Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sobolev spaces and the Cayley transform

Authors: Francesca Astengo and Bianca Di Blasio
Journal: Proc. Amer. Math. Soc. 134 (2006), 1319-1329
MSC (2000): Primary 43A80; Secondary 43A85, 43A15
Published electronically: October 4, 2005
MathSciNet review: 2199175
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The generalised Cayley transform  $ \mathcal{C}$ from an Iwasawa $ N$-group into the corresponding real unit sphere  $ \mathbb{S}$ induces isomorphisms between suitable Sobolev spaces $ \mathcal{H}^\alpha(\mathbb{S})$ and $ \mathcal{H}^\alpha(N)$. We study the differential of  $ \mathcal{C}$, and we obtain a criterion for a function to be in  $ \mathcal{H}^\alpha(\mathbb{S})$.

References [Enhancements On Off] (What's this?)

  • 1. F. Astengo, M. Cowling, - B. Di Blasio, The Cayley transform and uniformly bounded representations, J. Funct. Anal. 213 (2004), 241-269. MR 2078626 (2005e:22013)
  • 2. A. D. Banner, Some properties of boundaries of symmetric spaces of rank one, Geom. Dedicata 88 (2001), 113-133. MR 1877212 (2003a:22008)
  • 3. M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, in: Harmonic analysis and group representations, C.I.M.E., 49-128, Liguori, Napoli, 1982. MR 0777340 (86h:22012)
  • 4. M. Cowling, A. H. Dooley, A. Korányi, - F. Ricci, $ H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1-41. MR 1102963 (92e:22017)
  • 5. M. Cowling, A. H. Dooley, A. Korányi, - F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. 8 (1998), 199-237. MR 1705176 (2000m:53071)
  • 6. J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981), 69-70. MR 0619983 (82k:22009)
  • 7. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. MR 0494315 (58:13215)
  • 8. D. Geller, The Laplacian and the Kohn Laplacian for the sphere, J. Differential Geom. 15 (1980), 417-435. MR 0620896 (82i:35132)
  • 9. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, 80. Academic Press, Inc., New York, London, 1978. MR 0514561 (80k:53081)
  • 10. A. Kaplan, Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. MR 0554324 (81c:58059)
  • 11. A. Korányi - H.  M.  Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309-338. MR 0788413 (86m:32035)
  • 12. G. D. Mostow Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press, Princeton, N.J., 1973. MR 0385004 (52:5874)
  • 13. P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989), 1- 60. MR 0979599 (90e:53058)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 43A80, 43A85, 43A15

Retrieve articles in all journals with MSC (2000): 43A80, 43A85, 43A15

Additional Information

Francesca Astengo
Affiliation: Dipartimento di Matematica, Università di Genova, 16146 Genova, Italia

Bianca Di Blasio
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italia

Keywords: Heisenberg type groups, Sobolev spaces
Received by editor(s): November 18, 2004
Published electronically: October 4, 2005
Additional Notes: The authors thank the School of Mathematics of the University of the New South Wales and the Italian G.N.A.M.P.A. for their support
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society