Primary summand functions on three-dimensional compact solvmanifolds

Pfeffer, Carolyn

doi:10.1090/S0002-9939-1992-1112499-1

Primary summand functions on three-dimensional compact solvmanifolds
HTML articles powered by AMS MathViewer

by Carolyn Pfeffer

Proc. Amer. Math. Soc. 116 (1992), 213-217

DOI: https://doi.org/10.1090/S0002-9939-1992-1112499-1

PDF | Request permission

Abstract:

Leonard Richardson has shown that for a certain class of three-dimensional compact solvmanifolds, projections onto $\pi$-primary summands of ${L^2}\left ( M \right )$ do not preserve the continuity of functions on $M$. It is shown here that if the $\pi$-primary projection of a continuous function is ${L^\infty }$ then it is actually continuous. From this it follows that there are continuous functions on $M$ whose $\pi$-primary projections are essentially unbounded.

References

Flows on homogeneous spaces

L. Auslander and J. Brezin, Uniform distribution in solvmanifolds, Advances in Math. 7 (1971), 111–144. MR 301137, DOI 10.1016/0001-8708(71)90044-2
Jonathan Brezin, Geometry and the method of Kirillov, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975, pp. 13–25. MR 0387484
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. Basic theory and examples. MR 1070979
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 0233772
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
Leonard F. Richardson, A class of idempotent measures on compact nilmanifolds, Acta Math. 135 (1975), no. 1-2, 129–154. MR 486324, DOI 10.1007/BF02392017
Leonard F. Richardson, $N$-step nilpotent Lie groups with flat Kirillov orbits, Colloq. Math. 52 (1987), no. 2, 285–287. MR 893545, DOI 10.4064/cm-52-2-285-287
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776