## Abstract

We extend the solution of Burkholder's conjecture for products of conditional expectations, obtained by Delyon and Delyon for L_{2} and by Cohen for L_{p}, 1<p<∞, to the context of Badea and Lyubich: Let T be a finite convex combination of operators T_{j} which are products of finitely many conditional expectations. Then T^{n}f converges a.e. for every f∈L_{p}, 1<p<∞, with sup_{n}|T^{n}f|∈L_{p}. The proof uses the work of Le Merdy and Xu on positive L_{p} contractions satisfying Ritt's resolvent condition. As another application of the work of Le Merdy and Xu, we extend a result of Bellow, Jones and Rosenblatt, proving that if a probability {ak}k∈Z has bounded angular ratio, then for every positive invertible isometry S of an L_{p} space (1<p<∞), the operator T=∑k∈ZakSk is a positive L_{p} contraction such that for every f∈L_{p}, T^{n}f converges a.e. and sup_{n}|T^{n}f|∈L_{p}. If {a_{k}} is supported on N, the same result is true when S is only a positive contraction of L_{p}. Similar results are obtained for μ-averages of bounded continuous representations of a σ-compact LCA group by positive operators in one L_{p} space, 1<p<∞. For a positive contraction T on L_{p} which satisfies Ritt's condition and f∈(I-T)^{α}L_{p} (0<α<1) we prove that n^{α}T^{n}f→0 a.e., and sup_{n}n^{α}|T^{n}f|∈L_{p}.

Original language | English |
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Pages (from-to) | 1129-1153 |

Number of pages | 25 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 420 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2014 |

## Keywords

- Almost everywhere convergence
- Convolution powers of group actions
- Fractional coboundaries
- Positive contractions
- Products of conditional expectations
- Ritt operators