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Harmonic analysis on solvable extensions of H-type groups

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Abstract

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

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References

  1. Ancona, A. Théorie du potentiel sur les graphes et les varietés. Preprint, Cours de l’École d’Été de Saint Flour, Août (1988).

  2. Boggino, J. Generalized Heisenberg groups and solvmanifolds naturally associated. Rend. Sem. Mat. Univ. Polit. Torino43, 529–547 (1985).

    MathSciNet  MATH  Google Scholar 

  3. Brelot, M. Axiomatique et frontière de Martin. J. Math. Pures Appl.35, 297–335 (1956).

    MathSciNet  MATH  Google Scholar 

  4. Brelot, M. Axiomatique de fonctions harmoniques. Montreal: Les Presses de l’Univ. de Montreal 1969.

    Google Scholar 

  5. Cowling, M. Harmonic analysis on some nilpotent groups (with applications to the representation theory of some semisimple Lie groups). In: Topics in Modern Harmonic Analysis, Vol. I, 81–123. Ist. Naz. Alta Mat. Roma 1983.

  6. Cowling, M., Dooley, A. H., Korányi, A., and Ricci, F. H-type groups and Iwasawa decompositions. Adv. Math.87, 1–41 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  7. Cowling, M., Dooley, A.H., Korányi, A., Ricci, F. In preparation.

  8. Damek, E. Geometry of a semidirect extension of a Heisenberg type nilpotent group. Coll. Math.53, 255–268 (1987).

    MathSciNet  MATH  Google Scholar 

  9. Damek, E. Curvature of a semidirect extension of a Heisenberg type nilpotent group. Coll. Math.,53, 249–253 (1987).

    MathSciNet  MATH  Google Scholar 

  10. Damek, E. A Poisson kernel on Heisenberg type nilpotent groups. Coll. Math.,53, 239–247 (1987).

    MathSciNet  MATH  Google Scholar 

  11. Damek, E. Harmonic functions on a semidirect extension of type H nilpotent groups. Trans. Am. Math. Soc.290, 375–384 (1985).

    Article  MathSciNet  Google Scholar 

  12. Damek, E., and Ricci, F. A class of non-symmetric harmonic Riemannian spaces. Bull. Am. Math. Soc., to appear.

  13. Davies, E. B. Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press 1989.

    MATH  Google Scholar 

  14. Doob, J. Classical Potential Theory and its Probabilistic Counterpart. New York: Springer 1984.

    MATH  Google Scholar 

  15. Éîdelman, S. D. On fundamental solutions of parabolic systems. Mat. Sbornik38(80), 51–92 (1956).

    Google Scholar 

  16. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, G. Higher Transcendental Functions. New York: McGraw-Hill 1953.

    Google Scholar 

  17. Faraut, J. Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques. In: Analyse Harmonique. Nice: C.I.M.P.A. 1983.

    Google Scholar 

  18. Friedman, A. Classes of solutions of linear systems of partial differential equations of parabolic type. Duke Math. J.24, 433–442 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  19. Godement, R. A theory of spherical functions I. Trans. Am. Math. Soc.73, 496–556 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  20. Guivarc’h, Y. Sur la représentation intégrale des fonctions harmoniques et des fonctions propres positives dans un espace Riemannien symétrique. Bull. Sci. Math.108, 373–392 (1984).

    MathSciNet  MATH  Google Scholar 

  21. Helgason, S. Groups and Geometric Analysis. New York: Academic Press 1984.

    MATH  Google Scholar 

  22. Hervé, M., and Hervé, M.-R. Les fonctions surharmoniques dans l’axiomatique de M. Brelot associées à un opérateur elliptique dégénéré. Ann. Inst. Fourier Gren.22, 131–145 (1972).

    MATH  Google Scholar 

  23. Hulanicki, A. Subalgebra of L1 (G) associated with Laplacian on a Lie group. Coll. Math.31, 259–287 (1974).

    MathSciNet  MATH  Google Scholar 

  24. Kadison, R., and Ringrose, J. Fundamentals of the Theory of Operator Algebras. New York: Academic Press 1986.

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Kaplan, A. Riemannian nilmanifolds attached to Clifford modules. Geom. Ded.11, 127–136 (1981).

    MATH  Google Scholar 

  27. Kaplan, A. On the geometry of groups of Heisenberg type. Bull. London Math. Soc.15, 35–42 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  28. Kaplan, A., and Ricci, F. Harmonic analysis on groups of Heisenberg type. In: Harmonic Analysis, Lecture Notes in Mathematics992, pp. 416–435. New York: Springer 1983.

    Google Scholar 

  29. Karpelevic, F. I. The geometry of geodesics and the eigenfunctions of the Laplace-Beltrami operator on symmetric spaces. Trans. Moscow Math. Soc.14, 51–199 (1965).

    MathSciNet  Google Scholar 

  30. Korányi, A. Some applications of Gelfand pairs in classical analysis. In: Harmonic Analysis and Group Representations. Napoli: C.I.M.E. 1982.

    Google Scholar 

  31. Korányi, A. Geometric properties of Heisenberg-type groups. Adv. Math.56, 28–38 (1985).

    Article  MATH  Google Scholar 

  32. Martin, R. S. Minimal positive harmonic functions. Trans. Am. Math. Soc.49, 137–172 (1941).

    Article  MATH  Google Scholar 

  33. Ricci, F. Commutative algebras of invariant functions on groups of Heisenberg type. J. London Math. Soc.32, 265–271 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  34. Szabó, Z. The Lichnerowicz conjecture on harmonic manifolds. J. Diff. Geom.31, 1–28 (1990).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Instytut Matematyczny Universytetu Wroclawskiego, pl. Grunwaldzki 2/4, 50-384, Wroclaw, Poland

    Ewa Damek

  2. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Fulvio Ricci

Authors
  1. Ewa Damek
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  2. Fulvio Ricci
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Additional information

Communicated by Guido Weiss

This work has been partially supported by the Italian Consiglio Nazionale delle Ricerche. We profited from conversation with various colleagues, including P. Biler, J. Faraut, W. Hebisch, A. Korányi, A. Lunardi, A. Hulanicki, M. Pavone, and T. Pytlik.

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Damek, E., Ricci, F. Harmonic analysis on solvable extensions of H-type groups. J Geom Anal 2, 213–248 (1992). https://doi.org/10.1007/BF02921294

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