Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 889-901
University of Edinburgh,
School of Mathematics
JCMB, Kings Buildings,
Mayfield Road, Edinburgh,
EH9 3JZ, Scotland; j.azzam 'at' ed.ac.uk
Abstract. We show that if 0 < t < s ≤ n – 1, Ω ⊂ Rn with lower s-content regular complement, and z ∈ Ω, there is a chord-arc domain Ωz ⊂ Ω with center z so that Ht∞(∂Ωz ∩ ∂Ω) ≥t dist(z,Ωc)t. This was originally shown by Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when n = 2, s = 1, and Ω is a simply connected planar domain.
Domains satisfying the conclusion of this result support (p,β)-Hardy inequalities for β < p \ndash; n + t by a result of Koskela and Lehrbäck; Lehrbäck also showed that s-content regularity of the complement for some s > n – p + β was necessary. Thus, the combination of these results characterizes when a domain supports a pointwise (p,β)-Hardy inequality for β < p – 1 in terms of lower content regularity.
2010 Mathematics Subject Classification: Primary 28A75, 46E35, 26D15.
Key words: Chord-arc domain, visual boundary, John domain, Hardy inequality.
Reference to this article: J. Azzam: Accessible parts of the boundary for domains with lower content regular complements. Ann. Acad. Sci. Fenn. Math. 44 (2019), 889-901.
https://doi.org/10.5186/aasfm.2019.4458
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