Almost everywhere convergence of powers of some positive Lp contractions

Guy Cohen, Christophe Cuny, Michael Lin

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We extend the solution of Burkholder's conjecture for products of conditional expectations, obtained by Delyon and Delyon for L2 and by Cohen for Lp, 1<p<∞, to the context of Badea and Lyubich: Let T be a finite convex combination of operators Tj which are products of finitely many conditional expectations. Then Tnf converges a.e. for every f∈Lp, 1<p<∞, with supn|Tnf|∈Lp. The proof uses the work of Le Merdy and Xu on positive Lp 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 Lp space (1<p<∞), the operator T=∑k∈ZakSk is a positive Lp contraction such that for every f∈Lp, Tnf converges a.e. and supn|Tnf|∈Lp. If {ak} is supported on N, the same result is true when S is only a positive contraction of Lp. Similar results are obtained for μ-averages of bounded continuous representations of a σ-compact LCA group by positive operators in one Lp space, 1<p<∞. For a positive contraction T on Lp which satisfies Ritt's condition and f∈(I-T)αLp (0<α<1) we prove that nαTnf→0 a.e., and supnnα|Tnf|∈Lp.

Original languageEnglish
Pages (from-to)1129-1153
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Volume420
Issue number2
DOIs
StatePublished - 15 Dec 2014

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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