Tangents, rectifiability, and corkscrew domains
In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of σ-finite length have tangents on a set of positive H 1-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if Σ ⊆ Rd+1...
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:182687 |
| Acceso en línea: | https://ddd.uab.cat/record/182687 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6211808 |
| Access Level: | acceso abierto |
| Palabra clave: | Harmonic measure Absolute continuity Corkscrew domains Uniform rectifiability Tangent Contingent Semmes surfaces |
| Sumario: | In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of σ-finite length have tangents on a set of positive H 1-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if Σ ⊆ Rd+1 has the property that each ball centered on Σ contains two large balls in different components of Σc and Σ has σ-finite H d-measure, then it has d-dimensional tangent points in a set of positive H d-measure. As an application, we show that if the dimension of harmonic measure for an NTA domain in Rd+1 is less than d, then the boundary domain does not have σ-finite H d-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if Ω ⊆ Rd+1 is an exterior corkscrew domain whose boundary has locally finite H d-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary. |
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