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

By Matthew Badger. 2010

55 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d9e43d1d6d4689337ea440b97be28ae64164a386

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$ (2012, result): “ which satisﬁes the “interior big UNIFORM RECTIFIABILITY AND HARMONIC MEASURE I 5 pieces” condition, but whose harmonic measure fails to be doubling). We mention also in this context the recent paper [Ba], in which the geometric conclusion of [DJ], namely the existence of “interior big pieces” of Li”
Rectifiability of harmonic measure (2015, unknown): “Se]. To be precise, [Da], [DJ], [Se] establish a quantitative scale-invariant result, theA∞ property of harmonic measure, which in the planar case was proved by Lavrent’ev [Lv]. See also [HM1], [BL], [Ba], [Az], [AMT1], [Mo], [ABHM] along with [AHMNT] in this context. We shall not give a precise deﬁ”
A new characterization of chord-arc domains (2014, unknown): “sions, for Lipschitz domains by B. Dahlberg [Dah]; for chord-arc domains by G. David and D. Jerison [DJ], and independently, by S. Semmes [Sem]; for NTA domains without an ADR hypothesis by M. Badger [Bad]; for the Riesz measure associated to the p-Laplacian in chord-arc domains, by J. Lewis and the”
An analyst’s traveling salesman theorem for sets of dimension larger than one (2016, background): “If E has locally finiteH n−1-measure, then one can show that, for any ball B centered on E , there is a Lipschitz graph so thatH n−1( ∩B∩E) ≥ rn−1 B with Lipschitz constant depending on H n−1(B ∩ E) (see [32] and [4]).”