Timeline for Control the oscillation of a function by its total variation
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 4, 2019 at 14:49 | answer | added | Piotr Hajlasz | timeline score: 3 | |
| Jun 4, 2019 at 14:32 | comment | added | Daniele Tampieri | I am not sure on how to proceed, but intuitively I would try to show that $u\in BV$ (or $BV_\mathrm{loc}$) implies $u$ is essentially bounded, i.e. $u\in L^\infty_\mathrm{loc}$: then I would show that seestial bondedness imply a control of the (essential) oscillation in terms of the $BV$ norm. I do not post a full answer since I am not yet skilled enough to work quickly with the extension of those concepts in dimension $N>1$. | |
| Jun 4, 2019 at 14:14 | comment | added | user140746 | @DanieleTampieri Thank you. How would the proof go in the general case? | |
| Jun 4, 2019 at 12:45 | comment | added | Daniele Tampieri | When $N=1$ and you use the classical definition of total variation $$V^a_b(f)=\sup_{\mathcal{P}} \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |$$ where $\mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\}|P\text{ is a partition of } [a,b] \right\}$, then this is certainly true since $u$ is bounded. In the case $N>1$ you should recur to a concept of essential (in the sense of measure theory) oscillation. | |
| Jun 4, 2019 at 11:30 | review | First posts | |||
| Jun 4, 2019 at 12:27 | |||||
| Jun 4, 2019 at 11:25 | history | asked | user140746 | CC BY-SA 4.0 |