Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, and Alexander Stokolos. "Transference of weak type bounds of multiparameter ergodic and geometric maximal operators." Fundamenta Mathematicae 218.3 (2012): 269-283. <http://eudml.org/doc/282848>.

@article{PaulHagelstein2012,
abstract = {Let $U₁, ..., U_\{d\}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ₊^\{d\}$ and the associated collection of rectangular parallelepipeds in $ℝ^d$ with sides parallel to the axes and dimensions of the form $n₁ × ⋯ × n_d$ with $(n₁,...,n_d) ∈ Γ.$ The associated multiparameter geometric and ergodic maximal operators $M_\{\}$ and $M_\{Γ\}$ are defined respectively on $L¹(ℝ^\{d\})$ and L¹(Ω) by $M_\{\}g(x) = sup_\{x ∈ R ∈ \} 1/|R| ∫_\{R\}|g(y)|dy$ and $M_\{Γ\}f(ω) = sup_\{(n₁, ..., n_\{d\}) ∈ Γ\} 1/\{n₁⋯ n_\{d\}\} ∑_\{j₁=0\}^\{n₁-1\} ⋯ ∑_\{j_\{d\}=0\}^\{n_\{d\}-1\} |f(U₁^\{j₁\} ⋯ U_\{d\}^\{j_\{d\}\}ω)|$. Given a Young function Φ, it is shown that $M_\{\}$ satisfies the weak type estimate $|\{x ∈ ℝ^d : M_\{\}g(x) > α\}| ≤ C_\{\}∫_\{ℝ^d\} Φ(c_\{\}|g|/α)$ for a pair of positive constants $C_\{\}$, $c_\{\}$ if and only if $M_\{Γ\}$ satisfies a corresponding weak type estimate $μ\{ω ∈ Ω : M_\{Γ\} f(ω) > α\} ≤ C_\{Γ\}∫_\{Ω\} Φ(c_\{Γ\}|f|/α)$. for a pair of positive constants $C_\{Γ\}$, $c_\{Γ\}$. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.},
author = {Paul Hagelstein, Alexander Stokolos},
journal = {Fundamenta Mathematicae},
keywords = {ergodic theory; maximal functions; transference},
language = {eng},
number = {3},
pages = {269-283},
title = {Transference of weak type bounds of multiparameter ergodic and geometric maximal operators},
url = {http://eudml.org/doc/282848},
volume = {218},
year = {2012},
}

TY - JOUR
AU - Paul Hagelstein
AU - Alexander Stokolos
TI - Transference of weak type bounds of multiparameter ergodic and geometric maximal operators
JO - Fundamenta Mathematicae
PY - 2012
VL - 218
IS - 3
SP - 269
EP - 283
AB - Let $U₁, ..., U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ₊^{d}$ and the associated collection of rectangular parallelepipeds in $ℝ^d$ with sides parallel to the axes and dimensions of the form $n₁ × ⋯ × n_d$ with $(n₁,...,n_d) ∈ Γ.$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{Γ}$ are defined respectively on $L¹(ℝ^{d})$ and L¹(Ω) by $M_{}g(x) = sup_{x ∈ R ∈ } 1/|R| ∫_{R}|g(y)|dy$ and $M_{Γ}f(ω) = sup_{(n₁, ..., n_{d}) ∈ Γ} 1/{n₁⋯ n_{d}} ∑_{j₁=0}^{n₁-1} ⋯ ∑_{j_{d}=0}^{n_{d}-1} |f(U₁^{j₁} ⋯ U_{d}^{j_{d}}ω)|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate $|{x ∈ ℝ^d : M_{}g(x) > α}| ≤ C_{}∫_{ℝ^d} Φ(c_{}|g|/α)$ for a pair of positive constants $C_{}$, $c_{}$ if and only if $M_{Γ}$ satisfies a corresponding weak type estimate $μ{ω ∈ Ω : M_{Γ} f(ω) > α} ≤ C_{Γ}∫_{Ω} Φ(c_{Γ}|f|/α)$. for a pair of positive constants $C_{Γ}$, $c_{Γ}$. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.
LA - eng
KW - ergodic theory; maximal functions; transference
UR - http://eudml.org/doc/282848
ER -