Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, and Paul Hagelstein. "Continuity of halo functions associated to homothecy invariant density bases." Colloquium Mathematicae 134.2 (2014): 235-243. <http://eudml.org/doc/286578>.

@article{OleksandraBeznosova2014,
abstract = {Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection 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 containing x whose diameters tend to 0. The geometric maximal operator $M_\{\}$ associated to is defined on L¹(ℝⁿ) by $M_\{\}f(x) = sup_\{x∈R∈\} 1/|R| ∫_\{R\} |f|$. The halo function ϕ of is defined on (1,∞) by $ϕ(u) = sup\{1/|A| |\{x ∈ ℝⁿ: M_\{\}χ_\{A\}(x) > 1/u\}|: 0 < |A| < ∞\}$ 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.},
author = {Oleksandra Beznosova, Paul Hagelstein},
journal = {Colloquium Mathematicae},
keywords = {maximal functions; differentiation of integrals; halo conjecture},
language = {eng},
number = {2},
pages = {235-243},
title = {Continuity of halo functions associated to homothecy invariant density bases},
url = {http://eudml.org/doc/286578},
volume = {134},
year = {2014},
}

TY - JOUR
AU - Oleksandra Beznosova
AU - Paul Hagelstein
TI - Continuity of halo functions associated to homothecy invariant density bases
JO - Colloquium Mathematicae
PY - 2014
VL - 134
IS - 2
SP - 235
EP - 243
AB - Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection 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 containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{}f(x) = sup_{x∈R∈} 1/|R| ∫_{R} |f|$. The halo function ϕ of is defined on (1,∞) by $ϕ(u) = sup{1/|A| |{x ∈ ℝⁿ: M_{}χ_{A}(x) > 1/u}|: 0 < |A| < ∞}$ 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.
LA - eng
KW - maximal functions; differentiation of integrals; halo conjecture
UR - http://eudml.org/doc/286578
ER -