Vladimir Chilin, Doğan Çömez, Semyon Litvinov

Research output: Contribution to journalArticlepeer-review


Given a σ-finite infinite measure space (Ω, µ), it is shown that any Dun-ford–Schwartz operator T: L1(Ω) → L1(Ω) can be uniquely extended to the space L1(Ω) + L(Ω). This allows one to find the largest subspace Rµ of L1(Ω) + L(Ω) such that the ergodic averages n−1 ∑n−1 k=0T k (f) converge almost uniformly (in Egorov’s sense) for every f ∈ Rµ and every Dunford–Schwartz operator T. Utilizing this result, almost uniform convergence of the averages n−1 ∑n−1 k=0βkTk (f) for every f ∈ Rµ, any Dunford– Schwartz operator T and any bounded Besicovitch sequence {βk} is established. Further, given a measure preserving transformation τ: Ω → Ω, Assani’s extension of Bourgain’s Return Times theorem to σ-finite measures is employed to show that for each f ∈ Rµ there exists a set Ωf ⊂ Ω such that µ(Ω \ Ωf) = 0 and the averages n−1 ∑n−1 k=0βkf(τkω) converge for all ω ∈ Ωf and any bounded Besicovitch sequence {βk}. Applications to fully symmetric subspaces E ⊂ Rµ are outlined.

Original languageEnglish (US)
Pages (from-to)219-238
Number of pages20
JournalColloquium Mathematicum
Issue number2
StatePublished - 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics


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