Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 241–262 | Cite as

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

  • Matthew Badger


We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.


Harmonic measure Absolute continuity Big pieces of Lipschitz graphs Corkscrew condition NTA domain Hausdorff dimension Wolff snowflake 

Mathematics Subject Classification (2000)

28A75 31A15 


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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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