Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 889-901

ACCESSIBLE PARTS OF THE BOUNDARY FOR DOMAINS WITH LOWER CONTENT REGULAR COMPLEMENTS

Jonas Azzam

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 Ω_z ⊂ Ω with center z so that H^t_∞(∂Ω_z ∩ ∂Ω) ≥_t dist(z,Ω^c)^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.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4458