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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).

Mathmo
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    Just a remark, a compactly supported function with compactly supported Fourier Transform must be $0$. Hence the Fourier Transform of a mollifier cannot be compactly supported. – nicomezi Oct 09 '19 at 09:14
  • Is $\phi_\epsilon$ an approximation of identity ? – nicomezi Oct 09 '19 at 09:21
  • Ah sorry for confusion on both fronts - not assuming $\phi$ to be compactly supported (which seems to be the usual definition of mollifier) - just that it decays fast enough for moments to exist. Yes $\phi_\epsilon(x)=\phi(x/\epsilon)/\epsilon$. – Mathmo Oct 09 '19 at 09:39
  • its proven for $H^s$ spaces (more or less; i didnt check the specifics of the definition of a mollifier) in Majda & Bertozzi's book on fluids (vorticity and incompressible flow, part of lemma 3.5) – Calvin Khor Oct 09 '19 at 10:23
  • Thanks, that's helpful. The case of $p=2$ is the easiest one though - the $p=1$ case is the one I am most interested in. – Mathmo Oct 09 '19 at 11:28
  • I don't have a reference, but I think the easiest way to prove this estimate (let's say for $k=1$) is to write $\delta_0-\phi_\epsilon=(1_{(0,\infty)}-\Phi_\epsilon)'$ with $\Phi_\epsilon(x)=\int_{-\infty}^x \phi_\epsilon(t),dt$ and use Young's inequality to get $|f-\phi_\epsilon\ast f|p\leq |1{(0,\infty)}-\Phi_\epsilon|_1|f'|_p$. Of course, $\phi$ has to decay fast enough to make the $L^1$-norm on the right side finite, but then it scales $\sim \epsilon$. – MaoWao Oct 09 '19 at 13:11

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