Abstract
Let T be a power-bounded operator on Lp(μ), 1 < p < ∞. We use a sublinear growth condition on the norms {∥ k=1n Tk 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 Lp. Combining them with results of Marcus and Pisier, we show that if {gn} is independent with zero expectation and uniformly bounded, then almost surely any realization {bn} has the property that for every γ > 3/4, any contraction T on L2(μ) and f ∈ L2(μ), the series Σk=1∞ b kTk f(x)/kγ converges μ-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of L 1(μ) of a probability space and f ∈ Lp(μ), 1 < p < ∞, the series Σk=1∞ b kTk f(x)/kγ 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