The Structure Estimate at a Maximiser for Operators within a Cost Defect of a Reference Operator

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$ with closed score along couplings, whose penalty domain $\mathcal{D}$ has the map property. Let $F$, the reference operator, be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair, that satisfies the shift-coercivity condition and the shift-semicontinuity condition. We write $\tfrac{\alpha}{2}$ for the product of $\alpha\in\mathbb{R}$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) and $\alpha^{-1}$ for the multiplicative inverse of a positive $\alpha\in\mathbb{R}$; $|s|$ is the absolute value of $s\in\mathbb{R}$, and $M_{2}(\nu)$ the second moment of $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

Let $F_{1}$ and $F_{2}$ be second-order equation operators over $\mathcal{D}_{\Sigma}$, let $H\in\mathbb{R}$ be nonnegative, and let $h_{1},h_{2}:\mathcal{D}_{\Sigma}\to\mathbb{R}$ satisfy $0\le h_{1}(\nu)\le H$ and $0\le h_{2}(\nu)\le H$ for every $\nu\in\mathcal{D}_{\Sigma}$ and

for every $(\nu,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$, every $r\in\mathbb{R}$ and every $Y\in\mathcal{S}(d)$. No semicontinuity or other regularity of $h_{1}$, $h_{2}$ is assumed.

Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{D}$; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, $u$ has penalty-subordinate growth from above and $v$ from below. Assume that $u$ is a viscosity subsolution of $F_{1}$ and that $v$ is a viscosity supersolution of $F_{2}$, both relative to the penalty pair. For positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ of $u$ and $v^{+}_{\delta}$ of $v$, functions on $\mathcal{D}$, are then defined. Fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$, as provided by Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below.

Let $\delta,\alpha\in\mathbb{R}$ satisfy $0<\delta<1$ and $1<\alpha$, let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be the function with value

at $(\mu,\nu)$, and let $M(\delta,\alpha)$ be the supremum of its values, a real number by Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions §maximiser. Assume $0\le M(\delta,\alpha)$. Let $B,R\in\mathbb{R}$ satisfy

Let $\lambda$ be a properness constant for $F$ at $R$, and let $(\omega_{1},\omega_{2})$ be a second-order structure pair for $F$ at $R$.

Let $m\in\mathbb{R}$ be nonnegative with $M_{2}(\mu)\le m$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le\delta^{-1}B$; such an $m$ exists by Basic Properties of a Wasserstein-Coercive Penalty Pair §moment (applied with $c=\delta^{-1}B$, the number provided there being replaced by its absolute value). Put

a positive real number, and let $C\in\mathbb{R}$ be a score bound for $F$ at $(\delta,R_{0}+2H)$, which exists by The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity because $0<\delta<1$ and $0<R_{0}+2H$. Let

where $\lVert\cdot\rVert_{\nu}$ is the norm of $L^{2}(\nu;\mathbb{R}^{d})$, which contains $\Sigma(\nu)$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair; and let $H_{1},H_{2}\in\mathbb{R}$ satisfy $h_{1}(\nu)\le H_{1}$ and $h_{2}(\nu)\le H_{2}$ for every $\nu\in K$ (for instance $H_{1}=H_{2}=H$). The numbers $m$, $R_{0}$, $C$ and the set $K$ are determined by the penalty pair, $F$, $b$, $b'$, $e_{0}$, $B$, $\delta$, $\alpha$ and $H$ alone; they do not involve $u$, $v$, $F_{1}$, $F_{2}$, $h_{1}$ or $h_{2}$.

(The structure estimate at a maximising pair, with defects)¶ Then there is $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$ with $\Psi_{\delta,\alpha}(\rho^{*},\sigma^{*})=M(\delta,\alpha)$ and