Timeline for Structure of the Cantor part of the derivative of a BV function
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2019 at 10:54 | vote | accept | Romeo | ||
| Dec 29, 2017 at 13:34 | comment | added | Gro-Tsen | The thing is, I don't know how to properly justify all my claims about the Hausdorff measures of the Cantor-like set I described, which is why I wrote a comment (suggesting you consider it) rather than an answer. I'm fairly sure it works (and more generally, Cantor-like sets with appropriate growth rates can be used as counterexamples for a lot of things), but this is really not my specialty. | |
| Dec 29, 2017 at 9:07 | comment | added | Romeo | @Gro-Tsen Uh, I see. That's really interesting how nasty Hausdorff measures (and dimension) can be! If you want to write your comments as an answer I will mark it as the accepted answer. Thanks a lot! | |
| Dec 28, 2017 at 13:59 | comment | added | Gro-Tsen | No, I think the derivative of the staircase in the example I propose will not be $\mathscr{H}^\beta$ for any $\beta$: it is, in a certain sense, intermediate between $\mathscr{H}^\alpha$ for $\alpha=\log2/\log3$ and $\mathscr{H}^{\alpha-\varepsilon}$ for every $\varepsilon>0$, because the constructed Cantor set has measure $0$ for the H-measure of dimension $\alpha$ but infinity for the H-measure of any smaller dimension. The H-dimension is a very coarse classification, which can be indefinitely subdivided (if I understand correctly what is going on!). | |
| Dec 28, 2017 at 13:30 | comment | added | Romeo | @Gro-Tsen Thanks a lot for your comment, it makes perfectly sense. I think you are right: the Cantor set you constructed has still H-dimension $\alpha= \log 2 / \log 3$, nevertheless its staircase has not $\mathscr H^{\alpha}$ as derivative. Btw, one more question: the derivative of its staircase will still be some $\mathscr H^{\beta}$ (though not with $\beta = \alpha$), am I right? So in the end, is it true that the Cantor part of the derivative of a BV function is always (a.c. w.r.t. some) fractional Hausdorff measure? Thanks again. | |
| Dec 28, 2017 at 10:54 | comment | added | Gro-Tsen | What happens if you construct a Cantor-like set in $[0,1]$ by repeatedly cutting a middle interval, but such that each of the $2^n$ parts after the $n$-th cut has a length $\ell_n$ which tends to $0$ much faster than $(1/3)^n$ but much slower than $(1/3+\varepsilon)^n$ for any fixed $\varepsilon>0$? The resulting set would have H-dimension $\log2/\log3$ but would still be infinitely small w.r.t. H-measure of that dimension, so I think the derivative of its "staircase" would be a counterexample to your question. | |
| Dec 28, 2017 at 9:52 | history | edited | Romeo | CC BY-SA 3.0 |
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| Dec 27, 2017 at 13:39 | history | edited | Romeo | CC BY-SA 3.0 |
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| Dec 25, 2017 at 16:49 | answer | added | Jean Duchon | timeline score: 9 | |
| Dec 25, 2017 at 16:03 | history | asked | Romeo | CC BY-SA 3.0 |