 

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, KakutaniRokhlin lemma, nonperiodic transformation 
Subject: 
Primary 37A25, Secondary: 42B25 


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 Z^{n} by
\tilde{M}_{S}f(m) :=
sup_{0 ∈ R ⊂ Rn}(1/#(R ∩ Z^{n}))∑_{ j∈ R ∩ Zn} f(m+j), m∈ Z^{n},
where the supremum is taken over all open rectangles in R^{n} 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 ∈ Z^{n}: \tilde{M}_{S}ϗ_{E}(m) > α},
is Hölder continuous of order 1/n. Moreover, letting U_{1}, ..., 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^{∗}_{S}f(ω) :=
sup_{0 ∈ R ⊂ Rn} (1/#(R ∩ Z^{n}))∑_{(j1, ..., jn)∈ R}f(U_{1}^{j1}... U_{n}^{jn}ω), 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^{∗}_{S}ϗ_{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.


Acknowledgements
P. H. is partially supported by a grant from the Simons Foundation (#521719 to Paul Hagelstein). I. P. is supported by grant MTM201453850 of the Ministerio de Economía y Competitividad (Spain), grant IT64113 of the Basque Government, and IKERBASQUE


Author information
Paul Hagelstein:
Department of Mathematics, Baylor University, Waco, Texas 76798
paul_hagelstein@baylor.edu
Ioannis Parissis:
Departamento de Matemáticas, Universidad del Pais Vasco, Aptdo. 644, 48080 Bilbao, Spain and Ikerbasque, Basque Foundation for Science, Bilbao, Spain
ioannis.parissis@ehu.es

