Injections of Sobolev Spaces

Question

I am studying Sobolev Spaces and appear the following question, that look simple, by I do not know how answer it:

If $T>0$ and $k\geq 2$, with $k\in \mathbb{N}$, then $ H^{k-1}(0,T) \subset W^{k,\infty}(0,T)$.

Why it is true ?

Answer 1

It is not true. Counterexample: For $x>0$, let $\mathscr I f(x) = \int_0^xf(y)dy$ and consider $G:=\mathscr I^{k-1}g$ where $g(x)=\frac1{x^{1/4}}$. Note $G^{(k-1)}=g\in L^2((0,T))\setminus L^\infty((0,T))$. Easy to check that $G^{(j)}\in L^\infty$ for all $j\le {k-2}$. So $G$ is a function that is in $H^{k-1}((0,T))$ for every $T$, but not even in $W^{k-1,\infty}((0,T))$, so its not in $W^{k,\infty}((0,T))$.