Estiamate $\sum_{z\in \epsilon\mathbb Z^d \cap \Omega}\Big(f(z)\big(\eta((x-z)/\epsilon)-\eta((y-z)/\epsilon)\big)\Big)^2$ with Holder norm

Question

Let us consider a bouded domain $\Omega$ a function $\eta \in C^\infty_c(\mathbb R^n)$ and a function $f \in W^{k,p}$. How can we estimate $$\sum_{z\in \epsilon\mathbb Z^d \cap \Omega}\Big(f(z)\big(\eta((x-z)/\epsilon)-\eta((y-z)/\epsilon)\big)\Big)^2$$ for $x,y \in \Omega$?

My guess is that it should be something like $\le C|x-y|^{2\alpha}|\eta((x-z)/\epsilon)|$ where $\alpha$ is the Holder exponent of the embedding $W^{k,p} \hookrightarrow C^{0,\alpha}$. Is this true?