Trace theorem says that there exists unique continous linear mapping $\gamma : W^{k,p}(\varOmega) \rightarrow L^q(\varOmega)$ that $\gamma(u) = u|_{\partial \varOmega}$.
It is correct to write $||u||_{L_q(\partial \varOmega)} = ||\gamma(u)||_{W^{k,p}(\varOmega)}$? I think that that is what the theorem says, if I understand right. By the way, I am trying to prove $||u||_{L^q(\partial \varOmega)} \leq c ||u||_{W^{k,p}(\varOmega)}$ and I am using the equality above to prove.