A Paley-Wiener theorem on NA harmonic spaces

Francesca Astengo; Bianca di Blasio

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 2, page 211-233
  • ISSN: 0010-1354

Abstract

top
Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.

How to cite

top

Astengo, Francesca, and di Blasio, Bianca. "A Paley-Wiener theorem on NA harmonic spaces." Colloquium Mathematicae 80.2 (1999): 211-233. <http://eudml.org/doc/210713>.

@article{Astengo1999,
abstract = {Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.},
author = {Astengo, Francesca, di Blasio, Bianca},
journal = {Colloquium Mathematicae},
keywords = {nilpotent Lie group; Paley-Wiener type theorem; Helgason-Fourier transform},
language = {eng},
number = {2},
pages = {211-233},
title = {A Paley-Wiener theorem on NA harmonic spaces},
url = {http://eudml.org/doc/210713},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Astengo, Francesca
AU - di Blasio, Bianca
TI - A Paley-Wiener theorem on NA harmonic spaces
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 2
SP - 211
EP - 233
AB - Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.
LA - eng
KW - nilpotent Lie group; Paley-Wiener type theorem; Helgason-Fourier transform
UR - http://eudml.org/doc/210713
ER -

References

top
  1. [ADY] J.-P. Anker, E. Damek and C. Yacoub, Spherical analysis on harmonic AN groups, Ann. Scuola Norm. Sup. Pisa 33 (1996), 643-679. Zbl0881.22008
  2. [ACD] F. Astengo, R. Camporesi and B. Di Blasio, The Helgason Fourier transform on a class of nonsymmetric harmonic spaces, Bull. Austral. Math. Soc. 55 (1997), 405-424. Zbl0894.43003
  3. M. Cowling, A. H. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal., to appear. Zbl0966.53039
  4. [CH] M. G. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507-549. Zbl0681.43012
  5. [Da1] E. Damek, A Poisson kernel on Heisenberg type nilpotent groups, Colloq. Math. 53 (1987), 239-247. Zbl0661.53035
  6. [Da2] E. Damek, Curvature of a semidirect extension of a Heisenberg type nilpotent group, ibid., 249-253. 
  7. [Da3] E. Damek, Geometry of a semidirect extension of a Heisenberg type nilpotent group, ibid., 255-268. 
  8. E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248. Zbl0788.43008
  9. E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139-142. Zbl0755.53032
  10. [Di] B. Di Blasio, Paley-Wiener type theorems on harmonic extensions of H-type groups, Monatsh. Math. 123 (1997), 21-42. Zbl0887.43001
  11. [DoZ] A. Dooley and G. Zhang, Spherical functions on H-type groups, preprint. 
  12. [EbO] P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. 
  13. A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, Higher Transcendental Functions, Vols. I, II, McGraw-Hill, New York, 1953. Zbl0051.30303
  14. [F] J. Faraut, Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un, J. Funct. Anal. 49 (1982), 230-268. Zbl0526.43005
  15. [GV] R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergeb. Math. Grenzgeb. 101, Springer, Berlin and New York, 1988. Zbl0675.43004
  16. [HC] Harish-Chandra, Discrete series for semisimple Lie groups, Acta Math. 116 (1966), 1-111. Zbl0199.20102
  17. [He] S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys Monographs 39, Amer. Math. Soc., Providence, R.I., 1994. 
  18. [H] A. Hulanicki, Subalgebra of L 1 ( G ) associated with laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287. Zbl0316.43005
  19. [HR] A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence for bounded harmonic functions on balls in C n , Invent. Math. 62 (1980), 325-331. Zbl0449.31008
  20. [Ka1] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. 
  21. [Ka2] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127-136. Zbl0495.53046
  22. [Ko1] A. Korányi, Some applications of Gelfand pairs in classical analysis, in: Harmonic Analysis and Group Representations, C.I.M.E., Liguori, Napoli, 1980, 333-348. 
  23. [Ko2] A. Korányi, Geometric properties of Heisenberg type groups, Adv. Math. 56 (1985), 28-38. Zbl0589.53053
  24. [Ri1] F. Ricci, Harmonic analysis on groups of type H, preprint. 
  25. [Ri2] F. Ricci, The spherical transform on harmonic extensions of H-type groups, Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), 381-392. Zbl0829.43021
  26. [RWW] H. S. Ruse, A. G. Walker and T. J. Willmore, Harmonic Spaces, Edizioni Cremonese, Roma, 1961. 
  27. [V] V. S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Lecture Notes in Math. 576, Springer, Berlin, 1977. Zbl0354.43001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.