Weakly porous sets and Muckenhoupt Ap distance functions

We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight w(x)=dist(x,E)−α belongs to the Muckenhoupt class A1, for some α>0, if and only if E⊂Rn is weakly porous. We also give a precise quantitative version of this characterization...

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Bibliographic Details
Published inJournal of functional analysis Vol. 287; no. 8
Main Authors Anderson, Theresa C., Lehrbäck, Juha, Mudarra, Carlos, Vähäkangas, Antti V.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.10.2024
Subjects
Distance function
Fractals
Muckenhoupt weight
Weak porosity
42B99
28A75
42B25
Distance function
28A80
Weak porosity
Muckenhoupt weight
Fractals
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Summary:We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight w(x)=dist(x,E)−α belongs to the Muckenhoupt class A1, for some α>0, if and only if E⊂Rn is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of w∈Ap, for 1<p<∞, as well. At the end of the paper, we give an example of a set E⊂R which is not weakly porous but for which w∈Ap∖A1 for every 0<α<1 and 1<p<∞.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2024.110558