Timeline for Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'
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| Jun 18, 2016 at 0:00 | comment | added | Matt Rosenzweig | @Brian: Would you please elaborate on how you obtain the inequality $$K^{(2)}(\sum_{\phi : \delta^{1/2}-cap}\sum_{\theta \in J(\phi)}||\Theta_{\theta} \ast (\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}})||_{p}^2)^{1/2} \leq K^{(2)}(\sum_{\phi : \delta^{1/2}-cap}||\widehat{g_{\phi}d\sigma}w_{B_{\delta^{-1}}}||_{p}^2)^{1/2}$$ It seems that like you're using Young's inequality and Holder's inequality together with the uniform boundedness of $J(\phi)$. If so, in your application of Young's inequality, how do you know that $\|\Theta_{\theta}\|_{1}$ is finite? | |
| Jul 17, 2015 at 10:38 | comment | added | Brian | I completely understand what you say! Your comments are of great service to me. Thank you for your time! | |
| Jul 17, 2015 at 7:42 | comment | added | Terry Tao | I think $\widehat{g_{t,\theta} d\sigma_t}$ is just the convolution of $f_\theta$ with $\widehat{d\sigma_t}$, so one basically just needs to estimate $\widehat{d\sigma_t}$ on $B_{\delta^{-1}}$ (it should be concentrated on the dual $\delta^{-1/2} \times \delta^{-1}$ tube). | |
| Jul 17, 2015 at 2:15 | comment | added | Brian | Thanks for your comments. But how to control the $l^2$ norm of $L^{p}$ norm of $g_{t,\theta}d\sigma_{t}$ via that of $f_{\theta}$? That is the most difficult part. In Problem 2.2, this part can be easily resolved. How to use Schur's test? How to take integral operator? | |
| Jul 15, 2015 at 13:13 | comment | added | Terry Tao | This seems similar to Problem 2.2 of my lecture notes de.arxiv.org/pdf/math/0311181.pdf . I think one should first convolve $f$ by some suitable approximation to the identity $\varphi$ Fourier concentrated near $B_{\delta^{-1}}$, then disintegrate $\hat f*\varphi$ into measures $g_t\ d\sigma_t = \sum_\theta g_{t,\theta}\ d\sigma_t$) supported on vertical translates $P^{n-1}+te_n$ of the paraboloid $P^{n-1}$, apply the $K^{(1)}$ bound, then use something like Schur's test to control the $L^p$ norm of $\widehat{g_{t,\theta}\ d\sigma_t}$ by that of $f_\theta$. | |
| Jul 15, 2015 at 10:00 | history | edited | Brian |
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| Jul 15, 2015 at 9:55 | review | First posts | |||
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| Jul 15, 2015 at 9:50 | history | asked | Brian | CC BY-SA 3.0 |