Basic Properties of the Delta-Envelopes on the Wasserstein Space

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 $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and let $\delta\in\mathbb{R}$ be positive. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, which is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, so that $\mathcal{D}$ 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 §measures; that a real-valued function is bounded above, or below, near each point of a nonempty subset, its upper and lower semicontinuous envelopes, and the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of $u$ relative to the penalty pair, functions on $\mathcal{D}$, are as fixed there, always for this metric; the functions $u-\delta\mathcal{E}$ and $u+\delta\mathcal{E}$ on $\mathcal{D}$ are likewise those of The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair. Being upper semicontinuous and lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ is understood in that metric space, and that a function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is continuous is as fixed there. The function $-u$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ has the value $-u(\nu)$ at $\nu$. Then the following hold.

1. (Semicontinuity and bounds)¶ If $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ and

If $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $u^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ and $u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$.

2. (Duality)¶ If $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $-u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $(-u)^{+}_{\delta}(\nu)=-\,u^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$. If $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $-u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $(-u)^{-}_{\delta}(\nu)=-\,u^{+}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$.

3. (Exact envelopes)¶ Suppose that $u$ is continuous and that $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$. Then $u$ is bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, so that both $\delta$-envelopes are defined, and