$$ H^s_p=\left\{f\in \mathcal{S}'(\mathbb{R}^d):\ (1-\Delta)^{s/2} f\in L^p, \ p>1\right\} . $$ $$ \|f\|_{H^s_p}:=\|(1-\Delta)^{s/2} f\|_p. $$ Let $\rho\geq 0, \rho\in C^\infty_c$,$\int\rho=1$ and $\rho_n(x)=n^d \rho(nx)$.
My Question is: suppose $f\in \cap_{s\geq 0}H_p^s,\ \ f_n(x)=f*\rho_n(x)=\int f(x-y)\rho_n(y)\ dy$ and $\alpha\in (0,1)$, is the following inequality: $$ \| \ \ {|f_n|} \ \ \|_{H^{-\alpha}_p}\leq C \|f\|_{H^{-\alpha}_p} \ \ \ \left(\mbox{or}\ \leq C \|f\|_{H^{-\beta}_q}\\ \mbox{for some suitable index}\ \beta<1, \ q>1.\right) $$ reasonable? And how to prove it?