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.
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References
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Lacey, M.T. Ergodic averages on circles. J. Anal. Math. 67, 199–206 (1995). https://doi.org/10.1007/BF02787789
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DOI: https://doi.org/10.1007/BF02787789