Radon transforms on Siegel-type nilpotent Lie groups

doi:10.1007/s11464-019-0787-3

Front. Math. China

2019, Vol. 14

Issue (5) : 855-866 https://doi.org/10.1007/s11464-019-0787-3

RESEARCH ARTICLE

Radon transforms on Siegel-type nilpotent Lie groups

Xingya FAN^1,², Jianxun HE²(

), Jinsen XIAO³, Wenjun YUAN²

¹. School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
². School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
³. School of Sciences, Guangdong University of Petrochemical Technology, Maoming 525000, China

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Abstract

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

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

Corresponding Author(s): Jianxun HE

Issue Date: 22 November 2019

Cite this article:

Xingya FAN,Jianxun HE,Jinsen XIAO, et al. Radon transforms on Siegel-type nilpotent Lie groups[J]. Front. Math. China, 2019, 14(5): 855-866.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0787-3
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I5/855

1	V Bargmann. On a Hilbert space of analytic functions and an associated integral transform. Comm Pure Appl Math, 1961, 14: 187–214 https://doi.org/10.1002/cpa.3160140303
2	V Bargmann. 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 https://doi.org/10.1002/cpa.3160200102
3	E Carlton. A Plancherel formula for idyllic nilpotent Lie groups. Trans Amer Math Soc, 1976, 224: 1–42 https://doi.org/10.2307/1997413
4	E Damek, A Hulanicki, D Müller, M M Peloso. Pluriharmonic H2 functions on symmetric irreducible Siegel domains. Geom Funct Anal, 2000, 10: 1090–1117 https://doi.org/10.1007/PL00001648
5	A Dooley, G K Zhang. Algebras of invariant functions on the Shilov boundaries of Siegel domains. Proc Amer Math Soc, 1998, 126: 3693–3699 https://doi.org/10.1090/S0002-9939-98-05051-5
6	J Faraut, A Korányi. Analysis on Symmetric Cones. Oxford: Oxford Univ Press, 1994
7	R Felix. Radon-transformation auf nilpotenten Lie-gruppen. Invent Math, 1993, 112: 413–443 https://doi.org/10.1007/BF01232441
8	D Geller, E M Stein. Estimates convolution operators on the Heisenberg group. Bull Amer Math Soc (N S), 1984, 267: 99–103 https://doi.org/10.1007/BF01458467
9	J X He, H P Liu. Inversion of the Radon transform associated with the classical domain of type one. Internat J Math, 2005, 16: 875–887 https://doi.org/10.1142/S0129167X05003120
10	J X He, H P Liu. Admissible wavelets and inverse Radon transform associated with the affine homogeneous Siegel domains of type II. Comm Anal Geom, 2007, 15: 1–28 https://doi.org/10.4310/CAG.2007.v15.n1.a1
11	J X He, H P Liu. Wavelet transform and Radon transform on the quaternion Heisenberg group. Acta Math Sin (Engl Ser), 2014, 30: 619–636 https://doi.org/10.1007/s10114-014-2361-y
12	S Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. New York-London: Academic Press, 1978
13	S Helgason. Integral Geometry and Radon Transforms. New York: Springer, 2011 https://doi.org/10.1007/978-1-4419-6055-9
14	R Howe. On the role of the Heisenberg group in harmonic analysis. Bull Amer Math Soc (N S), 1980, 3: 821–843 https://doi.org/10.1090/S0273-0979-1980-14825-9
15	L K Hua. Harmonic Analysis of Functions of Several Complex Variables on the Classical Domains. Providence: Amer Math Soc, 1963 https://doi.org/10.1090/mmono/006
16	A Korányi, A Wolf. Realization of Hermitian symmetric spaces as generalized halfplanes. Ann of Math, 1965, 81(2): 265–288 https://doi.org/10.2307/1970616
17	C Moore, A Wolf. Square integrable representations of nilpotent groups. Trans Amer Math Soc, 1973, 185: 445–462 https://doi.org/10.2307/1996450
18	R Ogden, S Vági. Harmonic analysis of a nilpotent group and function theory of Siegel domains of type II. Adv Math, 1979, 33: 31–92 https://doi.org/10.1016/S0001-8708(79)80009-2
19	B Ørsted, G K Zhang. Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ Math J, 1994, 43: 551–583 https://doi.org/10.1512/iumj.1994.43.43023
20	J Peetre. The Weyl transform and Laguerre polynomials. Matematiche (Catania), 1972, 27: 301–323
21	L Z Peng, G K Zhang. Radon transform on H-type and Siegel-type nilpotent groups. Internat J Math, 2007, 18: 1061–1070 https://doi.org/10.1142/S0129167X07004412
22	F Ricci, E M Stein. Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds. J Funct Anal, 1989, 86: 360–389 https://doi.org/10.1016/0022-1236(89)90057-8
23	H Rossi, M Vergne. 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 https://doi.org/10.2140/pjm.1976.65.193
24	B Rubin. The Radon transform on the Heisenberg group and the transversal Radon transform. J Funct Anal, 2012, 262: 234–272 https://doi.org/10.1016/j.jfa.2011.09.011
25	I Satake. Algebraic Structures of Symmetric Domains. Princeton: Princeton Univ Press, 1980 https://doi.org/10.1515/9781400856800
26	B Shabat. Introduction to Complex Analysis, Part II: Functions of Several Variables. Providence: Amer Math Soc, 1992 https://doi.org/10.1090/mmono/110
27	R Strichartz. Lpharmonic analysis and Radon transforms on the Heisenberg group. J Funct Anal, 1991, 96: 350–406 https://doi.org/10.1016/0022-1236(91)90066-E
28	A Wolf. Harmonic Analysis on Commutative Spaces. Providence: Amer Math Soc, 2007 https://doi.org/10.1090/surv/142
29	A Wolf. Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions. Math Ann, 2011, 349: 263{299 https://doi.org/10.1007/s00208-010-0525-3
30	G K Zhang. Radon transform on symmetric matrix domains. Trans Amer Math Soc, 2009, 361: 1351–1369 https://doi.org/10.1090/S0002-9947-08-04658-8

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