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Norm-variation of triple ergodic averages for commuting transformations

Polona Durcik, Lenka Slavíková, Christoph Thiele

TL;DR

The paper establishes norm variation bounds with $r>4$ for triple ergodic averages along three commuting transformations, addressing the challenging case where cubical frequency structure is lost. By transferring the problem to a multi-parameter singular Brascamp-Lieb framework, the authors develop a hierarchical set of propositions handling on-diagonal and off-diagonal, Whitney and non-Whitney regimes, using lacunary cone decompositions, tensorization, and telescoping identities. Key innovations include a careful decomposition of the multiplier into components with different scale behavior and a new 3D-telescope argument that yields bounded Brascamp-Lieb forms even without full cubical symmetry. The results advance quantitative convergence theory for multi-parameter ergodic averages and delineate the sharp threshold $r>4$ for three commuting transformations, with open questions remaining for more than three.

Abstract

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations.

Norm-variation of triple ergodic averages for commuting transformations

TL;DR

The paper establishes norm variation bounds with for triple ergodic averages along three commuting transformations, addressing the challenging case where cubical frequency structure is lost. By transferring the problem to a multi-parameter singular Brascamp-Lieb framework, the authors develop a hierarchical set of propositions handling on-diagonal and off-diagonal, Whitney and non-Whitney regimes, using lacunary cone decompositions, tensorization, and telescoping identities. Key innovations include a careful decomposition of the multiplier into components with different scale behavior and a new 3D-telescope argument that yields bounded Brascamp-Lieb forms even without full cubical symmetry. The results advance quantitative convergence theory for multi-parameter ergodic averages and delineate the sharp threshold for three commuting transformations, with open questions remaining for more than three.

Abstract

We prove an -variation estimate, , in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any for more than three commuting transformations.
Paper Structure (11 sections, 15 theorems, 428 equations, 4 figures)

This paper contains 11 sections, 15 theorems, 428 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A collection of propositions on singular Brascamp-Lieb forms
  3. Proof of Theorem \ref{['thm:2']} from Proposition \ref{['prop:1']} and Corollaries \ref{['cor:1']} and \ref{['cor:2']}
  4. Proof of Proposition \ref{['prop:1']} using Propositions \ref{['prop:2']} and \ref{['firstcz']}
  5. Proof of Proposition \ref{['prop:2']} using Propositions \ref{['firstcz']} and \ref{['firststick']}
  6. Proof of Proposition \ref{['firstcz']} using Lemma 3 in
  7. Proof of Proposition \ref{['firststick']} using Propositions \ref{['thm:4']} and \ref{['secstickphi']}
  8. Proof of Proposition \ref{['thm:4']} using Propositions \ref{['secstickphi']}, \ref{['secstickgauss']}, and Theorem 1.1 in
  9. Proof of Propositions \ref{['secstickphi']} and \ref{['secstickgauss']}
  10. Proof of Proposition \ref{['secstickphi']}
  11. Proof of Proposition \ref{['secstickgauss']}

Key Result

Theorem 1.1

For all $r>4$, there exists a constant $C>0$ such that the following holds. Let $(X,\mathcal{F},\mu)$ be a $\sigma$-finite measure space, $T_0,T_1,T_2\colon~X\to X$ mutually commuting measure preserving transformations and let $J$ and $n_0<n_1<\cdots<n_J$ be positive integers. For any $f_0,f_1\in L^ where we have defined for almost every $x\in X$

Figures (4)

  • Figure 1: Structure of $m=\widehat{K}$ in Theorem \ref{['thm:2']}.
  • Figure 2: Structure of the proof of the main theorem
  • Figure 3: Lacunary cones
  • Figure 4: Three-dimensional multiplier

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: on-diagonal, Whitney, 2D
  • Corollary 2.2
  • Proposition 2.3: on-diagonal, non-Whitney, 2D
  • Corollary 2.4
  • Proposition 2.5: off-diagonal, Whitney, 2D
  • Proposition 2.6: off-diagonal, non-Whitney, 2D
  • Proposition 2.7: on-critical, non-Whitney, 3D
  • Proposition 2.8: off-critical, non-Whitney, 3D, discrete
  • ...and 9 more