Journal d’Analyse Mathématique

, Volume 98, Issue 1, pp 43–64

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

  • Qiao Yuying
  • Swanhild Bernstein
  • Sirkka-Liisa
  • John Ryan

DOI: 10.1007/BF02790269

Cite this article as:
Yuying, Q., Bernstein, S., Sirkka-Liisa et al. J. Anal. Math. (2006) 98: 43. doi:10.1007/BF02790269


We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • Qiao Yuying
    • 1
  • Swanhild Bernstein
    • 2
  • Sirkka-Liisa
    • 3
  • John Ryan
    • 4
  1. 1.Department of MathematicsHebei Normal UniversityShijazhuangP. R. China
  2. 2.Institute of Mathematics and PhysicsBauhaus University WeimarWeimarGermany
  3. 3.Department of MathematicsTampere University of TechnologyTampereFinland
  4. 4.Department of MathematicsUniversity of ArkansasFayettevilleUSA