Let $\Omega\subset\mathbb{R}^d$ be a compact subset of $\mathbb{R}^d$. For a Schwartz function $f:\mathbb{R}^d\to\mathbb{R}$ the Fourier Transform can be written as $$ \hat{f}(\xi) = \int_{\mathbb{R}^d}f(x)e^{ix^\intercal\xi}d\xi $$
If we look at an Schwartz arbitrary function $g:\Omega\to\mathbb{R}$, then this does not work directly. This is because the Fourier Transform is not defined for a compact subset of $\mathbb{R}^d$. According to two other posts, (1) and (2), it is possible to use the Paley Wiener theorem or some other approach to extend $g$ to a function from $\mathbb{R}^d\to\mathbb{R}$ when $g\in L^1$.
I am interested in the value of
$$
\gamma(f) = \int_{\mathbb{R}^d}||\xi||^{s}_2|\hat{f}(\xi)e^{ix^\intercal\xi}|d\xi=\int_{\mathbb{R}^d}||\xi||^{s}_2|\hat{f}(\xi)|d\xi
$$
for a function $f:\Omega\to\mathbb{R}$, in particular when $\gamma(f)$ is finite and well defined. What assumptions on $f$ and $\Omega$, aside from $f$ being a Schwartz function, are needed to make $\gamma$ finite and well defined? Does the boundary of $\Omega$ for example need to be smooth?
Dirier mentioned in the comments that $C^{\infty}$ and compactly supported on $\Omega$ should work. What if $f$ is less smooth or not compactly supported?
(1) Analog of the Fourier transform on a bounded domain?
(2) Fourier transform of function defined on subset of $\mathbb{R}^n$