New York Journal of Mathematics
Volume 23 (2017) 1219-1236


Paul Hagelstein and Ioannis Parissis

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

view    print

Published: September 13, 2017
Keywords: ergodic theory, maximal operators, Kakutani-Rokhlin lemma, nonperiodic transformation
Subject: Primary 37A25, Secondary: 42B25

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) := sup0 ∈ 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) := supE ⊂ Zn, 0 < #E < ∞ (1/#E)#{m ∈ Zn: \tilde{M}SϗE(m) > α},
is Hölder continuous of order 1/n. Moreover, letting U1, ..., Un 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
MSf(ω) := sup0 ∈ R ⊂ Rn (1/#(R ∩ Zn))∑(j1, ..., jn)∈ R|f(U1j1... Unjnω)|, ω∈Ω.
Then the corresponding Tauberian constant CS(α), defined by
CS(α) := supE ⊂ Ω, μ(E) > 0 (1/μ(E))μ({ω ∈ Ω : MSϗE(ω) > α}),
also satisfies CS ∈ C1/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

Author information

Paul Hagelstein:
Department of Mathematics, Baylor University, Waco, Texas 76798

Ioannis Parissis:
Departamento de Matemáticas, Universidad del Pais Vasco, Aptdo. 644, 48080 Bilbao, Spain and Ikerbasque, Basque Foundation for Science, Bilbao, Spain