Let us consider a smooth function $u$ defined on some open set containing $\bar{\Omega}$ which satisfies $u=g$ on $\partial \Omega$ when $\Omega$ is Lipschitz (You may assume that $\Omega$ is an rectangle in $\mathbb{R}^2$.). After some study, I got to know that there exists some results related to the inverse trace theorem. For example, if $g\in B^{1-1/p, p}(\partial \Omega)$, then there exists a continuous extension $E$ from $B^{1-1/p, p}(\partial \Omega)$ to $W^{1,p}(\Omega)$ such that $Tr(E(g))=g$ on $\partial \Omega$. (If $p=1$, then Trace map is a surjective map between $W^{1,p}(\Omega)$) and $L^{1}(\partial \Omega)$.)
My question is "Is there some different extension whose image is bounded when $g$ lies in the image of smooth functions for trace map?". More specifically, the previously well known fact is that there exists a constant $C$ such that $||E(g)||_{W^{1,p}(\Omega)}\le C ||g||_{B^{1-1/p, p} \hspace{0.2cm}(\partial \Omega)}$.
Is there any other Extension or any other bound which is corresponding to the right hand side of above equation when $g$ is a trace of some smooth function $u$?
For example, is there exist a extension satisfying $||E(g)||_{W^{k,p}(\Omega)}$ $\le$$||g||_{L^2(\partial \Omega)}\cdot C(\Omega)$ where $C(\Omega)$ is a constant depending only on $\Omega$ or dimension. In 1 d problem, linear intepolation gives such extension. (The right hand side could be nonzero when $g=0$. The extension need not be continuous, but only boundedness of image.)
I tried to use cutoff function or mollifier and even something related to partition of unity to reduce the interior norm of $||u||_{W^{k,p}(\Omega)}$, but the result was not good.
Thanks in advance.
