Timeline for How to use Stein-Tomas theorem to check to following inequality?
Current License: CC BY-SA 4.0
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| Sep 13, 2019 at 22:54 | comment | added | Willie Wong | (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. | |
| Sep 13, 2019 at 22:50 | comment | added | Willie Wong | 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 $T^*T$. | |
| S Sep 13, 2019 at 16:11 | history | rollback | Tao |
Rollback to Revision 1 - Edit approval overridden by post owner or moderator
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| Sep 13, 2019 at 14:09 | history | suggested | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing: norm sign scaling+ $exp\, \to\, \exp$.
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| Sep 13, 2019 at 12:38 | review | Suggested edits | |||
| S Sep 13, 2019 at 16:11 | |||||
| Sep 13, 2019 at 12:10 | history | asked | Tao | CC BY-SA 4.0 |