# BMO Solvability and Absolute Continuity of Harmonic Measure

@article{Hofmann2016BMOSA, title={BMO Solvability and Absolute Continuity of Harmonic Measure}, author={Steve Hofmann and Phi Long Le}, journal={The Journal of Geometric Analysis}, year={2016}, volume={28}, pages={3278-3299} }

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 absolute continuity (the weak-$$A_\infty $$A∞ property) of elliptic-harmonic measure with respect to surface measure on $$\partial \Omega $$∂Ω. We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual… Expand

#### 20 Citations

BMO Solvability and $$A_{\infty }$$A∞ Condition of the Elliptic Measures in Uniform Domains

- Mathematics
- 2018

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO… Expand

BMO Solvability and Absolute Continuity of Caloric Measure

- Mathematics
- 2020

We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-
$A_{\infty }$
property) of caloric measure with respect to surface measure, for an… Expand

A geometric characterization of the weak-$A_\infty$ condition for harmonic measure

- Mathematics
- 2018

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,… Expand

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

- Mathematics
- Inventiones mathematicae
- 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,… Expand

A Weak Reverse Hölder Inequality for Caloric Measure

- Mathematics
- The Journal of Geometric Analysis
- 2019

Following a result of Bennewitz–Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Hölder inequality on an open set $$\Omega… 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

Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress

- Mathematics
- Acta Mathematica Sinica, English Series
- 2019

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary… 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 from Carleson measure estimates and ε-approximability of bounded harmonic functions

- Mathematics
- Duke Mathematical Journal
- 2018

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

BMO Solvability and $$A_{\infty }$$A∞ Condition of the Elliptic Measures in Uniform Domains

- Mathematics
- 2018

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO… Expand

BMO solvability and the $A_\infty$ condition for elliptic operators

- Mathematics
- 2010

We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an… 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 from Carleson measure estimates and ε-approximability of bounded harmonic functions

- Mathematics
- Duke Mathematical Journal
- 2018

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded… 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

The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability

- Mathematics
- 2017

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… 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

BMO Solvability and the A∞ Condition for Elliptic Operators

- Mathematics
- 2010

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an… Expand

Square Functions and the $$A_\infty $$A∞ Property of Elliptic Measures

- Mathematics
- 2014

In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, nonsymmetric divergence form… Expand

Wiener's criterion for the heat equation

- Mathematics
- 1982

This paper establishes a necessary and sufficient condition for a boundary point of an arbitrary bounded open subset of R "+1 to be regular for the heat equation. Our criterion is, in the sense… Expand