Timeline for Is there a generalization of Sobolev spaces for certain locally compact groups?
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11 events
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| Jan 14, 2015 at 14:47 | comment | added | Juhana Siljander | For further references about Sobolev spaces in an arbitrary metric space, I would look for this book: ems-ph.org/books/book.php?proj_nr=141. Furthermore, the theory has been generalized even more in here: liu.diva-portal.org/smash/…. | |
| Aug 17, 2014 at 15:52 | comment | added | Terry Tao | There is some literature on Sobolev spaces in arbitrary metric measure spaces: www2.pitt.edu/~hajlasz/OriginalPublications/… . For a locally compact group, one can use Haar measure for the measure, and if the group is second countable one can use Birkhoff-Kakutani to get a metric (but the choice of metric is not unique, and this can lead to different Sobolev spaces, e.g. a Riemannian metric gives different results to a Carnot-Caratheodory metric). | |
| Jun 18, 2014 at 12:28 | comment | added | timur | I heard that the analogy of tempered distributions in this setting is called Harish-Chandra functions. To define them, one must introduce some kind of decay conditions, so it might give something useful if you Google this term. | |
| Jun 18, 2014 at 12:12 | comment | added | Johannes Hahn | Even another way to do this is to define fractional sobolev spaces as interpolation spaces between integer-order sobolov spaces. | |
| Jun 18, 2014 at 11:02 | answer | added | user53114 | timeline score: 1 | |
| Nov 18, 2011 at 21:59 | comment | added | B R | In another direction, you can ask about $p$-adic Sobolev spaces. | |
| Nov 18, 2011 at 21:51 | comment | added | B R | Christopher, for a compact Lie group $G$, you can define a Sobolev space on $G$ via charts, reducing it to $\mathbb R^n$, or you can define it analogously to $\mathbb R^n$ by using the Plancherel theorem for $G$ (replacing the Laplacian on $\mathbb R^n$ with the Casimir operator on $G$ and using a sum over the dual space of $G$, rather than an integral over $\mathbb R^n$). When $G$ is non-abelian, I don't think these are equivalent, though they might be. | |
| Nov 18, 2011 at 20:48 | comment | added | Christopher A. Wong | It seems like those developments are based on the fact that such groups can be made into Riemannian manifolds, in some way reducing it to a known case. Am I wrong in this assessment? | |
| Nov 18, 2011 at 20:02 | comment | added | Deane Yang | I believe that if you google "Sobolev spaces compact Lie groups", you'll find lots of hits and if you look at them as well as references contained in them, you'll find what you want. | |
| Nov 18, 2011 at 19:51 | history | edited | Christopher A. Wong | CC BY-SA 3.0 |
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| Nov 18, 2011 at 19:39 | history | asked | Christopher A. Wong | CC BY-SA 3.0 |