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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fourier and Radon transform on harmonic $ NA$ groups


Authors: Swagato K. Ray and Rudra P. Sarkar
Journal: Trans. Amer. Math. Soc. 361 (2009), 4269-4297
MSC (2000): Primary 43A85; Secondary 22E30
Published electronically: March 16, 2009
MathSciNet review: 2500889
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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.


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Additional Information

Swagato K. Ray
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India
Email: skray@iitk.ac.in

Rudra P. Sarkar
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Rd., Calcutta 700108, India
Email: rudra@isical.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04800-4
PII: S 0002-9947(09)04800-4
Keywords: Harmonic $NA$ groups, Radon transform
Received by editor(s): September 14, 2007
Published electronically: March 16, 2009
Additional Notes: This work was supported by research grant no. 48/1/2006-R&DII/1488 of National Board for Higher Mathematics, India.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.