no. 4

Spherical analysis on harmonic AN groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 23 (1996) no. 4, pp. 643-679. 
@article{ASNSP_1996_4_23_4_643_0,
     author = {Anker, Jean-Philippe and Damek, Ewa and Yacoub, Chokri},
     title = {Spherical analysis on harmonic $A N$ groups},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {643--679},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 23},
     number = {4},
     year = {1996},
     zbl = {0881.22008},
     mrnumber = {1469569},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1996_4_23_4_643_0/}
}
Anker, Jean-Philippe; Damek, Ewa; Yacoub, Chokri. Spherical analysis on harmonic $A N$ groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 23 (1996) no. 4, pp. 643-679. http://www.numdam.org/item/ASNSP_1996_4_23_4_643_0/

[An1] J.-Ph. Anker, Le noyau de la chaleur sur les espaces symétriques U(p, q) / U( p) x U(q) , In: "Harmonic Analysis, Luxembourg 1987", P. Eymard - J.-P. Pier (eds.), Lecture Notes in Mathematics 1359, Springer, 1988, pp. 60-82 | MR 974304 | Zbl 0669.43009

[An2] J.-Ph. Anker, The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi and Varadarajan, J. Funct. Anal. 96 (1991), 331-349. | MR 1101261 | Zbl 0732.43006

[An3] J.-Ph. Anker, L p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. 132 (1990), 597-628 | MR 1078270 | Zbl 0741.43009

[An4] J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257-297. | MR 1150587 | Zbl 0764.43005

[An5] J.-Ph. Anker, Analyse sphérique sur les espaces hyperboliques, Cours D.E.A. 1992-1993, Université Nancy I.

[AJ] J.-Ph. Anker, - L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, in preparation. | Zbl 0942.43005

[AMPS] J.-Ph. Anker - P. Martinot - E. Pedon - A. Setti, The wave equation on harmonic AN groups, in preparation.

[AS] J.-Ph. Anker - A. Seeger, Lp estimates for functions of the Laplacian on noncompact manifolds, in preparation.

[As] F. Astengo, Asymptotic expansions for spherical functions and multipliers on solvable extensions of H-type groups, J. Lie Theory 5 (1995), 147-164. | EuDML 231976 | Zbl 0849.43001

[ACD] F. Astengo - R. Camporesi - B. Di Blasio, The Helgason Fourier transform on a class of nonsymmetric harmonic spaces, preprint (1996). | MR 1456271

[Bo] J. Boggino, Generalized Heisenberg groups and solvmanifolds naturally associated, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), 529-547. | MR 884876 | Zbl 0625.53050

[BTV] J. Berndt - F. Tricerri - L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Mathematics 1598, Springer, 1995. | MR 1340192 | Zbl 0818.53067

[CS] J.-L. Clerc - E.M. Stein, Lp-multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. USA 71 (1974), 3911-3912. | MR 367561 | Zbl 0296.43004

[Co1] M.G. Cowling, The Kunze-Stein phenomenon, Ann. of Math. 107 (1978), 209-234. | MR 507240 | Zbl 0363.22007

[Co2] M.G. Cowling, Unitary and uniformly bounded representations of some simple Lie groups, In: "Harmonic analysis and group representations, C.I.M.E. 1980", Liguori, Napoli, 1982, pp. 49-128. | MR 777340

[Co3] M.G. Cowling, Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisimple Lie groups), In: "Topics in modem harmonic analysis (Proc. Sem. Torino Milano, May-June 1982)" vol. I, Ist. Naz. Alta Mat. F. Severi, Roma, 1983, pp. 81-123. | MR 748862 | Zbl 0541.22005

[CDKR1] M.G. Cowling - A.H. Dooley - A. Korányi - F. Ricci, H-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1-41. | MR 1102963 | Zbl 0761.22010

[CDKR2] M.G. Cowling - A.H. Dooley - A. Korányi - F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. (to appear). | MR 1705176 | Zbl 0966.53039

[CF] M.G. Cowling - J.J.F. Fournier, Inclusions and non-inclusions of spaces of convolution operators, Trans. Amer. Math. Soc. 221 (1976), 59-95. | MR 493164 | Zbl 0331.43007

[CGGM] M.G. Cowling - G.I. Gaudry - S. Giulini - G. Mauceri, Weak type (1,1) estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc. 323 (1991), 637-649. | MR 967310 | Zbl 0722.22006

[CGM] M.G. Cowling - S. Giulini - S. Meda, Lp-Lq estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces I, Duke Math. J. 72 (1993), 109-150. | MR 1242882 | Zbl 0807.43002

[CH] M.G. Cowling - U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. math. 96 (1989), 507-549. | MR 996553 | Zbl 0681.43012

[Dam1] E. Damek, Geometry ofa semi-direct extension ofa Heisenberg type nilpotent group, Collect. Math. 53 (1987), 255-268. | MR 924070 | Zbl 0661.53033

[Dam2] E. Damek, Curvature of a semi-direct extension of a Heisenberg type nilpotent group, Collect. Math. 53 (1987), 249-253. | MR 924069 | Zbl 0661.53034

[Dam3] E. Damek, A Poisson kernel on Heisenberg type nilpotent groups, Collect. Math. 53 (1987), 239-247. | MR 924068 | Zbl 0661.53035

[Dam4] E. Damek, Harmonic functions on a semidirect extension of type H nilpotent groups, Trans. Amer. Math. Soc. 290 (1985), 375-384. | MR 787971 | Zbl 0593.43010

[DR1] E. Damek - F. Ricci, A class ofnonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139-142. | MR 1142682 | Zbl 0755.53032

[DR2] E. Damek - F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248. | MR 1164603 | Zbl 0788.43008

[Dav] E.B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, 1989. | MR 990239 | Zbl 0699.35006

[DM] E.B. Davies - N. Mandouvalos, Heat kernels bounds on hyperbolic space and Kleinian groups, Proc. London Math. Soc. 57 (1988), 182-208. | MR 940434 | Zbl 0643.30035

[Di1] B. Di Blasio, Analisi di Fourier sferica su estensioni risolubili di gruppi di Heisenberg generalizzati, Tesi di Dottorato, Politecnico di Torino (1996).

[Di2] B. Di Blasio, Positive definite spherical functions on harmonic spaces NA, Boll. Unione Mat. Ital. (to appear). | MR 1489047 | Zbl 0898.43005

[Di3] B. Di Blasio, Paley-Wiener type theorems on harmonic extensions of H-type groups, Monatsh. Math. (to appear). | MR 1428881 | Zbl 0887.43001

[Di4] B. Di Blasio, An extension of the theory of Gelfand pairs to radial functions on Lie groups, Boll. Unione Mat. Ital. (to appear). | MR 1479515 | Zbl 0885.22016

[DZ] A.H. Dooley - G. Zhang, Spherical functions on H-type groups, preprint (1995).

[Fa] J. Faraut, Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, In: "Analyse Harmonique", J.-L. Clerc et al. (eds.), C.I.M.P.A., Nice 1983, pp. 315-446.

[FK] M. Flensted-Jensen - T.H. Koornwinder, Positive definite spherical functions on a non-compact, rank one symmetric space, In: "Analyse Harmonique sur les Groupes de Lie II, Séminaire Nancy-Strabourg 1976-78", P. Eymard et al. (eds.), Lecture Notes in Mathematics 739, Springer, 1979, pp. 249-282. | MR 560841 | Zbl 0433.43014

[GM] S. Giulini - G. Mauceri, Analysis of a distinguished Laplacian on solvable Lie groups, Math. Nachr. 163 (1993), 151-162. | MR 1235064 | Zbl 0801.43002

[Hel] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984. | MR 754767 | Zbl 0543.58001

[Her] C. Herz, Problems of extrapolation and spectral synthesis on groups, In: "Conference on harmonic analysis, College Park 1971", D. Gulick - R. L. Lipsman (eds.), Lecture Notes in Mathematics 266, Springer, 1972, pp. 157-166. | MR 394054 | Zbl 0246.43004

[Ka1] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by compositions of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. | MR 554324 | Zbl 0393.35015

[Ka2] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Ded. 11 (1981), 127-136. | MR 621376 | Zbl 0495.53046

[Ka3] A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35-42. | MR 686346 | Zbl 0521.53048

[KR] A. Kaplan - F. Ricci, Harmonic analysis on groups of Heisenberg type, Lecture Notes in Mathematics 992, Springer, 1983, pp. 416-435. | MR 729367 | Zbl 0521.43008

[Koo] T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, In: "Special functions: Group theoretical aspects and applications", R. A. Askey et al. (eds.), Reidel, 1984, pp. 1-85. | MR 774055 | Zbl 0584.43010

[Kor] A. Korányi, Geometric properties of Heisenberg-type groups, Adv. Math. 56 (1985), 28-38. | MR 782541 | Zbl 0589.53053

[Li] R.L. Lipsman, An indicator diagram II. The Lp convolution theorem for connected unimodular groups, Duke. Math. J. 37 (1970), 459-466. | MR 267045 | Zbl 0201.17601

[Lo] N. Lohoué, Inégalités de Sobolev pour les sous Laplaciens de certains groupes unimodulaires, Geom. Funct. Anal. 2 (1992), 394-420. | MR 1191567 | Zbl 0807.22007

[LRy] N. Lohoué - Th. Rychener, Die Resolvente von Δ auf symmetrischen Räumen vom nichtkompakten Typ, Comm. Math. Helv. 57 (1982), 445-468. | Zbl 0505.53022

[LZ] N. Lohoué - Zhu Fu Liu, Estimation faible de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I 302 (1986), 303-305. | MR 838580 | Zbl 0583.42006

[LRo] G. Lorang - B. Roynette, Etude d'unefonctionnelle liée au pont de Bessel, Ann. Inst. Henri Poincaré Section B (Prob. Stat.) 32 (1996), 107-133. | Numdam | MR 1373728 | Zbl 0842.60076

[MW] R.J. Miatello - N.R. Wallach, The resolvent ofthe Laplacian on locally symmetric spaces, J. Diff. Geom. 36 (1992), 663-698. | MR 1189500 | Zbl 0766.53044

[Mu] A. Munch, Comportement exact du noyau de la chaleur sur les espaces hyperboliques, Mémoire D.E.A., Université Nancy I 1993.

[Ric1] F. Ricci, Commutative algebras of invariant functions on groups of Heisenberg type, J. London Math. Soc. 32 (1985), 265-271. | MR 811785 | Zbl 0592.22009

[Ric2] F. Ricci, The spherical transform on harmonic extensions of H-type groups, Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), 381-392. | MR 1261450 | Zbl 0829.43021

[Rie] C. Riehm, The automorphism group of a composition of quadratic forms, Trans. Amer. Math. Soc. 269 (1982), 403-414. | MR 637698 | Zbl 0483.10021

[ST] R.J. Stanton - P.A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251-276. | MR 511124 | Zbl 0411.43014

[Ste] E.M. Stein, Analytic continuation of group representations, Adv. Math. 4 (1970), 172-207. | MR 263985 | Zbl 0194.16201

[Str] J.-O. Strömberg, Weak type L1 estimates for maximal functions on non-compact symmetric spaces, Ann. of Math. 114 (1981), 115-126. | MR 625348 | Zbl 0472.43010

[Sz1] Z.I. Szabó, The Lichnerowicz conjecture on harmonic manifolds, J. Differential Geom. 31 (1990), 1-28. | MR 1030663 | Zbl 0686.53042

[Sz2] Z.I. Szabó, Spectral theory for operator families on Riemannian manifolds, In: "Differential geometry, UCLA 1990", R. Greene - S. T. Yau (eds.) Proc. Symp. Pure Math. 54, 3, 1993, pp. 615-665. | MR 1216651 | Zbl 0791.58100

[Ta] M.E. Taylor, Lp estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793. | MR 1016445 | Zbl 0691.58043

[Vr] L. Vretare, On Lp Fourier multipliers on certain symmetric spaces, Math. Scand. 37 (1975),111-121. | MR 415211 | Zbl 0315.43005

[Ya] C. Yacoub, Analyse radiale sur des variétés harmoniques AN et d'opérateurs différentiels généralisant l'opérateur de Jacobi, Thèse, Université Paris-Sud / Orsay, 1994.