Timeline for A Sobolev embedding theorem for functions on spheres
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2022 at 12:49 | vote | accept | Dapao Zhang | ||
| Mar 30, 2022 at 12:49 | |||||
| Mar 30, 2022 at 12:16 | answer | added | Dapao Zhang | timeline score: 0 | |
| Mar 30, 2022 at 7:55 | answer | added | pavel | timeline score: 2 | |
| Mar 28, 2022 at 17:22 | comment | added | Christian Remling | When $f=1$, this works because of the decay of $\widehat{\sigma}$, which is perhaps restating what Willie already said above. (Or it's completely unhelpful since this doesn't give one any idea what one would do with a very irregular $f$.) | |
| Mar 28, 2022 at 15:35 | history | edited | Dapao Zhang | CC BY-SA 4.0 |
Add an idea using Fourier restriction.
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| Mar 28, 2022 at 2:12 | comment | added | Dapao Zhang | Those $f$ may be considered as distributions supported on the sphere, as the inequality says. | |
| Mar 28, 2022 at 0:37 | comment | added | paul garrett | For the questioner: do you extend functions from the sphere to the ambient Euclidean spaces by degree-zero homogeneity? Or just to consider them as distributions supported on the sphere? Could you clarify? | |
| Mar 28, 2022 at 0:29 | comment | added | Willie Wong | IIRC how spherical harmonics transform under Fourier transform is discussed in Stein & Weiss, Introduction for Fourier Analysis on Euclidean Spaces (Chapter 4). But the sort of decay results that guarantees embedding into $H^{-s}$ are more related to Fourier restriction theorems, and I don't think the spherical harmonics expansion help that much. | |
| Mar 27, 2022 at 19:01 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing and formatting
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| Mar 27, 2022 at 17:12 | comment | added | Daniel Shapero | Maybe the wiki on plane wave expansion is helpful here? + some asymptotics of Bessel functions | |
| Mar 27, 2022 at 15:58 | history | asked | Dapao Zhang | CC BY-SA 4.0 |