sect. 7.2), while we need the convolution kernel to be contained in $B_{1/2}(\bm{0}) \subset Q$, $B_r(\bm{x})$ means an open ball centers in $\bm{x}$ with radius $r$.

\[lem:3.1\] Let $u \in H^1({\mathbb{R}}^d)$. Then $\partial_i {\mathbf{S}}_\epsilon(u)={\mathbf{S}}_\epsilon(\partial_i u)$, $${\left\lVert {\mathbf{S}}_\epsilon u \right\rVert}_{0, {\mathbb{R}}^d} \leq {\left\lVert u \right\rVert}_{0, {\mathbb{R}}^d} ,$$ and there exists a constant $C$ only depends on $d$, such that: $${\left\lVert {\mathbf{S}}_\epsilon u - u \right\rVert}_{0, {\mathbb{R}}^d} \leq \epsilon C  {\left\lVert \nabla u \right\rVert}_{0, {\mathbb{R}}^d} .$$

\[lem:3.2\] Let $f \in L^2_{\text{loc}}({\mathbb{R}}^d)$ be a 1-periodic function. Then there exists a constant $C$ only depends on $d$, such that for any $u \in L^2({\mathbb{R}}^d)$, $${\left\lVert f^\epsilon {\mathbf{S}}_\epsilon u \right\rVert}_{0, {\mathbb{R}}^d} \leq C{\left\lVert f \right\rVert}_{0, Q}{\left\lVert u \right\rVert}_{0,{\mathbb{R}}^d} .$$

Denote $\tilde{\Omega}_{\epsilon}=\{\bm{x} \in {\mathbb{R}}^d: \text{dist}(\bm{x}, \partial \Omega) < \epsilon \}$ as a boundary strip to $\Omega$ with width $2\epsilon$, and $\Omega_{\epsilon}=\{\bm{x} \in \Omega: \text{dist}(\bm{x}, \partial \Omega) < \epsilon \}$. We have an estimation:

\[lem:3.4\] Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb{R}}^d$ and $f \in L^2_{\text{loc}}({\mathbb{R}}^d)$ a 1-periodic function. Then there exists a constant $C$ only depends on $\Omega$, such that for any $u \in H^1({\mathbb{R}}^d)$ $$\int_{\tilde{\Omega}_{\epsilon}} {\left\lvert f^\epsilon \right\rvert}^2 {\left\lvert {\mathbf{S}}_\epsilon(u) \right\rvert}^2 \leq C \epsilon {\left\lVert f \right\rVert}_{0,Q}^2{\left\lVert u \right\rVert}^2_{1,{\mathbb{R}}^d} .$$

We can obtain 