The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth


Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by \(h > 0\) the mean curvature of horospheres in X, and set \(\rho = h/2\). Fixing a basepoint \(o \in X\), for \(\xi \in \partial X\), denote by \(B_{\xi }\) the Busemann function at \(\xi \) such that \(B_{\xi }(o) = 0\). Then for \(\lambda \in \mathbb {C}\) the function \(e^{(i\lambda - \rho )B_{\xi }}\) is an eigenfunction of the Laplace–Beltrami operator with eigenvalue \(-(\lambda ^2 + \rho ^2)\). For a function f on X, we define the Fourier transform of f by

$$\begin{aligned} \tilde{f}(\lambda , \xi ) := \int _X f(x) e^{(-i\lambda -\rho )B_{\xi }(x)} \mathrm{{d}}vol(x) \end{aligned}$$

for all \(\lambda \in \mathbb {C}, \xi \in \partial X\) for which the integral converges. We prove a Fourier inversion formula

$$\begin{aligned} f(x) = C_0 \int _{0}^{\infty } \int _{\partial X} \tilde{f}(\lambda , \xi ) e^{(i\lambda - \rho )B_{\xi }(x)} \mathrm{{d}}\lambda _o(\xi ) |c(\lambda )|^{-2} \mathrm{{d}}\lambda \end{aligned}$$

for \(f \in C^{\infty }_c(X)\), where c is a certain function on \(\mathbb {R} - \{0\}\), \(\lambda _o\) is the visibility measure on \(\partial X\) with respect to the basepoint \(o \in X\) and \(C_0 > 0\) is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.

This is a preview of subscription content, access via your institution.


  1. 1.

    Astengo, F., Camporesi, R., Di Blasio, B.: The Helgason Fourier transform on a class of nonsymmetric harmonic spaces. Bull. Aust. Math. Soc. 55, 405–424 (1997)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Besse, A.L.: Manifolds all of Whose Geodesics are Closed. Ergebnisse u.i. Grenzgeb. Math., vol. 93. Springer, Berlin (1978)

    Google Scholar 

  4. 4.

    Benoist, Y., Foulon, P., Labourie, F.: Flots d’Anosov à distributions stable et instable differéntiables. J. Am. Math. Soc. 5(1), 33–74 (1992)

    Google Scholar 

  5. 5.

    Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics, vol. 20. Walter de Gruyter, Berlin (1995)

    Google Scholar 

  6. 6.

    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der mathematischen Wissenschaften, p. 319. ISSN 0072-7830 (1999)

  7. 7.

    Bourdon, M.: Sur le birapport au bord des CAT(-1) espaces. Inst. Hautes Etudes Sci. Publ. Math. 83, 95–104 (1996)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Buyalo, S., Schroeder, V.: Elements of Asymptotic Geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2007)

    Google Scholar 

  9. 9.

    Bloom, W.R., Xu, Z.: The Hardy–Littlewood maximal function for Chebli–Trimeche hypergroups. Contemp. Math. 183, 45–69 (1995)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chebli, H.: Operateurs de translation generalisee et semi-groupes de convolution. Theorie de potentiel et analyse harmonique. Lecture Notes in Mathematics, pp. 35–59. Springer, Berlin (1974)

    Google Scholar 

  11. 11.

    Chebli, H.: Theoreme de Paley–Wiener associe a un operateur differentiel singulier sur \((0, \infty )\). J. Math. Pures Appl. 9(58), 1–19 (1979)

    MathSciNet  Google Scholar 

  12. 12.

    Copson, E.T., Ruse, H.S.: Harmonic Riemannian spaces. Proc. R. Soc. Edinb. 60, 117–133 (1940)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Druţu, C., Kapovich, M.: Geometric Group Theory. American Mathematical Society Colloquium Publications, vol. 63. American Mathematical Society, Providence, RI (2018)

    Google Scholar 

  14. 14.

    Damek, E., Ricci, F.: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc. N.S. 27(1), 139–142 (1992)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math. 46, 45–109 (1973)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Foulon, P., Labourie, F.: Sur les variétés compactes asymptotiquement harmoniques. Invent. Math. 109(1), 97–111 (1992)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Heber, J.: On harmonic and asymptotically harmonic homogeneous spaces. Geom. Funct. Anal. 16(4), 869–890 (2006)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Helgason, S.: Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs, vol. 39. American Mathematical Society, Providence RI (1994)

    Google Scholar 

  19. 19.

    Kaplan, A.: Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147–153 (1980)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Knieper, G.: Hyperbolic dynamics and Riemannian geometry. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, vol. 1A, pp. 453–545. Elsevier, Amsterdam (2002)

    Google Scholar 

  21. 21.

    Knieper, G.: New results on noncompact harmonic manifolds. Comment. Math. Helv. 87, 669–703 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Knieper, G.: A survey on noncompact harmonic and asymptotically harmonic manifolds. In: Geometry, Topology, and Dynamics in Negative Curvature. London Mathematical Society Lecture Note Series, vol. 425, pp. 146–197. Cambridge University Press, Cambridge (2016)

  23. 23.

    Koornwinder, T.H.: Jacobi functions and analysis on noncompact semisimple Lie groups. In: Askey, R.A., et al. (eds.) Special Functions: Group Theoretical Aspects and Applications, pp. 1–85. Reidel, Kufstein (1984)

    Google Scholar 

  24. 24.

    Knieper, G., Peyerimhoff, N.: Noncompact harmonic manifolds. Oberwolfach Preprints (2013).,

  25. 25.

    Knieper, G., Peyerimhoff, N.: Harmonic functions on rank one asymptotically harmonic manifolds. J. Geom. Anal. 26(2), 750–781 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Lichnerowicz, A.: Sur les espaces Riemanniens completement harmoniques. Bull. Soc. Math. Fr. 72, 146–168 (1944)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Nikolayevsky, Y.: Two theorems on harmonic manifolds. Comment. Math. Helv. 80, 29–50 (2005)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Peyerimhoff, N., Samiou, E.: Integral geometric properties of non-compact harmonic spaces. J. Geom. Anal. 25(1), 122–148 (2015)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ranjan, A., Shah, H.: Harmonic manifolds with minimal horospheres. J. Geom. Anal. 12(4), 683–694 (2002)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Ranjan, A., Shah, H.: Busemann functions in a harmonic manifold. Geom. Dedicata 101, 167–183 (2003)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ruse, H., Walker, A., Willmore, T.: Harmonic Spaces. Edizioni Cremonese, Roma (1961)

    Google Scholar 

  32. 32.

    Szabo, Z.: The Lichnerowicz conjecture on harmonic manifolds. J. Differ. Geom. 31, 1–28 (1990)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Trimeche, K.: Transformation integrale de Weyl et theoreme de Paley–Wiener associe a un operateur differentiel singulier sur \((0, \infty )\). J. Math. Pures Appl. 60, 51–98 (1981)

    MathSciNet  Google Scholar 

  34. 34.

    Trimeche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach, Amsterdam (1997)

    Google Scholar 

  35. 35.

    Trimeche, K.: Inversion of the Lions transmutation operators using generalized wavelets. Appl. Comput. Harmon. Anal. 4(1), 97–112 (1997)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Walker, A.C.: On Lichnerowicz’s conjecture for harmonic 4-spaces. J. Lond. Math. Soc. 24, 317–329 (1948)

    MathSciNet  Google Scholar 

  37. 37.

    Willmore, T.J.: Riemannian Geometry. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1993)

    Google Scholar 

  38. 38.

    Xu, Z.: Harmonic analysis on Chebli–Trimeche hypergroups. Ph.D. thesis, Murdoch University, Australia (1994)

Download references


The first author would like to thank Swagato K. Ray and Rudra P. Sarkar for generously sharing their time and knowledge over the course of numerous educative and enjoyable discussions. The other two authors like to thank the MFO for hospitality during their stay in the “Research in Pairs” program in 2019 and the SFB/TR191 “Symplectic structures in geometry, algebra and dynamics”. This article generalizes an earlier version by the first author in the case of negatively curved harmonic manifolds.

Author information



Corresponding author

Correspondence to Kingshook Biswas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Biswas, K., Knieper, G. & Peyerimhoff, N. The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth. J Geom Anal 31, 126–163 (2021).

Download citation


  • Harmonic manifolds
  • Fourier transform
  • Busemann functions

Mathematics Subject Classification

  • 53C25