The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ as in Hilbert Triples: Standing Notation and Background §open-sets, let $\mathrm{Sym}(H)$ be as in Hilbert Triples: Standing Notation and Background §restriction, and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$. For a real $\alpha$ and $x,y\in H$, $\alpha(x-y)$ denotes the scalar multiple by $\alpha$ of the difference $x-y$ in $H$, and $\alpha^{2}$ is the natural power.

1. (Structure triple at a level)¶ Let $R\in\mathbb{R}$ be positive and let $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ be moduli of continuity. We say that $(\omega_{1},\omega_{2},\omega_{3})$ is a structure triple for $F$ at $R$ if

for all $x,y\in W$, every $r\in\mathbb{R}$ with $-R\le r\le R$, all $X,Y\in\mathrm{Sym}(H)$ and all $\alpha,\delta\in\mathbb{R}$ with $1<\alpha$ and $0<\delta<1$. The three arguments of the moduli are nonnegative, being $|x-y|_{H}$ and multiples of $|x-y|_{H}^{2}$ by the nonnegative reals $\alpha$ and $\delta\alpha^{2}$.

2. (The first-order structure condition)¶ The operator $F$ satisfies the first-order structure condition if for every positive $R\in\mathbb{R}$ there is a structure triple for $F$ at $R$.