Vol. 10, No. 3, 2017

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The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

Steve Hofmann, Phi Le, José María Martell and Kaj Nyström

Vol. 10 (2017), No. 3, 513–558
Abstract

Let E n+1, n 2, be an Ahlfors–David regular set of dimension n. We show that the weak-A property of harmonic measure, for the open set Ω := n+1 E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < .

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Keywords
harmonic measure and $p$-harmonic measure, Poisson kernel, uniform rectifiability, Carleson measures, Green function, weak-$A_\infty$
Mathematical Subject Classification 2010
Primary: 31B05, 31B25, 35J08, 42B25, 42B37
Secondary: 28A75, 28A78
Milestones
Received: 12 February 2016
Accepted: 12 November 2016
Published: 17 April 2017
Authors
Steve Hofmann
Department of Mathematics
University of Missouri
Columbia, MO 65211
United States
Phi Le
Mathematics Department
Syracuse University
215 Carnegie Building
Syracuse, NY 13244
United States
José María Martell
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Consejo Superior de Investigaciones Científicas
C/ Nicolás Cabrera, 13-15
E-28049 Madrid
Spain
Kaj Nyström
Department of Mathematics
Uppsala University
SE-751 06 Uppsala
Sweden