Hi all. My question on M.SE is unanswered (https://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn) so I want to post it here. I changed it slightly.

If I have a function $f:\Omega \to \mathbb{R}$ in $H^k(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is a compact surface, then what is known about the Fourier transform $\hat{f}$? What space does it lie in? 

Most importantly, I want to know if $\lVert f\rVert_{H^k(\Omega)}$ is equivalent to $\lVert \hat{f}(1+|\xi|^2)^{\frac k 2}\rVert_{L^2(K)}$ for some compact $K$ (recall that this is true if the domains are $\mathbb{R}^n$.)

Does this hold? Thanks for any help.