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.



Lower bounds on translates of a function
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 $0$.


