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Besicovitch-weighted ergodic theorems with continuous time

Semyon Litvinov

TL;DR

This work extends Besicovitch-weighted ergodic theorems to continuous time for semigroups of Dunford-Schwartz operators on L^p spaces over σ-finite spaces, proving almost uniform convergence in Egorov's sense under Besicovitch modulation. The authors establish a maximal ergodic inequality, prove a.u. convergence for both unweighted and weighted averages, and identify limits in the local regime; they further generalize the results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces, via Favà space methods and rearrangement arguments. The approach blends strong continuity in L^1, Bochner integrability, polynomial approximations of Besicovitch functions, and a robust extension framework to fully symmetric spaces, addressing infinite-measure settings and nonpositive semigroups. The findings provide a unified continuous-time, Besicovitch-perturbed ergodic theory with explicit a.u. limits and a pathway to broader function spaces, enhancing applications in ergodic analysis and related areas.

Abstract

Given $1\leq p<\infty$, we show that ergodic flows in the $L^p$-space over a $σ$-finite measure space generated by strongly continuous semigroups of Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly (in Egorov's sense). The corresponding local ergodic theorem is proved with identification of the limit. Then we extend these results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces.

Besicovitch-weighted ergodic theorems with continuous time

TL;DR

This work extends Besicovitch-weighted ergodic theorems to continuous time for semigroups of Dunford-Schwartz operators on L^p spaces over σ-finite spaces, proving almost uniform convergence in Egorov's sense under Besicovitch modulation. The authors establish a maximal ergodic inequality, prove a.u. convergence for both unweighted and weighted averages, and identify limits in the local regime; they further generalize the results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces, via Favà space methods and rearrangement arguments. The approach blends strong continuity in L^1, Bochner integrability, polynomial approximations of Besicovitch functions, and a robust extension framework to fully symmetric spaces, addressing infinite-measure settings and nonpositive semigroups. The findings provide a unified continuous-time, Besicovitch-perturbed ergodic theory with explicit a.u. limits and a pathway to broader function spaces, enhancing applications in ergodic analysis and related areas.

Abstract

Given , we show that ergodic flows in the -space over a -finite measure space generated by strongly continuous semigroups of Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly (in Egorov's sense). The corresponding local ergodic theorem is proved with identification of the limit. Then we extend these results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces.
Paper Structure (6 sections, 26 theorems, 104 equations)

This paper contains 6 sections, 26 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal ergodic inequality for the family $\{M_t(\mathcal{T},\beta)\}_{t>0}$
  4. Almost uniform convergence of non-weighted averages in $\mathcal{L}^p(\Omega,\mu)$
  5. A.u. convergence of the weighted averages in $\mathcal{L}^p(\Omega,\mu)$
  6. Extension to fully symmetric spaces

Key Result

Proposition 2.1

The space $\mathcal{L}^0$ is complete with respect to a.u. convergence

Theorems & Definitions (44)

  • Proposition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 34 more