Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions
نویسندگان
چکیده
منابع مشابه
Uniform Rectifiability, Carleson Measure Estimates, and Approximation of Harmonic Functions
Let E ⊂ Rn+1, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := Rn+1 \ E satisfy Carleson measure estimates, and are “ε-approximable”. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute con...
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The Mergelyan and Ahlfors-Beurling estimates for the Cauchy transform give quantitative information on uniform approximation by rational functions with poles off K. We will present an analogous result for an integral transform on the unit sphere in C2 introduced by Henkin, and show how it can be used to study approximation by functions that are locally harmonic with respect to the Kohn Laplacia...
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In this paper it is shown that if μ is an n-dimensional Ahlfors-David regular measure in R which satisfies the so-called weak constant density condition, then μ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n = 1, 2 and d − 1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in R. I...
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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 < ∞.
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 2016
ISSN: 0012-7094
DOI: 10.1215/00127094-3477128