Riesz transform on exterior Lipschitz domains and applications

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, and 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 the boundedness of the Riesz transforms $\nabla{\mathscr{L}}_D^{-1/2}$, $\nabla {\mathscr{L}}_N^{-1/2}$ in $L^p$ spaces. 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 $1<p<\infty$. The proof can be generalized to show the boundedness of the Riesz transforms, for operators with VMO coefficients on exterior Lipschitz or $C^1$ domains. The estimates can be also applied to the inhomogeneous Dirichlet and Neumann problems. These results are new even for the Dirichlet and Neumann of the Laplacian operator on the exterior Lipschitz and $C^1$ domains.