Compact weighted maximum and the actual boundary term

For the fixed $\mu >0$, the power function is continuous at every positive argument by [F4]. To check continuity at zero, let $\epsilon >0$ and put $\delta ={\epsilon }^{1\mathrm{/}\mu }>0$ by [F3] with exponent $1\mathrm{/}\mu >0$. The iterated-power law and normalization in [F4] give ${\delta }^{\mu }=\epsilon$. If $0\le t<\delta$, [F3] gives $0\le t^{\mu }<{\delta }^{\mu }=\epsilon$. Since $0^{\mu }=0$, this proves continuity at zero relative to $[0,\mathrm{\infty })$. The same monotonicity gives $0\le {\tau }^{\mu }\le T^{\mu }$ on $P$.

Put $S=\mathrm{\Sigma }\cap \{\tau >0\}$. If $S$ is empty, its stipulated supremum is zero and the right-hand side is nonnegative by $T\ge 0$ and [F3], so the estimate holds, even when $P$ consists only of zero-height points. Otherwise $P$ is nonempty. Applying [F2] to the continuous $v$ on compact $P$ gives finite extrema; taking $M$ to be the larger of their absolute values and $1$ yields $\mathrm{\mid }v\mathrm{\mid }\le M$ on $P$. Extend $w$ by zero at zero height. By the restriction and subspace assertions of [F5], the continuous $\tau$ restricts to a continuous map $\tau {\mathrm{\mid }}_P:P\to [0,\mathrm{\infty })$, since its values are nonnegative by [F1]. Composing this map with the continuous power map $t\mapsto t^{\mu }$ from step 1.1 gives continuity of ${\tau }^{\mu }:P\to \mathbb{R}$ by [F5]. The product assertion of [F5], applied on the topological space $P$ to ${\tau }^{\mu }$ and $v$, makes the extended $w$ continuous on all of $P$. At a zero-height point, $\mathrm{\mid }w\mathrm{\mid }\le M{\tau }^{\mu }$ tends to zero by step 1.1 and continuity of $\tau$, proving continuity there too.

In the remaining case $P$ is nonempty and compact, and the extended $w:P\to \mathbb{R}$ is continuous. Applying [F2] with domain $P$ gives a point at which $w$ attains its maximum on $P$.

At a maximum point with $\tau =0$ the value is zero. At a positive-height point on $\mathrm{\partial }\mathrm{\Sigma }$ it is at most $B$. If the maximum point $p$ is interior to $\mathrm{\Sigma }$ and has positive height, then $U=S\cap (\mathrm{\Sigma }\setminus \mathrm{\partial }\mathrm{\Sigma })$ is an open neighborhood of $p$ in $\mathrm{\Sigma }$ by continuity of $\tau$. Since $U\subseteq P$, the maximum on $P$ gives $w(q)\le w(p)$ for every $q\in U$, exactly the local-maximum condition in the Statement. The hypothesis therefore gives $v(p)\le 2$, so $w(p)\le 2\tau (p{)}^{\mu }\le 2T^{\mu }$ by step 1.1. These exhaustive cases bound the maximum on $P$ by $\max \{2T^{\mu },B\}$ and hence bound the supremum on $S\subseteq P$, with no regularity required of the zero level.