Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let $$\tilde{K}^{\epsilon} := K^{\epsilon} \ast G^{\epsilon}$$ where $G^{\epsilon} := (\pi \epsilon)^{-3/2} \exp(-|x|^2/\epsilon)$. Does $$f(x)(K^{\epsilon}-\tilde{K}^{\epsilon})$$ converge to zero in $\mathcal{D}'(\mathbb{R}\times \mathbb{R}^3)=(C^{\infty}_0)'(\mathbb{R}\times \mathbb{R}^3)$, i.e. does
$$\iint f(x)(K^{\epsilon}-\tilde{K}^{\epsilon})\varphi dx dt \rightarrow 0$$
as $\epsilon \rightarrow 0$ where $\varphi \in C^{\infty}_0$?
The problem is that $f$ is in no $L^p$ with $p\leq 2$, otherwise one could use Plancherel, then Hölder and Hausdorff-Young.
