## Revista Matemática Iberoamericana

### The Singularity of Orbital Measures on Compact Lie Groups

#### Abstract

We find the minimal real number $k$ such that the $k$th power of the Fourier transform of any continuous, orbital measure on a classical, compact Lie group belongs to $l^{2}$. This results from an investigation of the pointwise behaviour of characters on these groups. An application is given to the study of $L^{p}$-improving measures.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 517-530.

Dates
First available in Project Euclid: 17 June 2004

https://projecteuclid.org/euclid.rmi/1087482025

Mathematical Reviews number (MathSciNet)
MR2073130

Zentralblatt MATH identifier
1052.43011

#### Citation

Hare, Kathryn E.; Yee, Wai Ling. The Singularity of Orbital Measures on Compact Lie Groups. Rev. Mat. Iberoamericana 20 (2004), no. 2, 517--530. https://projecteuclid.org/euclid.rmi/1087482025

#### References

• Bröcker, T. and Dieck, T.: Representations of compact Lie groups. Graduate Texts in Mathematics 98. Springer-Verlag, New York, 1985.
• Hare, K. E.: The size of characters of compact Lie groups. Studia Math. 129 (1998), 1-18.
• Hare, K. E., Wilson, D. and Yee, W. L.: Pointwise estimates of the size of characters of compact Lie groups. J. Austral. Math. Soc. Ser. A. 69 (2000), 61-84.
• Humphreys, J. E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics 9. Springer-Verlag, New York-Berlin, 1972.
• Ragozin, D. L.: Central measures on compact simple Lie groups. J. Functional Analysis 10 (1972), 212-229.
• Ricci, F. and Stein, E. M.: Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds. J. Funct. Anal. 86 (1989), 360-389.
• Ricci, F. and Travaglini, G.: $L^p-L^q$ estimates for orbital measures and Radon transform on compact Lie groups and Lie algebras. J. Funct. Anal. 129 (1995), 132-147.
• Stein, E. M. and Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton, N.J., 1971.
• Varadarajan, V. S.: Lie groups, Lie algebras, and their representations. Reprint of the 1974 edition. Graduate Texts in Mathematics 102. Springer-Verlag, New York, 1984.