$L^2(\mathbb{S}^{d-1})$ 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:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}
\int\limits\_{\mathbb{R}^d} \bigg|\int\limits\_{\mathbb{S}^{d-1}}f(\omega)e^{-2\pi i x\cdot \omega}\Dm\sigma(\omega)\bigg|^2\frac{\Dm x}{(1+|x|^2)^s}\lesssim \int\_{\mathbb{S}^{d-1}}|f(\omega)|^2 \Dm\sigma(\omega)
$$

This is a lemma in Luis Vega's article [1] (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?

Another idea from Willie Wong is to use Fourier restriction. By Tomas-Stein, we can deduce the conclusion when $s>\frac{d}{d+1}$.

**Reference**

[1] Luis Vega, "Schrödinger equations: Pointwise convergence to the initial data" (English) Proceeding of the American Mathematical Society 102, No. 4, 874-878 (1988), [DOI 10.2307/2047326](https://doi.org/10.2307/2047326), [MR0934859](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0934859), [Zbl 0654.42014](https://zbmath.org/?q=an%3A0654.42014).