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 $$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$ for $f\in \mathring {W}^{1,p}(\Omega )$ . Here, $\mathcal {K}_p(\mathcal {L}_D^{1/2})$ is the kernel of $\mathcal {L}_D^{1/2}$ in $\mathring {W}^{1,p}(\Omega )$ , which coincides with $\tilde {\mathcal {A}}^p_0(\Omega ):=\{f\in \mathring {W}^{1,p}(\Omega ):\ \mathcal {L}_Df=0\}$ and is a one-dimensional subspace. As an application, we provide a substitution of $L^p$ -boundedness of $\sqrt {t}\nabla e^{-t\mathcal {L}_D}$ which is uniform in t for $p\ge n$ and $p>2$ .