Timeline for Sobolev inequalities on manifolds: dependence of the constants on the Riemannian metric
Current License: CC BY-SA 4.0
9 events
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| Oct 10, 2020 at 20:18 | answer | added | Bernd Ammann | timeline score: 1 | |
| Oct 10, 2020 at 18:57 | comment | added | C M | @DeaneYang Thanks a lot! I'll look further in this direction. | |
| Oct 10, 2020 at 18:41 | comment | added | Deane Yang | You can probably get the $L^\infty$ bound from the sharp $L^2$ inequality using Moser iteration. My guess is that this is somewhere in a paper or book that requires elliptic estimates on a Riemannian manifold. If you know or learn the basic outline of what Moser iteration is, then you probably can work out the details yourself. You can see something like what you want in Appendix C of my paper Convergence of Riemannian manifolds with integral bounds on curvature. II. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 2, 179–199. | |
| S Oct 10, 2020 at 14:46 | history | suggested | gmvh |
Added top-level tag
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| Oct 10, 2020 at 14:15 | review | Suggested edits | |||
| S Oct 10, 2020 at 14:46 | |||||
| Oct 10, 2020 at 12:49 | comment | added | C M | @DeaneYang : thanks, though it seems to me that Hebey's books only deal with subcritical Sobolev embeddings, and the ones I'm interested in are mostly super-critical (Morrey-type) inequalities. | |
| Oct 9, 2020 at 23:39 | comment | added | Deane Yang | Try looking at the book Sobolev Spaces on Riemannian Manifolds by Hebey. | |
| Oct 9, 2020 at 17:05 | comment | added | Neal | I don't have time to look up the reference, but I recall that Taylor's three volumes on PDEs develops the theory over Riemannian manifolds and may have proofs that let you make dependence explicit. | |
| Oct 9, 2020 at 14:23 | history | asked | C M | CC BY-SA 4.0 |