Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

@article{Akman2015RectifiabilityAE,
  title={Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries},
  author={Murat Akman and Matthew Badger and Steve Hofmann and Jos'e Mar'ia Martell},
  journal={arXiv: Classical Analysis and ODEs},
  year={2015}
}
Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partial\Omega$ is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $\partial\Omega$ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $\partial\Omega$ can be covered $\mathcal{H}^n$-a.e… 

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References

SHOWING 1-10 OF 52 REFERENCES
Uniform domains with rectifiable boundaries and harmonic measure
We assume that $\Omega \subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial \Omega}$
Rectifiability of harmonic measure in domains with porous boundaries
We show that if $n\geq 1$, $\Omega\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite and positive Hausdorff $H^{n}$-measure upon
Rectifiability of harmonic measure
In the present paper we prove that for any open connected set $${\Omega\subset\mathbb{R}^{n+1}}$$Ω⊂Rn+1, $${n\geq 1}$$n≥1, and any $${E\subset \partial \Omega}$$E⊂∂Ω with
Singular sets for harmonic measure on locally flat domains with locally finite surface measure
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high
Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:=
Harmonic measure and approximation of uniformly rectifiable sets
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew
Absolute continuity between the surface measure and harmonic measure implies rectifiability
In the present paper we prove that for any open connected set $\Omega\subset{\mathbb R}^{n+1}$, $n\geq 1$, and any $E\subset \partial\Omega$ with $0<{\mathcal H}^n(E)<\infty$ absolute continuity of
Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$
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
The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability
Let $E\subset \ree$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set$\Omega:= \ree\setminus E$, implies u
$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets
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
...
1
2
3
4
5
...