$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: \begin{equation} \int_{\mathbb{R}^d} |\int_{\mathbb{S}^{d-1}}f(\omega)e^{-2\pi i x\cdot \omega}d\sigma(\omega)|^2\frac{dx}{(1+|x|^2)^s}\lesssim \int_{\mathbb{S}^{d-1}}|f(\omega)|^2 \end{equation}

This is a lemma in Luis Vega's article "Schrodinger Equations: Pointwise Convergence to the Initial Data" (Page 2). He gives a somewhat roundabout proof. Intuitively, it may help to expand $f$ into spherical harmonics. But I don't know the accurate behavior of spherical harmonics under Fourier transform.

Could you please present a proof or give a explicit expression of spherical harmonics' Fourier transform?