4.4 Locally convex function [00TQ]

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4.4 Locally convex function

In section 4.2 we produced a collection of local average functions $\bar{\phi}=\bar{\phi}_{m,w}$ on $U_{w}^{s}$ corresponding to various choices of $w$ and $m$ with $\langle m,w\rangle=1$ . But the local coordinates $x^{m_{1}},\ldots x^{m_{n}}$ are naturally interpreted also as coordinates on $\partial\Delta_{\lambda}^{\vee}$ (cf. section 3.2), so $\bar{\phi}_{m,w}$ can be alternatively viewed as a collection of convex functions on the charts $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ of $\partial\Delta_{\lambda}^{\vee}$ . (Notice these local functions are defined without the need to shrink the domain to $U^{\infty}_{w,\delta}$ ).

The intuition is that up to $C^{0}$ -small error, the differences of these local functions agree with the cocycle $\{m-m^{\prime}\}$ , or equivalently, up to some $C^{0}$ -small fuzziness $\bar{\phi}_{m,w}+\langle m,x\rangle$ glue to a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ in the sense of Definition 3.22. The more precise statement is

Lemma 4.12.

On overlapping charts of $\partial\Delta_{\lambda}^{\vee}$ ,

|\bar{\phi}_{m,w}-\bar{\phi}_{m^{\prime},w^{\prime}}+(m-m^{\prime})|\leq Cs^{-1/2}.

Proof.

Since we know the local $L^{1}$ -oscillation estimate holds in every local region, in a log scale in $U^{s}_{w}$ , not necessarily in the shrinked region $U^{s}_{w,\delta}$ ,

\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)|}|\varphi_{m}-\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\varphi_{m}|d\mu_{s}\leq Cs^{-1/2}.

Since $\bar{\phi}_{m,w}$ is convex, a local $L^{1}$ -bound implies a local $L^{\infty}$ -bound in a slightly shrinked region, so in the log scale,

|\bar{\phi}_{m,w}-\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\varphi_{m}|\leq Cs^{-1/2}.

Likewise for $\bar{\phi}_{m^{\prime},w^{\prime}}$ . By definition the local potentials differ by

\varphi_{m}-\varphi_{m^{\prime}}=\langle m^{\prime}-m,\text{Log}_{s}(z)\rangle

Notice that for a given point $P$ on $\partial\Delta_{\lambda}^{\vee}$ , the log scales on $U^{s}_{w}$ and $U^{s}_{w^{\prime}}$ around $P$ have a nontrivial percentage of overlapping measure. Thus

\begin{split}|\bar{\phi}_{m,w}-\bar{\phi}_{m^{\prime},w^{\prime}}+(m-m^{\prime})|&\lesssim s^{-1/2}+|\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\varphi_{m}-\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\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}+\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_{overlap}|\varphi_{m}-\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}.\end{split}

∎

Remark 4.13.

The tropical version $U^{\infty}_{w}$ of $U^{s}_{w}$ is in general larger than $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$ ; it typically contains also some subset stretching to infinity along the $w$ -direction. If we regard $\bar{\phi}_{m,w}$ as local functions on $\mathcal{A}_{\lambda}^{\infty}$ instead of $\partial\Delta_{\lambda}^{\vee}$ , then there is a delicate issue. The Lemma above does not imply that $\bar{\phi}_{m,w}+\langle m,x\rangle$ for various choices of $m,w$ glue approximately on overlapping regions far from $\partial\Delta_{\lambda}^{\vee}$ . The problem is that such overlapping regions have too small measure, which breaks down the proof.