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Continuity of halo functions associated to homothecy invariant density bases

100%

Beznosova O., Hagelstein P.

Colloquium Mathematicum

2014

tom 134

nr 2

235-243

Let 𝓑 be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ 𝓑 of arbitrarily small diameter containing x. The collection 𝓑 is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $lim_{k→∞} 1/|R_{k}| ∫_{R_{k}} χ_{A} = χ_{A}(x)$ for any sequence ${R_{k}}$ of sets in 𝓑 containing x whose diameters tend to 0. The geometric maximal operator $M_{𝓑}$ associated to 𝓑 is defined on L¹(ℝⁿ) by $M_{𝓑}f(x) = sup_{x∈R∈𝓑} 1/|R| ∫_{R} |f|$. The halo function ϕ of 𝓑 is defined on (1,∞) by $ϕ(u) = sup{1/|A| |{x ∈ ℝⁿ: M_{𝓑}χ_{A}(x) > 1/u}|: 0 < |A| < ∞}$ and on [0,1] by ϕ(u) = u. It is shown that the halo function associated to any homothecy invariant density basis is a continuous function on (1,∞). However, an example of a homothecy invariant density basis is provided such that the associated halo function is not continuous at 1.

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

100%

Hagelstein P., Stokolos A.

Fundamenta Mathematicae

2012

tom 218

nr 3

269-283

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 𝓑 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 $M_{𝓑}$ and $M_{Γ}$ are defined respectively on $L¹(ℝ^{d})$ and L¹(Ω) by $M_{𝓑}g(x) = sup_{x ∈ R ∈ 𝓑} 1/|R| ∫_{R}|g(y)|dy$ 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 $M_{𝓑}$ satisfies the weak type estimate $|{x ∈ ℝ^d : M_{𝓑}g(x) > α}| ≤ C_{𝓑}∫_{ℝ^d} Φ(c_{𝓑}|g|/α)$ for a pair of positive constants $C_{𝓑}$, $c_{𝓑}$ 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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Hagelstein P.

1 Beznosova O.

1 Stokolos A.

1 2014

1 2012

Wyniki wyszukiwania

Continuity of halo functions associated to homothecy invariant density bases

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