Given $u\in W^{1}_{p}(\omega)$ with $1\leq p\leq \infty$, and the mollifier $\rho\in C_0^{\infty}(R^d)$ with support $B_1$ is a unit ball centered at the origin, $\rho\geq 0$ and $\int_{B_1} \rho = 1$. Let $u_\epsilon(x) = \int_{\omega} u(x-y)\epsilon^{-d}\rho(\frac{x}{\epsilon})dy\ \forall \ x\in K$ where $K$ is a compact subset of $\omega$. Prove that for $\epsilon$ small enough, $||u-u_\epsilon||_{L^{p}(K)}\leq C\epsilon|u|_{W^{1}_{p}(\omega)}$ for some constant $C>0$
My thought: I have not been able to make much progress on proving this difficult inequality besides reading this quite technical proof here (https://math.stackexchange.com/questions/328697/convolution-error-estimate-reference-request). But the proof idea there could not be applied here, since it uses properties of Fourier transform, which we don't have here for our convolution.
My question: Could someone please help with this problem? Any thoughts would really be appreciated.
