Fix $s > 1/2.$ The trace operator is surjective from $H^{s}(\mathbb{R}^{n+1})$ to $H^{s-\frac{1}{2}}(\mathbb{R}^{n}).$
If $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ with smooth boundary, I know there is a bounded trace operator from $H^{1}(\Omega)$ to $L^{2}(\partial \Omega).$ Intuitively, the image of the trace should be $H^{\frac{1}{2}}(\partial \Omega),$ by the result I commented at the beginning... But my problem is, How can we define such a space? via local coordinates? I do not see why that will necessarily work, I cannot work out the details. Of course, I cannot compute inmediately the Fourier transform on the surface, so describing the space $H^{\frac{1}{2}}(\partial \Omega)$ seems like a very hard thing to me.