## Abstract

Let T be a power-bounded operator on L_{p}(μ), 1 < p < ∞. We use a sublinear growth condition on the norms {∥ _{k=1}^{n} T^{k} f∥_{p}} to obtain for f the pointwise ergodic theorem with rate, as well as a.e. convergence of the one-sided ergodic Hilbert transform. For μ finite and T a positive contraction, we give a sufficient condition for the a.e. convergence of the "rotated one-sided Hilbert transform"; the result holds also for p = 1 when T is ergodic with T1 = 1. Our methods apply to norm-bounded sequences in L_{p}. Combining them with results of Marcus and Pisier, we show that if {g_{n}} is independent with zero expectation and uniformly bounded, then almost surely any realization {b_{n}} has the property that for every γ > 3/4, any contraction T on L_{2}(μ) and f ∈ L_{2}(μ), the series Σ_{k=1}^{∞} b _{k}T^{k} f(x)/k^{γ} converges μ-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of L _{1}(μ) of a probability space and f ∈ L_{p}(μ), 1 < p < ∞, the series Σ_{k=1}^{∞} b _{k}T^{k} f(x)/k^{γ} converges a.e. for γ ∈ (max{3/4, p+1/2p}, 1].

Original language | English |
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Pages (from-to) | 997-1031 |

Number of pages | 35 |

Journal | Illinois Journal of Mathematics |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2003 |

## ASJC Scopus subject areas

- General Mathematics