Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riesz transform on exterior Lipschitz domains   Let $\mathscr{L}=-\text{div}A\nabla$ be a uniformly elliptic operator on
$\mathbb{R}^n$, $n\ge 2$. Let $\Omega$ be an exterior Lipschitz domain, let
$\mathscr{L}_D$ and $\mathscr{L}_N$ be the operator $\mathscr{L}$ on $\Omega$
subject to the Dirichlet and Neumann boundary values, respectively. We
establish characterization for the boundedness of the Riesz transforms
$\nabla\mathscr{L}_D^{-1/2}$, $\nabla \mathscr{L}_N^{-1/2}$ in the sharp range.
As a byproduct, we show the reverse inequality
$\|\mathscr{L}_D^{1/2}f\|_{L^p(\Omega)}\le C\|\nabla f\|_{L^p(\Omega)}$ holds
for any $1<p<\infty$. As applications, we deduce the sharp range for the
boundedness of the Riesz transforms, for operators with VMO coefficients, on
exterior Lipschitz and $C^1$ domains. Application to inhomogeneous Dirichlet
and Neumann problem is also presented. Our result is new even for the Dirichlet
and Neumann Laplacian on exterior Lipschitz and $C^1$ domains.

18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio