A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$, since every open ball of $(H,d_{H})$ is a subset of $H$; accordingly $W=D(A)\cap H=D(A)$ and $V\cap H=V$ in the notation of Hilbert Triples: Standing Notation and Background §open-sets. Let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$ that is locally strictly proper, satisfies the first-order structure condition and satisfies the shift-continuity condition.

Let $u,v:H\to\mathbb{R}$ and let $C\in\mathbb{R}$ satisfy

then $u$ is bounded above near each point of $H$ and $v$ is bounded below near each point of $H$ by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §bound. Assume that $u$ is a viscosity subsolution of $F$ on $H$ and that $v$ is a viscosity supersolution of $F$ on $H$. Then the following hold.

1. (Uniform comparison on nearby points)¶ For every positive $\epsilon\in\mathbb{R}$ there is a positive $\theta\in\mathbb{R}$ such that all $x,y\in V$ with $|x-y|_{H}\le\theta$ satisfy

2. (Comparison)¶ $u(x)\le v(x)$ for every $x\in V$.