Fourier transform of a function defined on a smooth surface in Rd

Question

I have a function $f$ defined on a $C^2$-boundary of a bounded open subset $D\subset\mathbb{R}^n$, with $$\int_{\partial D}|f|d\sigma<\infty,$$ $d\sigma$ denoting the surface measure. Does it make sense to define the Fourier transform of $f$ as $$\hat{f}(\omega):=\int_{\partial D}f(y)e^{-2\pi i y\cdot \omega}d\sigma(y)?$$ If so, is $\omega$ in all of $\mathbb{R}^n$ or just some subset of it?