It is given in Evans and almost all other PDE books that the trace operator for a bounded domain $\Omega$ with $C^1$ boundary
(*) $\gamma:H^1(\Omega)\rightarrow L^2(\partial\Omega)$
is continuous i.e, $\|\gamma u\|_{L^2(\partial\Omega)}\leq C \|u\|_{H^1(\Omega)}$.
I was wondering if this result could be extended to
$\gamma:H^2(\Omega)\rightarrow H^1(\partial\Omega)$
is continuous i.e, $\|\gamma u\|_{H^1(\partial\Omega)}\leq C \|u\|_{H^2(\Omega)}$.
The above result assertion seems correct to me as I have increased the regularity equally by 1 in (*). I was reading about fractional sobolev spaces. The result I have read is that for a Lipschitz domain $\Omega$ and $\frac{1}{2}< s<\frac{3}{2}$ we have the following continuous trace operator,
$\gamma:H^s(\Omega)\rightarrow H^{s-\frac{1}{2}}(\partial\Omega)$
Now if I can take $s=\frac{3}{2}$ then it would follow easily as ,
$\|\gamma u\|_{H^1(\partial\Omega)}\leq C \|u\|_{H^\frac{3}{2}(\Omega)}\leq C\|u\|_{H^2(\Omega)}$
However $s<\frac{3}{2}$.
I have just started reading about fractional sobolev spaces, traces and extensions operator. So I am not quite sure about some of the results. Thanks for any help.