$L^2(\mathbb{S}^{d-1})$ 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?

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, MR0934859, Zbl 0654.42014.

$L^2(\mathbb{S}^{d-1})$ 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, MR0934859, Zbl 0654.42014.

$L^2(\mathbb{S}^{d-1})$ 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?

$L^2(\mathbb{S}^{d-1})$ 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?

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, MR0934859, Zbl 0654.42014.