Fourier Transform of $L^2$ functions

Question

Let $\Omega\subset\mathbb{R}^3$ be a bounded regular domain. Take $f(x,t)$ to be a function defined on $\Omega\times (0,T)$. Suppose that $f\in L^2(\Omega)$, and let $\hat{f}$ be the Fourier transform of $f$ in time (being extended by $0$ outside $(0,T)$). Can we prove that $\hat{f}$ remains also in $L^2(\Omega)$?