Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 3 (2017), 513-558.

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

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

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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 < .

Article information

Anal. PDE, Volume 10, Number 3 (2017), 513-558.

Received: 12 February 2016
Accepted: 12 November 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31B05: Harmonic, subharmonic, superharmonic functions 31B25: Boundary behavior 35J08: Green's functions 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28A78: Hausdorff and packing measures

harmonic measure and $p$-harmonic measure Poisson kernel uniform rectifiability Carleson measures Green function weak-$A_\infty$


Hofmann, Steve; Le, Long; Martell, José María; Nyström, Kaj. The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability. Anal. PDE 10 (2017), no. 3, 513--558. doi:10.2140/apde.2017.10.513.

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