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

 

Characterization of almost $ L^p$-eigenfunctions of the Laplace-Beltrami operator


Authors: Pratyoosh Kumar, Swagato K. Ray and Rudra P. Sarkar
Journal: Trans. Amer. Math. Soc. 366 (2014), 3191-3225
MSC (2010): Primary 43A85; Secondary 22E30
Published electronically: September 26, 2013
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Abstract: In 1980, Roe proved that if a doubly-infinite sequence $ \{f_k\}$ of functions on $ \mathbb{R}$ satisfies $ f_{k+1}=(df_{k}/dx)$ and $ \vert f_{k}(x)\vert\leq M$ for all $ k=0,\pm 1,\pm 2,\cdots $ and $ x\in \mathbb{R}$, then $ f_0(x)=a\sin (x+\varphi )$, where $ a$ and $ \varphi $ are real constants. This result was extended to $ \mathbb{R}^n$ by Strichartz in 1993, where $ d/dx$ was substituted by the Laplacian on $ \mathbb{R}^n$. While it is plausible that this theorem extends to other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic $ 3$-space. This negative result can indeed be extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the $ p$-dependence of the $ L^p$-spectrum of the Laplacian on the hyperbolic spaces. Taking this into account we shall prove that for all rank one Riemannian symmetric spaces of noncompact type, and more generally for the harmonic $ NA$ groups, the theorem actually holds true when uniform boundedness is replaced by uniform ``almost $ L^p$ boundedness''. In addition we shall see that for the symmetric spaces this theorem can be used to characterize the Poisson transforms of $ L^p$ functions on the boundary, which somewhat resembles the original theorem of Roe on $ \mathbb{R}$.


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Additional Information

Pratyoosh Kumar
Affiliation: Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India
Email: pratyoosh@iitg.ernet.in

Swagato K. Ray
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Rd., Calcutta 700108, India
Email: swagato@isical.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/S0002-9947-2013-06004-7
PII: S 0002-9947(2013)06004-7
Keywords: Spectrum of Laplacian, eigenfunction of Laplacian, symmetric space, Damek-Ricci space
Received by editor(s): June 29, 2011
Received by editor(s) in revised form: November 3, 2012
Published electronically: September 26, 2013
Additional Notes: The second and third authors were partially supported by a research grant (No. 2/48(6)/2010-R&D II/10807) of the National Board for Higher Mathematics, India.
Dedicated: Dedicated to the memory of our teacher Somesh C. Bagchi.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.