Journal d’Analyse Mathématique

, Volume 98, Issue 1, pp 43–64 | Cite as

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

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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. V. Ahlfors,Möbius Transformations in Several Dimensions, Ordway Lecture Notes, University of Minnesota, 1981.Google Scholar
  2. [2]
    L. V. Ahlfors,Möbius transformations in R n expressed through 2×2 matrices of Clifford numbers, Complex Variables5 (1986), 215–224.MathSciNetGoogle Scholar
  3. [3]
    Ö. Akin and H. Leutwiler,On the invariance of the solutions of the Weinstein equation under Möbius transformations, inClassical and Modern Potential Theory and Applications (K. Gowrisankran et al. eds.), Kluwer Dodrecht, 1994, pp. 19–29.Google Scholar
  4. [4]
    D. Calderbank,Dirac operators and Clifford analysis on manifolds, Max Plank Institute for Mathematics, Bonn, preprint 96-131, 1996.Google Scholar
  5. [5]
    C. Cao and P. Waterman,Conjugacy invariants of Möbius groups, inQuasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 109–139.Google Scholar
  6. [6]
    P. Cerejeiras and J. Cnops,Hodge-Dirac operators for hyperbolic space, Complex Variables41 (2000), 267–278.MATHMathSciNetGoogle Scholar
  7. [7]
    J. Cnops,An Introduction to Dirac Operators on Manifolds, Progress in Mathematical Physics, Birkhäuser, Boston, 2002.MATHGoogle Scholar
  8. [8]
    S.-L. Eriksson-Bique,Möbius transformations and k-hypermonogenic functions, inClifford algebras and potential theory, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu, Joensuu, 2004, pp. 213–226.Google Scholar
  9. [9]
    S.-L. Eriksson,Integral formulas for hypermonogenic functions, Bull. Belg. Math. Soc.11 (2004), 705–707.MATHMathSciNetGoogle Scholar
  10. [10]
    S.-L. Eriksson-Bique,k-hypermonogenic functions, inProgress in Analysis (H. Begehr et al., eds.) World Scientific, New Jersey, 2003, pp. 337–348.Google Scholar
  11. [11]
    S.-L. Eriksson-Bique and H. Leutwiler,Hypermonogenic functions, inClifford Algebras and their Applications in Mathematical Physics, Volume 2, (J. Ryan and W. Sprößig, eds.), Birkhäuser, Boston, 2000, pp. 287–302.Google Scholar
  12. [12]
    S.-L. Eriksson and H. Leutwiler,Hypermonogenic functions and their Cauchy-type theorems, inTrends in Mathematics: Advances in Analysis and Geometry, Birkhäuser, Basel, 2003, pp. 1–16.Google Scholar
  13. [13].
    S.-L. Eriksson and H. Leutwiler,Some integral formulas for hypermonogenic functions, to appear.Google Scholar
  14. [14]
    G. Gaudry, R. Long and T. Qian,A martingale proof of L 2 -boundedness of Clifford valued singular integrals, Ann. Mat. Pura Appl.165 (1993), 369–394.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Gowrisankran and D. Singman,Minimal fine limits for a class of potentials, Potential Anal.13 (2000), 103–114.CrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Habib,Invariance des fonctions α-harmoniques par les transformations de Möbius, Exposition Math.13 (1995), 469–480.MATHMathSciNetGoogle Scholar
  17. [17]
    L. K. Hua,Starting with the Unit Circle, Springer-Verlag, Heidelberg, 1981.MATHGoogle Scholar
  18. [18]
    A. Huber,On the uniqueness of generalized axially symmetric potentials, Ann. of Math. (2),60 (1954), 351–358.CrossRefMathSciNetGoogle Scholar
  19. [19]
    V. Iftimie,Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie9 (1965), 279–332.MathSciNetGoogle Scholar
  20. [20]
    R. S. Krausshar and J. Ryan,Clifford and harmonic analysis on spheres and hyperbolas Revista Matemática Iberoamericana21 (2005), 87–110.MATHMathSciNetGoogle Scholar
  21. [21]
    R. S. Krausshar, J. Ryan and Q. Yuying,Harmonic, monogenic and hypermonogenic functions on some conformally flat manifolds in R n arising from special arithmetic groups of the Vahlen group, Contemporary Mathematics, Contemporary Mathematics370 (2005), 159–173.Google Scholar
  22. [22]
    R. S. Krausshar and J. Ryan,Some conformally flat spin manifolds, Dirac operators and automorphic forms, to appear in Journal of Mathematical Analysis and Applications.Google Scholar
  23. [23]
    H. Leutwiler,Best constants in the Harnack inequality for the Weinstein equation, Aequationes Math.34 (1987), 304–315.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    H. Leutwiler,Modified Clifford analysis, Complex Variables17 (1992), 153–171.MATHMathSciNetGoogle Scholar
  25. [25]
    H. Liu, and J. Ryan,Clifford analysis techniques for spherical pde's, J. Fourier Analysis Appl.8 (2002), 535–564.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    A. McIntosh,Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, inClifford Algebras in Analysis and Related Topics, (J. Ryan, ed.), CRC Press, Boca Raton, 1996, pp. 33–87.Google Scholar
  27. [27]
    M. Mitrea,Singular Integrals, Hardy Spaces, and Clifford Wavelets, Lecture Notes in Mathematics, No 1575, Springer-Verlag, Heidelberg, 1994.Google Scholar
  28. [28]
    M. Mitrea,Generalized Dirac operators on non-smooth manifolds and Maxwell's equations, J. Fourier Analysis Appl.7 (2001), 207–256.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    I. Porteous,Clifford Algebras and the Classical Groups, Cambridge University Press, cambridge, 1995.MATHGoogle Scholar
  30. [30]
    J. Ryan,Dirac operators on spheres and hyperbolae, Boletín Sociedad Matemática Mexicana.3 (1996), 255–270.Google Scholar
  31. [31]
    K. Th. Vahlen,Über Bewegungen und Complexe Zahlen, Math. Ann.55 (1902), 585–593.CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    P. Van Lancker,Clifford analysis on the sphere, inClifford Algebras and their Applications in Mathematical Physics (V. Dietrich et al., eds.), Kluwer, Dordrecht, 1998, pp. 201–215.Google Scholar
  33. [33]
    A. Weinstein,Generalized axially symmetric potential theory, Bull. Amer. Math. Soc.59 (1953), 20–38.MATHMathSciNetCrossRefGoogle Scholar

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

Personalised recommendations