Solyanik estimates in ergodic theory

TL;DR

This paper establishes Solyanik estimates for ergodic maximal operators linked to commuting measure-preserving transformations, showing their sharp behavior as the threshold approaches 1, extending harmonic analysis results to ergodic theory.

Contribution

The paper proves the first Solyanik estimates for ergodic strong maximal operators, demonstrating their asymptotic behavior near the critical threshold and extending harmonic analysis techniques to ergodic settings.

Findings

The limit of the sharp Tauberian constant as alpha approaches 1 is 1.

The difference from 1 is bounded by a power of (1 - 1/alpha) depending on dimension.

Similar estimates are established for ergodic Hardy-Littlewood maximal operators.

Abstract

Let $U_1, \ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(\Omega, \Sigma, \mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $\mathsf M ^\ast _{\mathsf S}$ given by \[ \mathsf M ^\ast _{\mathsf S} f(\omega) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n) \in R\cap \mathbb{Z}^n}\big|f(U_1^{j_1}\cdots U_n^{j_n}\omega)\big|, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < \alpha < 1$ we define the sharp Tauberian constant of $\mathsf M ^\ast _{\mathsf S}$ with respect to $\alpha$ by \[ \mathsf C ^\ast _{\mathsf S} (\alpha) := \sup_{\substack{E \subset \Omega \\ \mu(E) > 0}}\frac{1}{\mu(E)}\mu(\{\omega \in \Omega : \mathsf M ^\ast _{\mathsf S} \chi_E (\omega) > \alpha\}). \] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate \[ \lim_{\alpha \rightarrow 1}\mathsf C ^\ast _{\mathsf S}(\alpha) = 1 \] holds, and that in particular we have \[\mathsf C ^\ast _{\mathsf S}(\alpha) - 1 \lesssim_n (1 - \frac{1}{\alpha})^{1/n}\] provided that $\alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, \ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.