Abstract
LetT 1 andT 2 be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ2, letT t=T u1 T v2 . LetS 1 denote the unit circle in ℝ2 and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA t f(x) = ∫ s 1 f(T ts x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
[B] J. Bourgain,Averages in the plane over convex curves and maximal operators, J. Analyse Math.47 (1986), 69–85.
[J] R. L. Jones,Ergodic averages on spheres, J. Analyse Math.61 (1993), 29–45.
[M] G. Mockenhaupt, A. Seeger and C. D. Sogge,Wave front sets, local smoothing and Bourgain's circular maximal theorem, Ann. Math.136 (1992), 207–218.
[S] E. M. Stein,Maximal functions: Spherical means, Proc. Natl. Acad. Sci. U.S.A.73 (1976), 2174–2175.
[SW] E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lacey, M.T. Ergodic averages on circles. J. Anal. Math. 67, 199–206 (1995). https://doi.org/10.1007/BF02787789
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02787789