H\"older continuity of Tauberian constants associated with discrete and ergodic strong maximal operators

This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $\tilde{M}_S$ on $\mathbb{Z}^n$ by \[ \tilde{M}_S f(m) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{ j\in R \cap \mathbb{Z}^n} |f(m+j)|,\qquad m\in \mathbb{Z}^n, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $\tilde{C}_S(\alpha)$, defined by \[ \tilde{C}_S(\alpha) := \sup_{\substack{E \subset \mathbb{Z}^n \\ 0 < \#E < \infty} } \frac{1}{\#E}\#\{m \in \mathbb{Z}^n:\, \tilde{M}_S\chi_E(m) > \alpha\}, \] is H\"older continuous of order $1/n$. Moreover, letting $U_1, \ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(\Omega, \Sigma, \mu)$ we define the associated maximal operator $M_S^\ast$ by \[ M^\ast_{S}f(\omega) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n)\in R}|f(U_1^{j_1}\cdots U_n^{j_n}\omega)|,\qquad \omega\in\Omega. \] Then the corresponding Tauberian constant $C^\ast_S(\alpha)$, defined by \[ C^\ast_S(\alpha) := \sup_{\substack{E \subset \Omega \\ \mu(E) > 0}} \frac{1}{\mu(E)}\mu(\{\omega \in \Omega :\, M^\ast_S\chi_E(\omega) > \alpha\}), \] also satisfies $C^\ast_S \in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.