Advanced Search

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 $li{m}_{k\to \infty }1/\left|R_k\right|{\int }_{R_k}{\chi }_A={\chi }_A\left(x\right)$ 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 $M_{}f\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|f\right|$. The halo function ϕ of is defined on (1,∞) by $\varphi \left(u\right)=sup1/\left|A\right||x\in ℝⁿ:M_{}{\chi }_A\left(x\right)>1/u|:0<|A|<\infty$ and on [0,1] by ϕ(u) = u. It is shown that the halo...