Hölder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators

Abstract: 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.