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The weak-\(A_\infty\) property of harmonic and \(p\)-harmonic measures implies uniform rectifiability. (English) Zbl 1369.31006

In this interesting paper, the authors show that if \(\Omega\) is an open subset of \(\mathbb{R}^{n+1}\) satisfying an interior corkscrew condition and with boundary which is \(n\)-dimensional Ahlfors-David regular and if the harmonic measure \(\omega\) for \(\Omega\) is absolutely continuous with respect to \(\sigma\) and if the Poisson kernel \(k=d\omega/d\sigma\) verifies an appropriate scale invariant higher integrability estimate, then \(\partial \Omega\) is uniformly rectifiable.
This result implies, in particular, that if \(E \subset \mathbb{R}^{n+1}\) is an Ahlfors-David regular set of dimension \(n\), then the weak \(A_\infty\) property of the harmonic measure for \(\Omega \equiv \mathbb{R}^{n+1}\setminus E\) implies that \(E\) is uniformly rectifiable. Moreover, the authors prove a similar result involving the \(p\)-Laplace operator, \(p\)-harmonic functions, and the \(p\)-harmonic measure.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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