I think it is possible that if I will be able to show that g is in $L^2(\mathbb{R})$ and show that the integral of $|\hat{g}(\omega)|^2(1+\omega)^s d\omega$ over $\mathbb{R}$ is finite, then it is in the Sobolev space $W^{p,s}$ where $p=2$, i.e., represented as $H^s(\mathbb{R})$. But how will I show these? Thanks for the help!
$f \in L^2$ so its Fourier transform $h$ is $L^2$. The Fourier transform of $g$ is $h^{s+1}$. Now, up to scaling, $h(\omega)=2iSinc(\omega)$, where $Sinc(x)=\frac{\sin{x}}{x}$.
So, $|h^{s+1}(\omega)|^2(1+\omega^2)^s=O(\omega^{-2s-2+2s})=O(\omega^{-2})$, thus proving the integrability.