Timeline for Growth rate of Lipschitz constants for derivatives of $C^\infty$ functions
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2017 at 2:23 | vote | accept | Yair Carmon | ||
| Oct 7, 2017 at 1:09 | answer | added | Christian Remling | timeline score: 3 | |
| Oct 7, 2017 at 1:02 | comment | added | Christian Remling | You could start out with a function that is very large on a small set, but has small $L^1$ norm and integral zero, and make that the $n$th derivative of the $n$th bump. More specifically, make the $L^1$ norm so small that whatever the bounds on the first $n-1$ derivatives were so far, your $n$th bump will also stay below this with its first $n-1$ derivatives (more succinctly, a desire to make the $n$th derivative large in no way forces you to make any of the earlier derivatives large also). | |
| Oct 7, 2017 at 0:56 | comment | added | Yair Carmon | Thanks! Can you give some more details on how to construct the bumps such that all derivatives stay bounded overall? I tried to scale a single bump and couldn't get it to work. | |
| Oct 7, 2017 at 0:39 | comment | added | Christian Remling | On $\mathbb R$ it's easy to build counterexamples, just use bumps with disjoint supports such that the $n$th bump refutes the bound for $p=n$ with $C_f=n$ (and the lower order derivatives are not spectacularly large, so that they still stay bounded overall). If you assume compact support, then I'm not completely sure off the top of my head if this type of counterexample still works, though it well might. | |
| Oct 7, 2017 at 0:31 | history | asked | Yair Carmon | CC BY-SA 3.0 |