Recently, I am reading Rodnianski & Schlag Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem in Stein's version one has \begin{equation}\lVert \int_{\mathbb{R}^3} \frac{exp(i|x-y|)}{4 \pi |x-y|} f(y) dy\rVert_{L^4(\mathbb{R}^3)} \le C \lVert f \rVert_{L^{\frac{4}{3}}(\mathbb{R}^3)}. \end{equation} I have read some references about Stein-Tomas theorem (Fourier Restriction estimate), but I really don't know how to apply this theorem to the proof of the inequality? Can some one give me some tips?
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1$\begingroup$ The version of Stein-Tomas I remember would give exactly the control you write, except with $\exp$ replaced by $\sin$. You use the fact that $\sin(i|x|)/ 4\pi|x|$ is the Fourier transform of the surface measure of the unit sphere, so if $T$ is the mapping that sends a function $f$ on $\mathbb{R}^3$ to the surface measure on the sphere given by $\hat{f} \mu$ where $\mu$ is the standard surface area measure, then the $\sin$ version of your inequality would be estimating $T^*T f$. Since Stein-Tomas says $T$ is bounded from $L^{4/3}$ to $L^2$, so is the mapping for $T^*T$. $\endgroup$– Willie WongCommented Sep 13, 2019 at 22:50
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1$\begingroup$ (In the above, $\sin(i|x|)$ should obviously be $\sin(|x|)$, the $i$ is spurious.) As to the $\cos(|x|)$ part, I don't have a copy of Stein's Beijing lecture notes handy, so I cannot check whether he has a more applicable version of the Stein-Tomas theorem there. $\endgroup$– Willie WongCommented Sep 13, 2019 at 22:54
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