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Continuity of halo functions associated to homothecy invariant density bases

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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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Beznosova O.

1 Hagelstein P.

1 2014

Wyniki wyszukiwania

Continuity of halo functions associated to homothecy invariant density bases