The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $(\mathcal{D}_{0},\mathcal{E}_{0})$ be a free entropy penalty, let $R>0$ be real, and let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be its wall-confined free energy with radius $R$; suppose that $\mathcal{D}$ is nonempty. By The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, $\mathcal{D}\subseteq\Sigma_{d,R}$, and we fix a real $e$ with $e\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$, which exists by the same clause. $\mathcal{D}$ is regarded as a subset of the metric space $(\Sigma_{d,R},W_{2})$ of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points; $\Sigma^{2}_{d}$ and $\kappa_{d}$ are those of Square-Integrable Noncommutative Laws: Standing Notation §laws. Let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, with $|u|\le b$ for a real $b$, and let $\delta>0$ be real.

1. (Upper envelope)¶ The function $\mu\mapsto u(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)$ on $\mathcal{D}$ is bounded above by $b-\delta e$, hence bounded above near each point of $\mathcal{D}$. Its upper semicontinuous envelope is written

$$u^{-}_{\delta}=\bigl(u\circ\kappa_{d}-\delta\,\mathcal{E}\bigr)^{*}:\mathcal{D}\to\mathbb{R}.$$

2. (Lower envelope)¶ The function $\mu\mapsto u(\kappa_{d}(\mu))+\delta\,\mathcal{E}(\mu)$ on $\mathcal{D}$ is bounded below by $-b+\delta e$, hence bounded below near each point of $\mathcal{D}$. Its lower semicontinuous envelope is written

$$u^{+}_{\delta}=\bigl(u\circ\kappa_{d}+\delta\,\mathcal{E}\bigr)_{*}:\mathcal{D}\to\mathbb{R}.$$

¶$u^{-}_{\delta}$ and $u^{+}_{\delta}$ are the penalty envelopes of $u$ at level $\delta$.