Timeline for Energy in doubling measure metric spaces
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 4, 2018 at 22:15 | comment | added | BigM | I wrote a sketch. I can do it with a few rather strong assumptions added. The ambient space should be Euclidean. Measure has to be a.c with repstect to Lebeague measure with bounded derivative . As C. Remling pointed out atomic measures and discrete spaces give cheap counterexamples | |
| Sep 4, 2018 at 21:14 | comment | added | Stefano Gogioso | Out of curiosity: did it work out? | |
| Aug 31, 2018 at 16:52 | comment | added | BigM | It should work. But I'm going to sit detail and fill in the details..btw thanks for refering to Assouad's embedding theorem. I wasnt aware of that. Seems if we can prove it for R^n then we have it for arbitrary doubling measure spaces. | |
| Aug 31, 2018 at 16:38 | comment | added | Stefano Gogioso | This is roughly what I was thinking. Fix $R$, and write $U_R$ for the union of $x \in U$ such that the open ball of radius $R$ around $x$ is completely contained in $U$. For fixed $x \in U_R$, break the integral over $y \in U$ into a short range part (radius $r \in [0,R]$) and a long range part (radius $r>R$). The long range part is bounded because of relative compactness. The short range part is bounded because of the integral expression above. You then integrate over x, and get again something bounded by relative compactness. You then let $R$ go to zero. | |
| Aug 31, 2018 at 15:59 | comment | added | BigM | For the sake of simplicity, let's stick to R. Why does the double integral behave as you stated? | |
| Aug 31, 2018 at 11:22 | history | answered | Stefano Gogioso | CC BY-SA 4.0 |