## 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

#### Abstract

Let $E ⊂ ℝn+1$, $n ≥ 2$, be an Ahlfors–David regular set of dimension $n$. We show that the weak-$A∞$ property of harmonic measure, for the open set $Ω := ℝn+1 ∖ E$, 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

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

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

https://projecteuclid.org/euclid.apde/1510843446

Digital Object Identifier
doi:10.2140/apde.2017.10.513

Mathematical Reviews number (MathSciNet)
MR3641879

Zentralblatt MATH identifier
1369.31006

#### Citation

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. https://projecteuclid.org/euclid.apde/1510843446

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