Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Singular measures with absolutely
continuous convolution squares
on locally compact groups


Author: Antonis Bisbas
Journal: Proc. Amer. Math. Soc. 127 (1999), 2865-2869
MSC (1991): Primary 43A05
DOI: https://doi.org/10.1090/S0002-9939-99-04827-3
Published electronically: April 23, 1999
MathSciNet review: 1600100
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Saeki's result states that on any locally compact nondiscrete group there exist continuous singular measures, with respect to the left Haar measure, $\mu $ with $\mu * \mu $ in $L^{p}$ for all $p, \;1\leq p < \infty $. This paper gives a new and short proof of this using Rademacher-Riesz products.


References [Enhancements On Off] (What's this?)

  • [1] A. Bisbas and C. Karanikas, On the continuity of measures, Applicable Analysis, Vol 48 (1993), 23-35. MR 95e:43001
  • [2] A. H. Dooley and S. K. Gupta, Continuous singular measures with absolutely continuous convolution squares, Proceedings of the American Math. Soc. Vol. 124, 10, (1996), 3115 - 3122. MR 96m:43011
  • [3] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer-Verlag, New York 1979. MR 81d:43001
  • [4] E. Hewitt and H. Zuckerman, Singular measures with absolutely continuous convolution squares, Proc. Cambridge Phil. Soc. 62 (1966), 339-420; 63, (1967), 367-368. MR 33:1655
  • [5] C. Karanikas, Examples of Riesz products-type measures on metrizable groups, Boll. Un. Math. Ital. 7, 4-A (1990), 331-341. MR 92d:43002
  • [6] C. Karanikas and S. Koumandos, Continuous singular measures with absolutely continuous convolution squares on locally compact groups, Illinois J. of Math. 35 (3) (1991), 490-495. MR 92b:43004
  • [7] D. L. Ragozin, Central measures on compact simple Lie groups, J. of Functional Analysis 10 (1972), 212-229. MR 49:5715
  • [8] S. Saeki, Singular measures having absolutely continuous convolution powers, Illinois J. of Math. 21 (1977), 395-412. MR 58:6719
  • [9] N. Wiener and A. Wintner, Fourier-Stieltjes transforms and singular infinite convolutions, Amer. J. Math., vol. 60 (1938), 513-522.
  • [10] J. H. Williamson, A theorem on algebras of measures on topological groups, Proc. Edinburgh Math. Soc., vol. 11 (1958/59), 195 - 206. MR 22:2851

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A05

Retrieve articles in all journals with MSC (1991): 43A05


Additional Information

Antonis Bisbas
Affiliation: Department of Mathematics, University of the Aegean, Karlovasi 83200, Samos, Greece
Address at time of publication: Technological Education Institute of Kozani, School of Technological Applications, General Department, Kila 50100, Kozani, Greece
Email: bisbas@kozani.teikoz.gr

DOI: https://doi.org/10.1090/S0002-9939-99-04827-3
Keywords: Singular measures, Rademacher functions
Received by editor(s): June 25, 1997
Received by editor(s) in revised form: December 3, 1997
Published electronically: April 23, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society