Let $f\in C_c(\mathbb{R}^n)$ and $\rho_\epsilon$ be a mollifier with support in $B(0,\epsilon)$. Is it possible to get a lower bound on $||\rho_\epsilon*f||_{H^s}$ in terms of $||f||_{H^s}$ if one can choose $\epsilon$ small enough?
I know the convergence speed estimate$$ ||\rho_\epsilon*f-f||_{H^{s}}\leq C \epsilon^m||f||_{H^{s+m}}$$holds for all $m\in \mathbb{N}$ and can be proved using the definition of $H^s$ norm through Fourier transform. From the same proof, one can show this estimate holds for $0\leq m\leq1$ as well. In particular, for $m=0$, the left hand side can be rewritten as
$$ \int_{\mathbb{R}^n} (\hat \rho(\epsilon \xi)-1)^2 \hat f(\xi)^2(1+|\xi|^2)^s ,$$for any $\epsilon>0$ ($\hat f$ is the Fourier transform). However, it looks like $||(\hat \rho(\epsilon \xi)-1)||_{L^\infty}>1$, so this constant $C>1$ and thus triangular inequality does not give any useful lower bound. Is it possible to optimize this constant by choosing $\epsilon$ small enough?