Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ and let $u:\mathcal{D}\to\mathbb{R}$. Penalty-subordinate growth from above and from below, and, for positive $\delta\in\mathbb{R}$, 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 Domain of a Noise 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 $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space, $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line. The function $-u$ on $\mathcal{D}$ has the value $-u(\nu)$ at $\nu$. Then claims 1 to 4 hold for every positive $\delta\in\mathbb{R}$, and claims 5 to 7 hold under their additional hypothesis that the pair is noise-closed.

1. (Semicontinuity and bounds)¶ 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)¶ 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)¶ 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)¶ 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 $\nu\in\mathcal{D}$.

5. (Bounded functions have subordinate growth)¶ Suppose that the pair is noise-closed, and let $b\in\mathbb{R}$. If $u(\mu)\le b$ for every $\mu\in\mathcal{D}$, then $u$ has penalty-subordinate growth from above; if $b\le u(\mu)$ for every $\mu\in\mathcal{D}$, then $u$ has penalty-subordinate growth from below.

6. (Envelope bounds)¶ Suppose that the pair is noise-closed, let $b\in\mathbb{R}$ and let $\delta\in\mathbb{R}$ be positive. If $u(\mu)\le b$ for every $\mu\in\mathcal{D}$, then

$$u^{-}_{\delta}(\mu)\le b-\delta\,\mathcal{E}(\mu)\qquad\text{for every }\mu\in\mathcal{D};$$

if $b\le u(\mu)$ for every $\mu\in\mathcal{D}$, then $b+\delta\,\mathcal{E}(\mu)\le u^{+}_{\delta}(\mu)$ for every $\mu\in\mathcal{D}$.

7. (Monotonicity in the weight)¶ Suppose that the pair is noise-closed, and let $\delta,\delta'\in\mathbb{R}$ satisfy $0<\delta<\delta'$. If $u$ has penalty-subordinate growth from above, then

$$u^{-}_{\delta'}(\mu)+(\delta'-\delta)\,\mathcal{E}(\mu)\le u^{-}_{\delta}(\mu)\qquad\text{for every }\mu\in\mathcal{D};$$

if $u$ has penalty-subordinate growth from below, then $u^{+}_{\delta}(\mu)\le u^{+}_{\delta'}(\mu)-(\delta'-\delta)\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.