I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev space $H^{1}(C)$. If for a function $u\in H^{1}(\mathcal{C})$ we denote by $u(\cdot,0)$ the trace of $u$ over $\Omega$, is it true that the range of this trace operator coincides with the Sobolev space $H^{1/2}(\Omega)$, i.e. $$\left\{u(\cdot,0):u\in H^{1}(\mathcal{C})\right\}=H^{1/2}(\Omega)$$ and, if yes, where I can find the proof of such result?
Thank you very much for any help!