7
$\begingroup$

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| (1 + |t|^2)^{s/2} \hat{f}(t)\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?

Improve this question
$\endgroup$
8
  • 1
    $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Commented Nov 18, 2011 at 20:02
  • 1
    $\begingroup$ 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? $\endgroup$
    – Christopher A. Wong
    Commented Nov 18, 2011 at 20:48
  • 1
    $\begingroup$ 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. $\endgroup$
    – timur
    Commented Jun 18, 2014 at 12:28
  • 2
    $\begingroup$ 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). $\endgroup$
    – Terry Tao
    Commented Aug 17, 2014 at 15:52
  • 1
    $\begingroup$ 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/…. $\endgroup$
    – Juhana Siljander
    Commented Jan 14, 2015 at 14:47

1 Answer 1

Reset to default
1
$\begingroup$

Yes, look at the papers of Gorka and Reyes.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Aren't they limited to the LCA case? $\endgroup$
    – Yemon Choi
    Commented Aug 17, 2014 at 11:58

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .