enclosed by an annulus, so $\sup_{B(1)}\Phi$ is bounded above, and the mean value property applied to all balls $B(p,2)$ with $1<|p|\leq 2$ gives a lower bound on $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}\Phi$ . Thus the $L^{1}$ -bound in $B(1)$ is clear.

For general $n$ , notice by induction we can bound for each $i\leq n$ ,

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{\{1<|z_{i}|<4,|z_{j}|<4,\forall j\neq i\}}|\Phi|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{\{1<|z_{j}|<4,\forall j\}}|\Phi|,

so $\Phi$ is controlled in $L^{1}$ on an annulus enclosing $B(1)$ , and we can bound $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(1)}|\Phi|$ similar to the $n=1$ case. ∎

Corollary 4.11.

(Local $L^{1}$ -oscillation bound II) In the chart of boundary type $U_{P}$ , the local potential $\phi$ satisfies

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{U_{P}}|\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{U_{P}}\phi|d\mu_{s}\leq Cs^{-1/2}.