If I have a function $f:\Omega \to \mathbb{R}$ in $H^k(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is compact, then what is known about the Fourier transform $\hat{f}$? What space does it lie in? I want to say something like it's in $L^2(K)$ or $H^k(K)$ where $K$ is some other compact subset of $\mathbb{R}^n$.
I don't know if this makes sense... but essentially I want to bound $|(1+|\xi|^2)^k|$ where $\xi$ is the variable in $\hat{f}$, so I need to know how big $\xi$ can get..
Does this hold? Thanks for any help.