Timeline for Local Sobolev embedding on complete Riemannian manifold
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 19, 2019 at 8:28 | vote | accept | DLIN | ||
| Jan 24, 2019 at 1:39 | comment | added | Deane Yang | @PiotrHajlasz, my guess is that the best constant corresponds to the constant curvature cases. The exponential factor seems to arise from the volume of a hyperbolic ball with radius $r$. But this is all speculation. | |
| Jan 23, 2019 at 23:44 | comment | added | Piotr Hajlasz | @DeaneYang Relative isoperimetric inequality for Ricci bounded below was proved by Buser. Buser's inequality is equivalent to a Poincare inequality. Ricci implies that the measure is doubling (Bishop–Gromov comparison theorem) and doubling plus Poincare implies the right Sobolev inequality. However, I am not sure if this gives the best constant as in the theorem stated in my answer. | |
| Jan 23, 2019 at 20:58 | comment | added | Deane Yang | Federer-Fleming proved that the best local Sobolev constant is equal to the best constant for the local isoperimetric inequality. A local isoperimetric inequality is known to hold in a neighborhood of a point, where the constant depends on a lower bound of Ricci and a lower bound, as a function of $r < 0$, on the volume of a geodesic ball of radius $r$ centered at that point. Alas, I don't remember the references for this. | |
| Jan 23, 2019 at 1:56 | history | edited | Piotr Hajlasz |
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| Jan 16, 2019 at 15:22 | history | edited | Piotr Hajlasz | CC BY-SA 4.0 |
added 50 characters in body
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| Jan 16, 2019 at 13:06 | answer | added | Piotr Hajlasz | timeline score: 3 | |
| Jan 16, 2019 at 12:08 | history | edited | DLIN |
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| Jan 16, 2019 at 10:33 | history | asked | DLIN | CC BY-SA 4.0 |