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 Zⁿ by \tilde{M}_Sf(m) := sup_{0 ∈ R ⊂ Rⁿ}(1/#(R ∩ Zⁿ))∑ _{j∈ R ∩ Zⁿ} |f(m+j)|, m∈ Zⁿ, where the supremum is taken over all open rectangles in Rⁿ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant \tilde{C}_S(α), defined by \tilde{C}_S(α) := sup_{E ⊂ Zⁿ, 0 < #E < ∞} (1/#E)#{m ∈ Zⁿ: \tilde{M}_Sϗ_E(m) > α}, is Hölder continuous of order 1/n. Moreover, letting U₁, ..., U_n denote a nonperiodic collection of commuting invertible transformations on the nonatomic probability space (Ω, Σ, μ) we define the associated maximal operator M ^∗_S by M^∗_Sf(ω) := sup_{0 ∈ R ⊂ Rⁿ} (1/#(R ∩ Zⁿ))∑_{(j1, ..., jn)∈ R}|f(U₁^j₁... U_n^j_nω)|, ω∈Ω. Then the corresponding Tauberian constant C^∗_S(α), defined by C^∗_S(α) := sup_{E ⊂ Ω, μ(E) > 0} (1/μ(E))μ({ω ∈ Ω : M^∗_Sϗ_E(ω) > α}), also satisfies C^∗_S ∈ 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.

P. H. is partially supported by a grant from the Simons Foundation (#521719 to Paul Hagelstein). I. P. is supported by grant MTM2014-53850 of the Ministerio de Economía y Competitividad (Spain), grant IT-641-13 of the Basque Government, and IKERBASQUE