Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 114, 10 pages      arXiv:0711.3905
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Some Sharp L2 Inequalities for Dirac Type Operators

Alexander Balinsky a and John Ryan b
a) Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF 24 4AG, UK
b) Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA

Received August 31, 2007, in final form November 14, 2007; Published online November 25, 2007

We use the spectra of Dirac type operators on the sphere Sn to produce sharp L2 inequalities on the sphere. These operators include the Dirac operator on Sn, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in Rn.

Key words: Dirac operator; Clifford algebra; conformal Laplacian; Paenitz operator.

pdf (239 kb)   ps (177 kb)   tex (12 kb)


  1. Ahlfors L.V., Old and new in Möbius groups, Ann. Acad. Sci. Fenn. Math. 9 (1984), 93-105.
  2. Beckner W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. 138 (1993), 213-242.
  3. Bojarski B., Conformally covariant differential operators, in Proceedings of XXth Iranian Math. Congress, Tehran, 1989.
  4. Brackx F., Delanghe R., Sommen F., Clifford analysis, Pitman, London, 1982.
  5. Branson T., Ørsted B., Spontaneous generators of eigenvalues, J. Geom. Phys. 56 (2006), 2261-2278, math.DG/0506047.
  6. Calderbank D., Dirac operators and Clifford analysis on manifolds with boundary, Preprint no. 53, Institute for Mathematics, Syddansk University, 1997, available at
  7. Cnops J., Malonek H., An introduction to Clifford analysis, Textos de Matematica, Serie B, Universidade de Coimbra, Departmento de Matematica, Coimbra, 1995.
  8. Davies E.B., Hinz A., Explicit construction for Rellich inequalities, Math. Z. 227 (1998), 511-523.
  9. Erdos L., Solovej J.P., The kernel of Dirac operators on S3 and R3, Rev. Math. Phys. 10 (2001), 1247-1280, math-ph/0001036.
  10. Lieb E., Loss M., Analysis, Graduate Texts in Mathematics, Vol. 14, American Mathematical Society, Providence, 2001.
  11. Liu H., Ryan J., Clifford analysis techniques for spherical PDE, J. Fourier Anal. Appl. 8 (2002), 535-564.
  12. Ryan J., Iterated Dirac operators in Cn, Z. Angew. Math. Phys. 9 (1990), 385-401.
  13. Ryan J., Dirac operators on spheres and hyperbolae, Bol. Soc. Mat. Mexicana 3 (1996), 255-269.
  14. Sommen F., Spherical monogenic functions and analytic functionals on the unit sphere, Tokyo J. Math. 4 (1981), 427-456.
  15. Sudbery A., Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. (85) (1979), 199-225.
  16. Van Lanker P., Clifford analysis on the sphere, in Clifford Algebras and their Applications in Mathematical Physics, Editors V. Dietrich et al., Kluwer, Dordrecht, 1998, 201-215.

Previous article   Next article   Contents of Volume 3 (2007)