Convergence rate of mollifiers in Sobolev spaces (reference request).

Question

Suppose I have a function $f\in W^{k,p}(\mathbb{R})$ and a mollifier $\phi$ with sufficiently many vanishing moments (I do not assume it has compactly supported Fourier transform). I suspect (see this question: Convolution Error Estimate Reference Request) that there is an estimate of the form $$ \|f-\phi_{\epsilon}*f\|_{p}\leq C\epsilon^{k} \|f\|_{W^{k,p}}. $$ Does anyone have a good reference for this (also the case $p=1$). (NB: also realise I can probably replace the full Sobolev norm with just $L^p$ norm of highest order derivative).