Related papers: $\varepsilon$-Approximability of Harmonic Function…

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1}…

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 harmonic function on $\Omega$ is…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with…

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 estimates, and are "$\varepsilon$-approximable".…

We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $\Omega\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus…

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…

Let $\Omega\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partial\Omega$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc…

Let $\Omega$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $\Omega$ satisfies a corkscrew condition and if $\partial \Omega$ is $d$-Ahlfors regular, i.e. Hausdorff measure…

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 $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform…

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 condition, and whose connected components are…

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 to surface measure, along with higher…

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 $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect…

In the present paper we sketch the proof of the fact 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 the harmonic…

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 open set satisfying the corkscrew…

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 the harmonic measure $\omega$ with respect…

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 $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform…

Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $\Sigma = \partial \Omega$. Under minimal and natural background assumptions - namely, that $\Sigma$ is time-symmetrically parabolic Ahlfors--David regular and…

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 $\mathcal{H}^d|_{\partial \Omega}$ is locally finite, then the Hausdorff…

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 Hausdorff) measure. Namely, that…

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness…