Solyanik estimates in ergodic theory

Paul Hagelstein¹ ; Ioannis Parissis²

¹ Department of Mathematics Baylor University Waco, TX 76798, U.S.A.
² Departamento de Matemáticas Universidad del Pais Vasco Aptdo. 644 48080 Bilbao, Spain and Ikerbasque Basque Foundation for Science Bilbao, Spain

Colloquium Mathematicum, Tome 145 (2016) no. 2, pp. 193-207.

Résumé

Let $U_1, \ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(\varOmega, \varSigma, \mu)$. Associated with these transformations is the ergodic strong maximal operator $\mathsf M _{\mathsf S} ^*$ given by $$ \mathsf M _{\mathsf S} ^* f(\omega) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n) \in R\cap \mathbb{Z}^n}|f(U_1^{j_1}\cdots U_n^{j_n}\omega)|, $$ 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 $0 \lt \alpha \lt 1$ we define the sharp Tauberian constant of $\mathsf M _{\mathsf S} ^*$ with respect to $\alpha$ by $$ \mathsf C^* _{\mathsf S} (\alpha) := \sup_{\substack{E \subset \varOmega \\ \mu(E) \gt 0}}\frac{1}{\mu(E)}\mu(\{\omega \in \varOmega : \mathsf M _{\mathsf S} ^* \chi_E (\omega) \gt \alpha\}). $$ 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 $$ \lim_{\alpha \rightarrow 1}\mathsf C^* _{\mathsf S}(\alpha) = 1 $$ holds, and that in particular \[\mathsf C^* _{\mathsf S}(\alpha) - 1 \lesssim_n ({1}/{\alpha} - 1)^{1/n}\] 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.

Détail

DOI : 10.4064/cm6729-2-2016

Keywords: ldots collection commuting measure preserving transformations probability space varomega varsigma associated these transformations ergodic strong maximal operator mathsf mathsf * given mathsf mathsf * omega sup subset mathbb frac cap mathbb sum ldots cap mathbb cdots omega where supremum taken rectangles mathbb containing origin whose sides parallel coordinate axes alpha define sharp tauberian constant mathsf mathsf * respect alpha mathsf * mathsf alpha sup substack subset varomega frac omega varomega mathsf mathsf * chi omega alpha motivated previous work solyanik authors regarding solyanik estimates geometric strong maximal operator harmonic analysis solyanik estimate lim alpha rightarrow mathsf * mathsf alpha holds particular mathsf * mathsf alpha lesssim alpha provided alpha sufficiently close solyanik estimates centered uncentered ergodic hardy littlewood maximal operators associated ldots shown further directions research field ergodic solyanik estimates discussed