Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$

By S. Hofmann, J. M. Martell. 2012

110 citations recorded by the source corpus.

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $\Omega\subset \mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant $L^q$ bounds, with $q>1$, on the Poisson kernel.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/196b09157212a557cbd056a4211a6216d7f77a85

How researchers see this paper

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In Applied Mathematics

1. J. M. Martell
2. S. Hofmann
3. Kaj Nyström
4. Phi Le
5. P. Auscher
6. Xuan Thinh Duong
7. D. Cruz-Uribe
8. A. Mcintosh

In Computational Theory and Mathematics

1. J. M. Martell
2. Kaj Nyström
3. S. Hofmann
4. John L. Lewis
5. Juan Luis V'azquez
6. Sun-Sig Byun
7. Lihe Wang

Citations in the corpus

• The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability (2015, background): “By Poincaré’s inequality (see, e.g., [Hofmann and Martell 2014, Section 4] in this context), we obtain that σ(Q).”