Solyanik estimates in ergodic theory
Tom 145 / 2016
Streszczenie
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.
