Norm-variation of cubic ergodic averages

TL;DR

The paper addresses quantitative norm-variation convergence of cubic ergodic averages formed from $d$ commuting measure-preserving transformations. It develops a real-harmonic-analytic approach by modeling the problem with a Euclidean variant $A_t(\mathbf{f})$ and employing singular Brascamp-Lieb form estimates with cubical structure to control long- and short-variation components. A central result is a norm-variation bound $\sum_{i=1}^I \|A_{t_i}(\mathbf{f})-A_{t_{i-1}}(\mathbf{f})\|_q^q \le C I^{1-\frac{q}{2}}$ with $q=\frac{2^d}{2^d-1}$, from which ergodic variants follow via transference and interpolation. The methods extend previous $d=2,3$ results to all $d\ge 1$, though they do not establish almost-everywhere convergence. The work advances the quantitative understanding of cubic averages and highlights the role of cubical multilinear singular integrals in ergodic theory.

Abstract

We prove a quantitative result on norm convergence of cubic ergodic averages with respect to $d\geq 1$ commuting measure-preserving transformations. We use harmonic analysis techniques, a key tool being estimates for singular Brascamp-Lieb forms with cubical structure, which are used as a black box.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Theorem 1.1 in DST22
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5