Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Harmonic measure and quantitative connectivity: geometric
  characterization of the $L^p$ solvability of the Dirichlet problem. Part II   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 weak local John condition
is satisfied. This yields the first geometric characterization of the
weak-$A_\infty$ condition for harmonic measure, which is important because of
its connection with the Dirichlet problem for the Laplace equation.

23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org