Does $||f'||_\infty \leq \sqrt{t_F-t_0}\,||f'||_2$ hold for time-limited continuous functions $f(t)$ with $\sup_t |f'(t)|<\infty$?

Solution 1:

No, none of these bounds hold.

First remark, the derivative is a local object, so the fact to be time limited will not help to bound a derivative. The derivative at a point do not change if you change your function at other points.

Then, defining $g = f'$, you can see that what you are asking has (almost) nothing to do with derivatives and just with bounding the $L^\infty$ norm by the $L^2$ norm. This is in general false: there are a lot of unbounded functions that are square integrable, such as $g(t) = |t|^{-1/4}$ with $t\in[-1,1]$ and $g(t)=0$ on the complement.

The only thing that is special about $g$ being the derivative of a compactly supported function is the fact that you have to ensure that $∫_{t_0}^{t_f} g = 0$, and so you can replace the above example by $g(t) = |t|^{-5/4}t$ (for $t_0 = -1$ and $t_f=1$, but you can translate and dilate the coordinates to get any $t_0$ and $t_f$). In terms of $f$, it leads to $$ f(t) = \frac{4}{3} \,(t^{3/4} - 1) \ \text{ if } t\in[-1,1] $$ and $f(t)=0$ if $|t|>1$ as a counterexample to all your inequalities.