Uniform rectifiability and harmonic measure II: Poisson kernels in $L^p$ imply uniform rectifiability

Abstract

We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\ge 2$, for an ADR domain $Ω\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition, we show that absolute continuity of harmonic measure with respect to surface measure on $\partial \Omega$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partial \Omega$.

References in corpus (2)

• Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited
• Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains

Cited by in corpus (15)

• Rectifiability of harmonic measure
• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions
• The weak-$A_{\infty }$ property of harmonic and $p$-harmonic measures implies uniform rectifiability
• Uniform rectifiability from Carleson measure estimates and $\epsilon$-approximability of bounded harmonic functions
• Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
• Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates
• Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets
• Generalized local Tb Theorems for Square Functions, and applications
• Tangent measures of elliptic harmonic measure and applications
• Boundary rectifiability and elliptic operators with $W^{1,1}$ coefficients
• Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure
• The Green function with pole at infinity applied to the study of the elliptic measure
• Absolute continuity of degenerate elliptic measure
• A Green function characterization of uniformly rectifiable sets of any codimension