The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $u:\mathcal{D}\to\mathbb{R}$ and let $\delta\in\mathbb{R}$ be positive. The set $\mathcal{D}$ contains the set $\mathcal{D}_{\Sigma}$, which is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii), so $\mathcal{D}$ is nonempty; it is regarded as a subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (i). That a real-valued function on $\mathcal{D}$ is bounded above, or below, near each point of $\mathcal{D}$, the sets $A_{g}(\mu)$ and $B_{g}(\mu)$ entering that notion, and the upper and lower semicontinuous envelopes are those of that definition, for this metric; penalty-subordinate growth from above and from below are those of that definition. The functions $u-\delta\mathcal{E}$ and $u+\delta\mathcal{E}$ on $\mathcal{D}$ have the values $u(\nu)-\delta\,\mathcal{E}(\nu)$ and $u(\nu)+\delta\,\mathcal{E}(\nu)$ at $\nu$.

1. (The envelope $u^{-}_{\delta}$)(i) Suppose that $u$ has penalty-subordinate growth from above, and let $C\in\mathbb{R}$ be as in that definition for this $\delta$, so that $u(\nu)\le C+\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$. Adding $-\delta\,\mathcal{E}(\nu)$ to both sides, by the compatibility of the order with addition in the ordered field $\mathbb{R}$, gives $u(\nu)-\delta\,\mathcal{E}(\nu)\le C$ for every $\nu\in\mathcal{D}$. So for every $\mu\in\mathcal{D}$ the number $C$ lies in $A_{u-\delta\mathcal{E}}(\mu)$, with the positive radius $1$, and $u-\delta\mathcal{E}$ is bounded above near each point of $\mathcal{D}$. Its upper semicontinuous envelope is therefore defined, and we write

$$u^{-}_{\delta}=(u-\delta\mathcal{E})^{*}:\ \mathcal{D}\to\mathbb{R}.$$

2. (The envelope $u^{+}_{\delta}$)(ii) Suppose that $u$ has penalty-subordinate growth from below, and let $C\in\mathbb{R}$ be as in that definition for this $\delta$. Exactly as in clause 1, $-C\le u(\nu)+\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$, so $-C$ lies in $B_{u+\delta\mathcal{E}}(\mu)$ for every $\mu\in\mathcal{D}$ and $u+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$. Its lower semicontinuous envelope is therefore defined, and we write

$$u^{+}_{\delta}=(u+\delta\mathcal{E})_{*}:\ \mathcal{D}\to\mathbb{R}.$$

(iii) The two functions $u^{-}_{\delta}$ and $u^{+}_{\delta}$ are together called the $\delta$-envelopes of $u$ relative to the penalty pair.