Localized Hölder contraction estimate for stochastic homogenization

Let $x\in\mathbb{R}^d$, $\omega\in\Omega$, and $n\geq 0$. For $v>0$, let $\chi_v(y)=\chi(y/v)$, and let $\chi_{n,x}(y)=\chi_{30\sqrt{d}L_n}(y-x)$, where $\chi(y)=1\wedge(2-|y|)_+$. For $f\in C^{0,\beta}(\mathbb{R}^d)$, write

$$|f|_n=\|f\|_{L^\infty(\mathbb{R}^d)}+\sup_{x\neq y}L_n^\beta\frac{|f(x)-f(y)|}{|x-y|^\beta}.$$

Localized Hölder contraction estimate. For each $f\in C^{0,\beta}(\mathbb{R}^d)$,

$$|\chi_{n,x}S_nf|_n\leq L_n^{-\delta}|f|_n.$$

The estimate compares the localized solutions of the original and approximate problems in the rescaled global Hölder norms. The surrounding discussion states that it does not hold for every triple $(x,\omega,n)$; such controls are instead available for large $n$ with high probability and on a large portion of space, as in the cited work.