Characterization of eigenfunctions of the Laplacian by boundedness conditions

Strichartz, Robert S.

doi:10.1090/S0002-9947-1993-1108614-1

Characterization of eigenfunctions of the Laplacian by boundedness conditions
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by Robert S. Strichartz PDF

Trans. Amer. Math. Soc. 338 (1993), 971-979 Request permission

Abstract:

If ${\{ {f_k}(x)\} _{k \in \mathbb {Z}}}$ is a doubly infinite sequence of functions on ${\mathbb {R}^n}$ which are uniformly bounded and such that $\Delta {f_k} = {f_{k + 1}}$, then $\Delta {f_0} = - {f_0}$. This generalizes a theorem of Roe $(n = 1)$. The analogous statement is true on the Heisenberg group, but false in hyperbolic space.

References

Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
Jean-Pierre Gabardo, Tempered distributions with spectral gaps, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 143–162. MR 994086, DOI 10.1017/S0305004100068043
Jean-Pierre Gabardo, Tempered distributions supported on a half-space of $\textbf {R}^n$ and their Fourier transforms, Canad. J. Math. 43 (1991), no. 1, 61–88. MR 1108914, DOI 10.4153/CJM-1991-005-6
Daryl Geller, Fourier analysis on the Heisenberg group. I. Schwartz space, J. Functional Analysis 36 (1980), no. 2, 205–254. MR 569254, DOI 10.1016/0022-1236(80)90100-7
Daryl Geller, Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), no. 15, 1665–1677. MR 729197, DOI 10.1080/03605308308820319
I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Ralph Howard, A note on Roe’s characterization of the sine function, Proc. Amer. Math. Soc. 105 (1989), no. 3, 658–663. MR 942633, DOI 10.1090/S0002-9939-1989-0942633-5
Ralph Howard and Margaret Reese, Characterization of eigenfunctions by boundedness conditions, Canad. Math. Bull. 35 (1992), no. 2, 204–213. MR 1165169, DOI 10.4153/CMB-1992-029-x
J. Roe, A characterization of the sine function, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 69–73. MR 549299, DOI 10.1017/S030500410005653X
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
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
Robert S. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87 (1989), no. 1, 51–148. MR 1025883, DOI 10.1016/0022-1236(89)90004-9
Robert S. Strichartz, $L^p$ harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), no. 2, 350–406. MR 1101262, DOI 10.1016/0022-1236(91)90066-E