In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in order to prove \begin{equation} \lVert f * \hat\mu \rVert_{L^{p'}(\mathbb{R}^d)}\le C \lVert f \rVert_{L^p(\mathbb{R}^d)}, \quad \text{for } p=\frac{2d+2}{d+3}, \quad d\ge 3, \end{equation} where $\hat{\mu}\triangleq K$ is the Foureir transform of Lebesgue measure of surface with nonvanishing Gaussian curvature (and we may assume it is just the Fourier tranform of Lebesgue measure of unit sphere $\mathbb{S}^{d-1}$), he tore the coordinates into two pieces $x=(x',t)$, where $x'=(x_1,...,x_{d-1})$, then \begin{equation} f*\hat{\mu} (x) =\int_{\mathbb{R}}\int_{\mathbb{R}^{d-1}} K(x'-y',t-s) f(y',s) dy'ds. \end{equation} Thus we may restrict our attention on the behavior of $K(x',t)$ with respect to $x'$. More precisely, if we assume that $(Ug)(x')= \int_{\mathbb{R}^{d-1}} K(x'-y',t)dt$, then Schlag claimed that $U(t)$ satifies \begin{equation} \lVert U(t) \rVert_{L^1(\mathbb{R}^{d-1}) \to L^\infty(\mathbb{R}^{d-1})} \le C |t|^{d-1}, \quad \lVert U(t) \rVert_{L^2(\mathbb{R}^{d-1}) \to L^2(\mathbb{R}^{d-1})} \le C <\infty, \end{equation} where $C>0$ is independent of $t\in \mathbb{R}$, and then we can use Riesz-Thorin interpolation theorem and then use Hardy-Littlewood-Sobolev inequality to get the desired estimates.

And my question is how to check the second estimate (i.e. the uniform bound of $L^2 \to L^2$), Schlag said it suffices to check that $K(\hat{\cdot},t) \in L_{\xi'}^\infty L_t^\infty ( \mathbb{R}^{d-1} \times \mathbb{R})$, where $K(\hat{\cdot},t)$ means the Fourier transform of $K(x',t)$ w.r.t. $x'$. For example, in $d=3$, then the Fourier transform of unit sphere can be represented by $\hat{\sigma}(x)=\frac{\sin{|x|}}{|x|}$ explicitly, but how can I check that \begin{equation} K(\xi',t)= \int_{\mathbb{R}^2} e^{-2\pi i x' \cdot \xi'}\frac{\sin{|(x',t)|}}{|(x',t)|} dx' \in L_{\xi'}^\infty L_t^\infty ( \mathbb{R}^{2} \times \mathbb{R}) \quad? \end{equation}