# Quantitative Fatou Theorems and Uniform Rectifiability

@article{Bortz2019QuantitativeFT,
title={Quantitative Fatou Theorems and Uniform Rectifiability},
author={Simon Bortz and Steve Hofmann},
journal={Potential Analysis},
year={2019},
volume={53},
pages={329-355}
}
• Published 4 January 2018
• Mathematics
• Potential Analysis
We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.
4 Citations
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## References

SHOWING 1-10 OF 31 REFERENCES
Bounded Analytic Functions
Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.-
Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$
• Mathematics
• 2012
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
Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions
• Mathematics
• 2014
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure
Singular Integrals and Di?erentiability Properties of Functions
A plurality of disks include solid laserable material and are in parallel spaced apart relation within a transparent tubular enclosure. Spaces between the disks constitute portions of a collant fluid
Analysis of and on uniformly rectifiable sets
• Mathematics
• 1993
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant
Rectifiability, interior approximation and harmonic measure
• Mathematics
Arkiv för Matematik
• 2019
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional
A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations☆
• Mathematics
• 2000
In the late 1950s and early 1960s, the work of De Giorgi [DeGi] and Nash [N], and then Moser [Mo], initiated the study of regularity of solutions to divergence form elliptic equations with merely
$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets
• Mathematics
• 2013
We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the
Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability
• Mathematics
• 2016
Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with
Approximation of harmonic functions
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