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Polynomial ergodic theorems in the spirit of Dunford and Zygmund

Dariusz Kosz, Bartosz Langowski, Mariusz Mirek, Paweł Plewa

TL;DR

The paper advances multiparameter ergodic theory by proving norm and almost everywhere convergence for polynomial-trajectory averages along commuting $\mathbb{R}$-flows, extending the Dunford–Zygmund framework to genuinely multi-parameter polynomial orbits. The authors develop a flexible Fourier-based approach to address the 'parameters-gluing' obstruction and establish sharp uniform oscillation/variation bounds for multiparameter Radon-type operators, enabling norm and a.e. convergence results for $f\in L^p(X)$ with $p\in(1,\infty)$. A key technical ingredient is the multiparameter oscillation/variation machinery, including a two-tier decomposition into long and short variations, a multiparameter Rademacher–Menshov inequality, and a rank-based splitting that handles nontrivial interactions among parameters. The work connects to a multiparameter variant of the Bellow–Furstenberg problem and provides a robust framework for analyzing polynomial orbits, with potential implications for future progress in higher-parameter ergodic and harmonic analysis problems. Overall, the paper delivers a comprehensive methodological toolkit for quantitative convergence phenomena in multiparameter ergodic averages and related Radon-type operators with polynomial structures.

Abstract

The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called "parameters-gluing'' phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow-Furstenberg problem.

Polynomial ergodic theorems in the spirit of Dunford and Zygmund

TL;DR

The paper advances multiparameter ergodic theory by proving norm and almost everywhere convergence for polynomial-trajectory averages along commuting -flows, extending the Dunford–Zygmund framework to genuinely multi-parameter polynomial orbits. The authors develop a flexible Fourier-based approach to address the 'parameters-gluing' obstruction and establish sharp uniform oscillation/variation bounds for multiparameter Radon-type operators, enabling norm and a.e. convergence results for with . A key technical ingredient is the multiparameter oscillation/variation machinery, including a two-tier decomposition into long and short variations, a multiparameter Rademacher–Menshov inequality, and a rank-based splitting that handles nontrivial interactions among parameters. The work connects to a multiparameter variant of the Bellow–Furstenberg problem and provides a robust framework for analyzing polynomial orbits, with potential implications for future progress in higher-parameter ergodic and harmonic analysis problems. Overall, the paper delivers a comprehensive methodological toolkit for quantitative convergence phenomena in multiparameter ergodic averages and related Radon-type operators with polynomial structures.

Abstract

The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on , , for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called "parameters-gluing'' phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow-Furstenberg problem.
Paper Structure (30 sections, 11 theorems, 113 equations, 1 figure)

This paper contains 30 sections, 11 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Continuous averages vs discrete averages
  4. General reductions and strategy of the proof
  5. Structure of the paper
  6. In Memoriam
  7. Notation
  8. Basic notation
  9. Multiparameter notation
  10. Difference and shift operators
  11. Euclidean spaces
  12. Function spaces
  13. Fourier transform
  14. Coordinatewise order
  15. Oscillation and variation seminorms
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $(X,\mathcal{B}(X), \mu)$ be a $\sigma$-finite measure space equipped with a family of $k\in\mathbb{Z}_+$ measure-preserving flows $(T_1^{t_1})_{t_1\in\mathbb{R}},\ldots, (T_k^{t_k})_{t_k\in\mathbb{R}} \colon X\to X$, not necessarily commuting, and such that the mapping $X \times \mathbb{R}^k\ni exists $\mu$-almost everywhere on $X$ and in $L^p(X)$ norm.

Figures (1)

  • Figure 1: Symbolic display of rewriting $a_{s'_1, s'_2} - a_{s_1, s_2}$ and $a_{s_1", s_2"} - a_{s'_1, s'_2}$ by using difference expressions associated with disjoint rectangles. The three thick dots in each coordinate system represent the three points $(1,1) \prec (s_1,s_2) \prec (s_1',s_2') \prec (s_1",s_2") \preceq (1+2^{L}, 1+2^{L})$.

Theorems & Definitions (21)

  • Theorem 1.1: Dunford D and Zygmund Z
  • Theorem 1.5: Quantitative multiparameter ergodic theorem for flows
  • Conjecture 1.10
  • Example 1.12
  • Theorem 1.14: Uniform oscillation estimates for multiparameter Radon operators
  • Theorem 3.1: Abstract oscillation/variation theorem
  • Lemma 3.8: cf. duorubio
  • proof
  • Lemma 3.9
  • proof
  • ...and 11 more