Let us asssume that a sequence $(v_n)$ satisfies
(i) $v_n\in{\rm W}^{2,\infty}(0,1)$,
(ii) $||v_n||_{{\rm H}^{\theta}(0,1)}$ is bounded for some $0<\theta<1$.
Is it true that then the following holds:
(iii) $\liminf_{n\rightarrow+\infty}|v_n'(s)|<+\infty$ (a.e. $s\in (0,1)$)?
Counterexamples are welcome. Here we have in mind the following Slobodeckij seminorm: https://en.wikipedia.org/wiki/Sobolev_space $$ ||v||_{{\rm H}^{\theta}(0,1)}:= \Big(\int_0^1\int_0^1{{|v(s)-v(\sigma)|^2}\over{|s-\sigma|^{2\theta+1}}}dsd\sigma\Big)^{1\over 2}\;. $$ Obviously, if we allow the case $\theta=1$, then we have $||v_n'||_{{\rm L}^2(0,1)}\leq C$, so that, by Fatou's Lemma, we get $\liminf_{n\rightarrow+\infty}|v_n'(s)|<+\infty$ (a.e. $s\in (0,1)$).