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Absolutely continuous measures on non quasi-analytic curves with independent powers

Author: Mats Anders Olofsson
Journal: Proc. Amer. Math. Soc. 129 (2001), 515-524
MSC (2000): Primary 43A10; Secondary 26E10
Published electronically: August 28, 2000
MathSciNet review: 1797134
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Abstract | References | Similar Articles | Additional Information


We prove that every non quasi-analytic Carleman class contains functions whose graph supports measures that are absolutely continuous with respect to arc length measure and yet they have independent convolution powers in the measure algebra $M(\mathbb{R}^2)$. The proof relies on conditions which ensure that the canonical map between two Cantor sets can be extended to a function in an arbitrary prescribed non quasi-analytic Carleman class.

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

  • 1. J-E. Björk, Wiener Subalgebras of $M(\mathbb{R}^n)$ Generated by Smooth Measures Carried by Smooth Submanifolds of $\mathbb{R}^n$, preprint, Stockholm University, 1979.
  • 2. Jan Boman, Equivalence of generalized moduli of continuity, Ark. Mat. 18 (1980), no. 1, 73–100. MR 608328,
  • 3. I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Translated from the Russian, with a supplementary chapter, Chelsea Publishing Co., New York, 1964. MR 0205105
  • 4. L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990.
  • 5. Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., No. 1301, Hermann, Paris, 1963 (French). MR 0160065
  • 6. Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1998. Corrected reprint of the 1988 original. MR 1670244
  • 7. A. Olofsson, Nilpotent Measures and Wiener Subalgebras of $M(\mathbb{R}^n)$, paper IV in Topics in Real and Complex Analysis, doctoral thesis, Stockholm University, 2000.
  • 8. Walter Rudin, Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1038803
  • 9. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • 10. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
  • 11. N. Th. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465–516. MR 0163985

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Additional Information

Mats Anders Olofsson
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

Keywords: Measure algebras, Wiener-Pitt phenomenon, independent powers
Received by editor(s): April 29, 1999
Published electronically: August 28, 2000
Additional Notes: The author was supported by the G. S. Magnusson Fund of the Royal Swedish Academy of Sciences
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society