Some remarks on Riesz transform on exterior Lipschitz domains

Let $n\ge2$ 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 operator $\nabla \mathcal{L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$, even if $\mathcal{L}=-\Delta$ being the Laplace operator and $\Omega$ being 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 $$\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\mathcal{L}^{1/2}_D (f-\phi)\right\|_{L^p(\Omega)}$$ for $f\in \dot{W}^{1,p}_0(\Omega)$. Here $\mathcal{K}_p(\mathcal{L}_D^{1/2})$ is the kernel of $\mathcal{L}_D^{1/2}$ in $\dot{W}^{1,p}_0(\Omega)$, which coincides with $\tilde{\mathcal{A}}^p_0(\Omega):=\{f\in \dot{W}^{1,p}_0(\Omega):\,\mathcal{L}_Df=0\}$ 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 $p>2$.