Accessible parts of boundary for simply connected domains

Koskela, Pekka; Nandi, Debanjan; Nicolau, Artur

doi:10.1090/proc/13994

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…|} for all z in the image of γ. A first result without assuming uniformity of the domain is proven by Koskela, Nandi, and Nicolau in [10] where they use techniques from complex analysis to show that any bounded simply connected domain in the complex plane satisfies (1) for all 0 < t < 1.…”

Section: Introductionmentioning

confidence: 99%

Accessible parts of the boundary for domains in metric measure spaces

Gibara¹,

Korte²

2022

Ann. Fenn. Math.

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We prove in the setting of \(Q\)-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the \(s\)-dimensional Hausdorff content, then its visible boundary has large \(t\)-dimensional Hausdorff content for every \(0<t<s\leq Q-1\). The visible boundary is the set of points that can be reached by a John curve from a fixed point \(z_{0}\in \Omega\). This generalizes recent results by Koskela-Nandi-Nicolau (from \(\mathbb R^2\)) and Azzam (\(\mathbb R^n\)). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

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Section: Introductionmentioning

confidence: 99%

Accessible parts of the boundary for domains in metric measure spaces

Gibara¹,

Korte²

2022

Ann. Fenn. Math.

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“…A first result without assuming uniformity of the domain is proven by Koskela, Nandi, and Nicolau in [10] where they use techniques from complex analysis to show that any bounded simply connected domain in the complex plane satisfies (1) for all 0 < t < 1.…”

Section: Introductionmentioning

confidence: 99%

Accessible parts of the boundary for domains in metric measure spaces

Gibara,

Korte

2021

Preprint

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We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z 0 ∈ Ω. This generalizes recent results by Koskela-Nandi-Nicolau (from R 2 ) and Azzam (R n ). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

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“…Recently, Koskela, Nandi and Nicolau in [KNN18] showed (1.3) holds for simply connected planar domains Ω, when n = 2 and t < s = 1 using techniques from complex analysis.…”

Section: Introductionmentioning

confidence: 99%

Accessible parts of the boundary for domains with lower content regular complements

Azzam¹

2019

Ann. Acad. Sci. Fenn. Math.

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We show that if 0 < t < s ≤ n − 1, Ω ⊆ R n with lower s-content regular complement, and z ∈ Ω, there is a chord-arc domainThis 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 − 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.

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Accessible parts of boundary for simply connected domains

Cited by 4 publications

References 10 publications

Accessible parts of the boundary for domains in metric measure spaces

Accessible parts of the boundary for domains in metric measure spaces

Accessible parts of the boundary for domains in metric measure spaces

Accessible parts of the boundary for domains with lower content regular complements

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