Holder 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_{ jin R cap mathbb{Z}^n} |f(m+j)|,qquad min 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}_Schi_E(m) > alpha}, ] is Holder 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 omegainOmega. ] 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_Schi_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.