Is there a bound on second derivative in terms of third derivative?

Question

An argument similar to the one outlined here shows that the sup-norm of $f''$ on $[0,1]$ can be bounded in terms of the sup-norms of $f$, $f'$, and $f'''$. Can the sup-norm of $f''$ be bounded in terms of sup-norms of $f$ and $f'''$ alone, without assuming a bound on $f'$? Are there counterexamples?

Answer 1

Setting $p,q,r = \infty$, $n=1$, $j=2$, $m=3$ in the Gagliardo-Nirenberg interpolation inequality yields $$\sup |f''| \le C \sup |f'''|^{2/3} \sup |f|^{1/3}.$$

You can establish this pretty easily starting from the easier case $$\|f'\| \le c \|f''\|^{1/2} \|f\|^{1/2},$$

where I'm using $\| \cdot \|$ for the sup norm. This is a quantification of the following fact: if the velocity is large at some time, then the only way to make the displacement stay small is to impose a large acceleration. This holds without any condition on the domain (other than one-dimensionality).

Applying this twice (to the functions $f'$ and $f$) yields $$\|f''\| \le c \|f'''\|^{1/2} \|f'\|^{1/2}\le c \|f'''\|^{1/2} (c \|f''\|^{1/2} \|f\|^{1/2})^{1/2}= c^{3/2} \|f''\|^{1/4} \|f'''\|^{1/2}\|f\|^{1/4}.$$

Now just divide both sides by $\| f''\|^{1/4}$ to get $$\|f''\|^{3/4} \le c^{3/2} \|f'''\|^{2/4} \|f\|^{1/4},$$

which is the G-N inequality I stated with $C = c^2$.