Radon transforms on Siegel-type nilpotent Lie groups

Abstract

Let \mathfrak{N}:={{\displaystyle H}}_n\times {ℂ}^n be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where {{\displaystyle H}}_n denotes the set of all n\times n Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on \mathfrak{N} and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on \mathfrak{N} is a unitary operator from Sobolev space W^n;2 into L²(\mathfrak{N}):

Keywords

Siegel domain / Siegel-type nilpotent group / Fourier transform / Radon transform

References

[1] Bargmann V. On a Hilbert space of analytic functions and an associated integral transform. Comm Pure Appl Math, 1961, 14: 187–214

[2] Bargmann V. On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces. Application to distribution theory. Comm Pure Appl Math, 1967, 20: 1–101

[3] Carlton E. A Plancherel formula for idyllic nilpotent Lie groups. Trans Amer Math Soc, 1976, 224: 1–42

[4] Damek E, Hulanicki A, Müller D, Peloso M M. Pluriharmonic H2 functions on symmetric irreducible Siegel domains. Geom Funct Anal, 2000, 10: 1090–1117

[5] Dooley A, Zhang G K. Algebras of invariant functions on the Shilov boundaries of Siegel domains. Proc Amer Math Soc, 1998, 126: 3693–3699

[6] Faraut J, Korányi A. Analysis on Symmetric Cones. Oxford: Oxford Univ Press, 1994

[7] Felix R. Radon-transformation auf nilpotenten Lie-gruppen. Invent Math, 1993, 112: 413–443

[8] Geller D, Stein E M. Estimates convolution operators on the Heisenberg group. Bull Amer Math Soc (N S), 1984, 267: 99–103

[9] He J X, Liu H P. Inversion of the Radon transform associated with the classical domain of type one. Internat J Math, 2005, 16: 875–887

[10] He J X, Liu H P. Admissible wavelets and inverse Radon transform associated with the affine homogeneous Siegel domains of type II. Comm Anal Geom, 2007, 15: 1–28

[11] He J X, Liu H P. Wavelet transform and Radon transform on the quaternion Heisenberg group. Acta Math Sin (Engl Ser), 2014, 30: 619–636

[12] Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. New York-London: Academic Press, 1978

[13] Helgason S. Integral Geometry and Radon Transforms. New York: Springer, 2011

[14] Howe R. On the role of the Heisenberg group in harmonic analysis. Bull Amer Math Soc (N S), 1980, 3: 821–843

[15] Hua L K. Harmonic Analysis of Functions of Several Complex Variables on the Classical Domains. Providence: Amer Math Soc, 1963

[16] Korányi A, Wolf A. Realization of Hermitian symmetric spaces as generalized halfplanes. Ann of Math, 1965, 81(2): 265–288

[17] Moore C, Wolf A. Square integrable representations of nilpotent groups. Trans Amer Math Soc, 1973, 185: 445–462

[18] Ogden R, Vági S. Harmonic analysis of a nilpotent group and function theory of Siegel domains of type II. Adv Math, 1979, 33: 31–92

[19] Ørsted B, Zhang G K. Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ Math J, 1994, 43: 551–583

[20] Peetre J. The Weyl transform and Laguerre polynomials. Matematiche (Catania), 1972, 27: 301–323

[21] Peng L Z, Zhang G K. Radon transform on H-type and Siegel-type nilpotent groups. Internat J Math, 2007, 18: 1061–1070

[22] Ricci F, Stein E M. Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds. J Funct Anal, 1989, 86: 360–389

[23] Rossi H, Vergne M. Group representations on Hilbert spaces defined in terms of ∂‾b-cohomology on the Shilov boundary of a Siegel domain. Pacific J Math, 1976, 65: 193–207

[24] Rubin B. The Radon transform on the Heisenberg group and the transversal Radon transform. J Funct Anal, 2012, 262: 234–272

[25] Satake I. Algebraic Structures of Symmetric Domains. Princeton: Princeton Univ Press, 1980

[26] Shabat B. Introduction to Complex Analysis, Part II: Functions of Several Variables. Providence: Amer Math Soc, 1992

[27] Strichartz R. L^pharmonic analysis and Radon transforms on the Heisenberg group. J Funct Anal, 1991, 96: 350–406

[28] Wolf A. Harmonic Analysis on Commutative Spaces. Providence: Amer Math Soc, 2007

[29] Wolf A. Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions. Math Ann, 2011, 349: 263{299

[30] Zhang G K. Radon transform on symmetric matrix domains. Trans Amer Math Soc, 2009, 361: 1351–1369

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature