Timeline for Reverse Hausdorff Young for nonnegative functions
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 9, 2014 at 20:59 | vote | accept | PhoemueX | ||
| Oct 9, 2014 at 20:57 | comment | added | PhoemueX | Dear Professor Tao, thank you very much for your comments. I will definitely try to work through the paper by Christ, because I was (more or less exactly) looking for functions which make Hausdorff Young sharp and was just hoping (perhaps naively) that non-negative functions might already work. | |
| Oct 8, 2014 at 0:02 | comment | added | Terry Tao | In fact, it is very rare for the Hausdorff-Young inequality to be close to sharp; the function $f$ basically has to look like the indicator function of the sum of a generalised arithmetic progression and an ellipsoid. A precise formulation of this fact (which comes from additive combinatorics) can be found in Proposition 6.4 of this paper of Christ: arxiv.org/pdf/1406.1210.pdf | |
| Oct 7, 2014 at 23:55 | answer | added | Bill Johnson | timeline score: 5 | |
| Oct 7, 2014 at 23:30 | comment | added | Terry Tao | If the spatial domain is the integers (so the frequency domain is the unit circle), then one can take $f$ to be the indicator function of a lacunary sequence such as $1, 2, \dots, 2^n$ and use Rudin's inequality (see e.g. Lemma 4.33 of my book with Van Vu) to contradict the inequality; the case $p = 4/3$ can be worked out by hand for instance. One can then transfer to Euclidean spaces by standard methods (e.g. blurring each integer by an approximation to the identity to pass from ${\bf Z}$ to ${\bf R}$). | |
| Oct 7, 2014 at 23:10 | answer | added | Christian Remling | timeline score: 2 | |
| Oct 7, 2014 at 21:50 | review | First posts | |||
| Oct 7, 2014 at 22:13 | |||||
| Oct 7, 2014 at 21:48 | history | asked | PhoemueX | CC BY-SA 3.0 |