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 $u:\mathcal{D}\to\mathbb{R}$ and let $\delta\in\mathbb{R}$ be positive. Penalty-subordinate growth from above and from below, and the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of $u$, functions on $\mathcal{D}$, are those of the definitions cited, and the functions $u\mp\delta\mathcal{E}$ on $\mathcal{D}$ are those of The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair. Upper semicontinuity, lower semicontinuity and continuity of a real function on $\mathcal{D}$ are understood relative to $\mathcal{D}$ in 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), $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Numbers. The function $-u$ on $\mathcal{D}$ has the value $-u(\nu)$ at $\nu$. Then the following hold.

1. (Semicontinuity and bounds)(i) If $u$ has penalty-subordinate growth from above, then $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ and $u(\nu)-\delta\,\mathcal{E}(\nu)\le u^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$. If $u$ has penalty-subordinate growth from below, then $u^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}$ and $u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$.

2. (Duality)(ii) The function $u$ has penalty-subordinate growth from above if and only if $-u$ has penalty-subordinate growth from below, and then $(-u)^{+}_{\delta}(\nu)=-\,u^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$. The function $u$ has penalty-subordinate growth from below if and only if $-u$ has penalty-subordinate growth from above, and then $(-u)^{-}_{\delta}(\nu)=-\,u^{+}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$.

3. (Exact envelopes)(iii) Suppose that $u$ is continuous on $\mathcal{D}$ and that $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$. If $u$ has penalty-subordinate growth from above, then $u^{-}_{\delta}(\nu)=u(\nu)-\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$; if $u$ has penalty-subordinate growth from below, then $u^{+}_{\delta}(\nu)=u(\nu)+\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$.

4. (Bounds by a smaller multiple of the penalty)(iv) Suppose that $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$, and let $C,\eta\in\mathbb{R}$ satisfy $0\le\eta\le\delta$. If $u$ has penalty-subordinate growth from above and $u(\nu)\le C+\eta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$, then

$$u^{-}_{\delta}(\nu)\le C-(\delta-\eta)\,\mathcal{E}(\nu)\qquad\text{for every }\nu\in\mathcal{D}.$$

If $u$ has penalty-subordinate growth from below and $-C-\eta\,\mathcal{E}(\nu)\le u(\nu)$ for every $\nu\in\mathcal{D}$, then $-C+(\delta-\eta)\,\mathcal{E}(\nu)\le u^{+}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$.