# 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

#### 31 Citations

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

Quantitative Comparisons of Multiscale Geometric Properties

- Mathematics
- 2019

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and… Expand

Uniform rectifiability implies Varopoulos extensions

- Mathematics
- 2020

We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $\Omega \subset \mathbb{R}^{n+1}$ be an… Expand

Carleson perturbations of elliptic operators on domains with low dimensional boundaries

- Mathematics
- 2020

We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for the degenerate elliptic operators of David, Feneuil… Expand

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition.

- Mathematics
- 2021

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (i.e., quantitatively open and path-connected) satisfiying the capacity density condition. Let $L_0… Expand

Elliptic theory in domains with boundaries of mixed dimension

- Mathematics
- 2020

Take an open domain $\Omega \subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $\Omega$ can be a ball on $\mathbb R^3$, minus one of its diameters… Expand

Necessary condition for the $L^2$ boundedness of the Riesz transform on Heisenberg groups

- Mathematics
- 2020

Let $\mu$ be a Radon measure on the $n$-th Heisenberg group $\mathbb{H}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on $\mathbb{H}^n$ is… Expand

Accessible parts of the boundary for domains with lower content regular complements

- Mathematics
- Annales Academiae Scientiarum Fennicae Mathematica
- 2019

We show that if $0<t<s\leq n-1$, $\Omega\subseteq \mathbb{R}^{n}$ with lower $s$-content regular complement, and $z\in \Omega$, there is a chord-arc domain $\Omega_{z}\subseteq \Omega $ with center… Expand

Harmonic Measure and the Analyst's Traveling Salesman Theorem

- Mathematics
- 2019

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose… 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

#### References

SHOWING 1-10 OF 90 REFERENCES

Rectifiability, interior approximation and Harmonic Measure

- Mathematics
- 2016

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

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

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

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

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

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… Expand