# Harmonic measure and quantitative connectivity: geometric characterization of the $$L^p$$-solvability of the Dirichlet problem

@article{Hofmann2020HarmonicMA, title={Harmonic measure and quantitative connectivity: geometric characterization of the \$\$L^p\$\$-solvability of the Dirichlet problem}, author={Steve Hofmann and Jos'e Mar'ia Martell}, journal={Inventiones mathematicae}, year={2020} }

Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a suitable connectivity condition, namely the weak local John condition. Together with other previous results by Hofmann and Martell, this implies that the weak-$A_\infty$ condition for harmonic measure holds if and only if $\partial\Omega$ is uniformly $n$-rectifiable and the… Expand

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#### References

SHOWING 1-10 OF 90 REFERENCES

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… Expand

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… Expand

Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula

- Mathematics
- International Mathematics Research Notices
- 2021

Let $\Omega \subset{{\mathbb{R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence… Expand

A sufficient geometric criterion for quantitative absolute continuity of harmonic measure

- Mathematics
- 2017

Let $\Omega\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that harmonic measure for $\Omega$ is… Expand

Absolute continuity of harmonic measure for domains with lower regular boundaries

- Mathematics
- Advances in Mathematics
- 2019

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors… Expand

BMO Solvability and Absolute Continuity of Harmonic Measure

- Mathematics
- 2016

We show that for a uniformly elliptic divergence form operator L, defined in an open set $$\Omega $$Ω with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative… Expand

Rectifiability of harmonic measure

- Mathematics
- 2015

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… Expand

Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability

- Mathematics
- 2015

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:=… Expand

Semi-uniform domains and a characterization of the $A_{\infty}$ property for harmonic measure

- Mathematics
- 2017

For domains with Ahlfors-David regular boundaries, it is known that the $A_{\infty}$ property of harmonic measure implies uniform rectifiability of the boundary \cite{MT15,HLMN17}. Since… Expand

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… Expand