Some remarks on Riesz transforms on exterior Lipschitz domains

Abstract

Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$ . Given an exterior Lipschitz domain $\Omega $ , let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform $\nabla \mathcal {L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$ , even if $\mathcal {L}=\Delta $ is the Laplace operator and $\Omega $ is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \Omega $ is $C^1$ . We prove that for $p>2$ and $p\in [n,\infty )$ , it holds that $$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$

Keywords

• 35J25
• 35B65
• 42B35
• 42B37