Tangents, rectifiability, and corkscrew domains
Azzam, Jonas (University of Edinburgh. School of Mathematics)

Date:	2018
Abstract:	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.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Harmonic measure ; Absolute continuity ; Corkscrew domains ; Uniform rectifiability ; Tangent ; Contingent ; Semmes surfaces
Published in:	Publicacions matemàtiques, Vol. 62 Núm. 1 (2018) , p. 161-176 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/329932
DOI: 10.5565/PUBLMAT6211808

16 p, 424.3 KB

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