The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $u:\mathcal{D}\to\mathbb{R}$. 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 Intrinsic Calculus on the Wasserstein Space: Standing Notation §operators.

1. (Bounded functions have subordinate growth)¶ 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.

2. (Envelope bounds)¶ 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

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

3. (Monotonicity in the weight)¶ Let $\delta,\delta'\in\mathbb{R}$ satisfy $0<\delta<\delta'$. If $u$ has penalty-subordinate growth from above, then

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