The Journal of Geometric Analysis

, Volume 26, Issue 3, pp 1913–1924 | Cite as

Eigenfunctions of the Laplace–Beltrami Operator on Harmonic \(NA\) Groups

  • Ewa Damek
  • Pratyoosh Kumar


We characterize some \(L^p\)-type eigenfunctions of the Laplace–Beltrami operator on harmonic \(NA\) groups corresponding to the eigenvalue \((\rho ^2-\beta ^2)\) for all \(\beta >0\).


Eigenfunctions Poisson transform Harmonic \(NA\) group 

Mathematics Subject Classification

Primary 43A85 Secondary 22E25 



This work was completed while both authors were visiting IISc Bangalore India. They are grateful to Prof. S. Thangavelu for his invitation as well as to the staff of the Department of Mathematics there for their kind support and warm hospitality. The authors would like to thank the referee for the valuable comments and suggestions made.


  1. 1.
    Anker, J.-P., Damek, E., Yacoub, C.: Spherical analysis on harmonic \(AN\) groups. Ann. Sc. Norm. Super. Pisa Cl. Sci. 23(4), 643–679 (1996)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Astengo, F., Camporesi, R., Di Blasio, B.: The Helgason Fourier transform on a class of nonsymmetric harmonic spaces. Bull. Aust. Math. Soc. 55(3), 405–424 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cowling, M.G., Dooley, A.H., Korányi, A., Ricci, F.: H-type groups and Iwasawa decompositions. Adv. Math. 87, 1–41 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Damek, E.: Harmonic functions on semidirect extensions of type \(H\) nilpotent groups. Trans. Am. Math. Soc. 290(1), 375–384 (1985). MR0787971 (86i:43013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Damek, E.: Geometry of a semi-direct extension of a Heisenberg type nilpotent group. Coll. Math. 53, 255–268 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Damek, E.: A Poisson kernel on Heisenberg type nilpotent group. Coll. Math. 53, 239–247 (1987)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Damek, E., Ricci, F.: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc. (N.S.) 27(1), 139–142 (1992). MR1142682 (93b:53043)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes, 28. Princeton University Press, Princeton (1982)zbMATHGoogle Scholar
  9. 9.
    Furstenberg, H.: A Poisson formula for semi-simple groups. Ann. Math. 77, 335–386 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hashizume, M., Kowata, A., Minemura, K., Okamoto, K.: An integral representation of an eigenfunction of the Laplacian on the Euclidean space. Hiroshima Math. J. 2, 535–545 (1972)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Helgason, S.: A duality for symmetric spaces with applications to group representations. Adv. Math. 5, 1–154 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Helgason, S.: Geometric Analysis on Symmetric Spaces. Mathematical surveys and monographs, 2nd edn. American Mathematical Society, Providence (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T., Tanaka, M.: Eigenfunctions of invariant differential operators on a symmetric space. Ann. Math. 107, 1–39 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Knapp, A.W., Williamson, R.E.: Poisson integrals and semisimple groups. J. Anal. Math. 24, 53–76 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kumar, P., Ray, S.K., Sarkar, R.P.: The role of restriction theorems in harmonic analysis on harmonic \(NA\) groups. J. Funct. Anal. 258(7), 2453–2482 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Saïd, S.B., Oshima, T., Shimeno, N.: Fatou’s theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces. Int. Math. Res. Not. 2003(16), 915–931 (2003)CrossRefzbMATHGoogle Scholar

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© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Institute of MathematicsWrocław UniversityWrocławPoland
  2. 2.Department of MathematicsIndian Institute of Technology, GuwahatiGuwahatiIndia

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