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

TL;DR

This paper proves that Tauberian constants for discrete and ergodic strong maximal operators are Hölder continuous of order 1/n, revealing their smoothness properties in relation to the dimension and dynamics of transformations.

Contribution

It establishes the Hölder continuity of Tauberian constants for both discrete and ergodic maximal operators, extending understanding of their smoothness in higher dimensions.

Findings

Tauberian constants are Hölder continuous of order 1/n

Continuity holds for both discrete and ergodic settings

In the one-dimensional case, smoothness relates to orbit lengths of transformations

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.