Higher-Dimensional Moving Averages and Submanifold Genericity

TL;DR

This paper generalizes moving-ergodic averages to commuting measure-preserving actions of $R^d$ by studying box averages and their maximal operator. It gives necessary and sufficient conditions for pointwise convergence of continuous box averages and shows that a linear-growth cone condition yields strong type $(p,p)$ bounds, yielding a.e. convergence for test functions in $L^p$ while failure in any coordinate leads to sweeping-out phenomena in aperiodic actions. It applies these results to submanifold genericity, proving that for a broad class of submanifolds the limit is not generic for $L^ ablafty$ test functions, though convergence can hold for smooth test functions in homogeneous settings and mean ergodicity remains in $L^p$. The work thus clarifies the regularity requirements needed for submanifold ergodic theorems and highlights limitations for extending such theorems beyond smooth test functions in general aperiodic actions.

Abstract

We generalize results of Jones and Olsen on multi-parameter moving ergodic averages to measure-preserving actions of $\mathbb R^d$ for $d\geq 1$. In particular, we give necessary and sufficient conditions for the pointwise convergence of averages over families of boxes in $\mathbb R^d$. As an application of our characterization, we show that averages along dilates of "locally flat" submanifolds in $\mathbb R^d$ do not necessarily converge point-wise for bounded measurable functions. This is closely related to the concept of submanifold-genericity recently introduced in \cite{BFK25}.

Figures (1)

• Figure 1: Set $\Omega_i^{(1)}(\lambda)$ for some sequence $(n_{ki},l_{ki})$.

Theorems & Definitions (19)

• Remark 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• proof