Reverse Sobolev embedding for H^s

Question

Assume that we are on a bounded domain with a smooth boundary in $\mathbb{R}^n$. I know that for $2m > n$ we have the embedding $$ H^{m+j} \subset C^j,j=0,1,\dots,n. $$ Is there any way to show the following reverse relation? $$ C^k \subset H^{l} $$

Specifically, I am thinking of $n = 2$ and I want to show that $$ H^3 \subset C^1 \subset H^2. $$ (The first part is done from the first embedding.)

Thanks very much!

Answer 1

No, a counterexample is given by $f(x):=x^{3/2}$ on $[0,1]$ (it works for higher dimensions too, simply make it not depend on additional components of $x$)