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Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. I would like to compute (or at least lower bound) the following: $$ \inf_{\theta\ne\theta'}\frac{\Vert f_\theta - f_{\theta'}\Vert_p}{|\theta-\theta'|}. $$ In particular, I want to understand how this depends on $f$, and would like a bound that depends explicitly on $f$. This is also where the properties of $f$ come in: The weaker the assumptions the better, but e.g. if there a nice bound that depends (say) on the deriviatives of $f$, then we can assume the needed regularity.

My suspicion is that there is an easy counterexample to show this can be rather poorly behaved even for smooth functions, but I have not been creative enough so far.

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For any real $p\ge1$ and any real $t\ne0$, $$\frac{\|f_t-f_0\|_p}{|t-0|} \le\frac{\|f_t\|_p+\|f_0\|_p}{|t|} =\frac{2\|f\|_p}{|t|}\to0$$ as $|t|\to\infty$.

So, the least lower bound in question is always $0$.

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  • $\begingroup$ And what if $\theta$ lay in a compact set, say $\theta\in[a,b]$? $\endgroup$
    – tim622
    Commented Jul 22, 2021 at 21:03
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    $\begingroup$ @tim622 : That would be a very different question (to be posted separately), with a very different answer. $\endgroup$
    – Iosif Pinelis
    Commented Jul 22, 2021 at 21:15
  • $\begingroup$ Sure, done: mathoverflow.net/questions/398119/… $\endgroup$
    – tim622
    Commented Jul 22, 2021 at 21:25

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