Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova ; Paul Hagelstein

Colloquium Mathematicae, Tome 135 (2014), p. 235-243 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2014-01-01

Zbl 1293.42020

EUDML-ID : urn:eudml:doc:286578