Relations between $L^\infty$ and $H^2$

Question

I know that $H^2(0,T)\subset L^2(0,T)$, and also we know that $L^\infty(0,T)\subset L^2(0,T)$, so my question is:

Is there some relation between $H^2(0,T)$ and $L^\infty(0,T)$, for instance: $L^\infty(0,T)\subset H^2(0,T)?$

Thanks

Answer 1

$L^\infty(0,T)$ is not contained in $H^2(0,T)$. As an exercise you can show that $1_{(0,T/2)}\notin H^2(0,T)$ (it is not even in $H^1$). The converse inclusion holds as a consequence of the Sobolev embedding theorem. In fact, $H^2(0,T)\subset H^1(0,T)\subset C([0,T])$.