Fourier and Radon transform on harmonic 𝑁𝐴 groups

Ray, Swagato; Sarkar, Rudra

doi:10.1090/S0002-9947-09-04800-4

Fourier and Radon transform on harmonic $NA$ groups
HTML articles powered by AMS MathViewer

by Swagato K. Ray and Rudra P. Sarkar PDF

Trans. Amer. Math. Soc. 361 (2009), 4269-4297 Request permission

Abstract:

In this article we study the Fourier and the horocyclic Radon transform on harmonic $NA$ groups (also known as Damek-Ricci spaces). We consider the geometric Fourier transform for functions on $L^p$-spaces and prove an analogue of the $L^2$-restriction theorem. We also prove some mixed norm estimates for the Fourier transform generalizing the Hausdorff-Young and Hardy-Littlewood-Paley inequalities. Unlike Euclidean spaces the domains of the Fourier transforms are various strips in the complex plane. All the theorems are considered on these entire domains of the Fourier transforms. Finally we deal with the existence of the Radon transform on $L^p$-spaces and obtain its continuity property.

References

Jean-Philippe Anker, Ewa Damek, and Chokri Yacoub, Spherical analysis on harmonic $AN$ groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 643–679 (1997). MR 1469569
Francesca Astengo, A class of $L^p$ convolutors on harmonic extensions of $H$-type groups, J. Lie Theory 5 (1995), no. 2, 147–164. MR 1389425
Francesca Astengo, Multipliers for a distinguished Laplacean on solvable extensions of $H$-type groups, Monatsh. Math. 120 (1995), no. 3-4, 179–188. MR 1363136, DOI 10.1007/BF01294856
Francesca Astengo, Roberto Camporesi, and Bianca Di Blasio, The Helgason Fourier transform on a class of nonsymmetric harmonic spaces, Bull. Austral. Math. Soc. 55 (1997), no. 3, 405–424. MR 1456271, DOI 10.1017/S0004972700034079
Francesca Astengo and Bianca Di Blasio, A Paley-Wiener theorem on $NA$ harmonic spaces, Colloq. Math. 80 (1999), no. 2, 211–233. MR 1703838, DOI 10.4064/cm-80-2-211-233
A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J. 28 (1961), 301–324. MR 126155, DOI 10.1215/S0012-7094-61-02828-9
Oscar Blasco and Francisco Villarroya, Transference of bilinear multiplier operators on Lorentz spaces, Illinois J. Math. 47 (2003), no. 4, 1327–1343. MR 2037006
Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), no. 1, 1–41. MR 1102963, DOI 10.1016/0001-8708(91)90060-K
Michael Cowling, Anthony Dooley, Adam Korányi, and Fulvio Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. 8 (1998), no. 2, 199–237. MR 1705176, DOI 10.1007/BF02921641
Ewa Damek, The geometry of a semidirect extension of a Heisenberg type nilpotent group, Colloq. Math. 53 (1987), no. 2, 255–268. MR 924070, DOI 10.4064/cm-53-2-255-268
Ewa Damek and Fulvio Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 139–142. MR 1142682, DOI 10.1090/S0273-0979-1992-00293-8
Ewa Damek and Fulvio Ricci, Harmonic analysis on solvable extensions of $H$-type groups, J. Geom. Anal. 2 (1992), no. 3, 213–248. MR 1164603, DOI 10.1007/BF02921294
Bianca Di Blasio, Positive definite spherical functions on harmonic space $NA$, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 3, 759–767 (English, with Italian summary). MR 1489047
Bianca Di Blasio, An extension of the theory of Gelfand pairs to radial functions on Lie groups, Boll. Un. Mat. Ital. B (7) 11 (1997), no. 3, 623–642 (English, with Italian summary). MR 1479515
Bianca Di Blasio, Paley-Wiener type theorems on harmonic extensions of $H$-type groups, Monatsh. Math. 123 (1997), no. 1, 21–42. MR 1428881, DOI 10.1007/BF01316934
Masaaki Eguchi, Shin Koizumi, and Shohei Tanaka, A Hausdorff-Young inequality for the Fourier transform on Riemannian symmetric spaces, Hiroshima Math. J. 17 (1987), no. 1, 67–77. MR 886982
Masaaki Eguchi and Keisaku Kumahara, An $L^{p}$ Fourier analysis on symmetric spaces, J. Functional Analysis 47 (1982), no. 2, 230–246. MR 664337, DOI 10.1016/0022-1236(82)90106-9
Masaaki Eguchi and Keisaku Kumahara, A Hardy-Littlewood theorem for spherical Fourier transforms on symmetric spaces, J. Funct. Anal. 71 (1987), no. 1, 104–122. MR 879703, DOI 10.1016/0022-1236(87)90018-8
Grafakos, L. Classical and Modern Fourier Analysis. Pearson Education, Inc., New Jersey, 2004.
Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
N. Lohoué and Th. Rychener, Some function spaces on symmetric spaces related to convolution operators, J. Funct. Anal. 55 (1984), no. 2, 200–219. MR 733916, DOI 10.1016/0022-1236(84)90010-7
Parasar Mohanty, Swagato K. Ray, Rudra P. Sarkar, and Alladi Sitaram, The Helgason-Fourier transform for symmetric spaces. II, J. Lie Theory 14 (2004), no. 1, 227–242. MR 2040178
D. M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), no. 5, 641–650. MR 667786, DOI 10.1512/iumj.1982.31.31046
F. Ricci, The spherical transform on harmonic extensions of $H$-type groups, Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), no. 4, 381–392 (1993). Differential geometry (Turin, 1992). MR 1261450
François Rouvière, Espaces de Damek-Ricci, géométrie et analyse, Analyse sur les groupes de Lie et théorie des représentations (Kénitra, 1999) Sémin. Congr., vol. 7, Soc. Math. France, Paris, 2003, pp. 45–100 (French, with English and French summaries). MR 2038648
Boris Rubin, Reconstruction of functions from their integrals over $k$-planes, Israel J. Math. 141 (2004), 93–117. MR 2063027, DOI 10.1007/BF02772213
Cora Sadosky, Interpolation of operators and singular integrals, Monographs and Textbooks in Pure and Applied Mathematics, vol. 53, Marcel Dekker, Inc., New York, 1979. An introduction to harmonic analysis. MR 551747
Donald C. Solmon, A note on $k$-plane integral transforms, J. Math. Anal. Appl. 71 (1979), no. 2, 351–358. MR 548770, DOI 10.1016/0022-247X(79)90196-3
Robert J. Stanton and Peter A. Tomas, A note on the Kunze-Stein phenomenon, J. Functional Analysis 29 (1978), no. 2, 151–159. MR 578902, DOI 10.1016/0022-1236(78)90003-4
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
S. Thangavelu, On Paley-Wiener and Hardy theorems for $NA$ groups, Math. Z. 245 (2003), no. 3, 483–502. MR 2021567, DOI 10.1007/s00209-003-0547-6
Alberto Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1986 original [Dover, New York; MR0869816]. MR 2059284