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 $p>2$.
Categories: math.AP math.CA