Some remarks on Riesz transform 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 $\hat{I}\text{c◯}$, let ${\mathcal{L}}_D$ be the elliptic operator $\mathcal{L}$ on $\hat{I}\text{c◯}$ 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 $\mathcal{L}=-\hat{I}”$ being the Laplace operator and $\hat{I}\text{c◯}$ being a domain outside a ball. Suppose that $A$ are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \hat{I}\text{c◯}$ is $C^1$, we prove that for $p>2$ and $p\in [n,\infty )$, it holds $$\underset{\ddot{I}•\in {\mathcal{K}}_p({\mathcal{L}}_D^{1/2})}{\inf }{\Vert \nabla (f-\ddot{I}•)\Vert }_{L^p(\hat{I}\text{c◯})}\sim \underset{\ddot{I}•\in {\mathcal{K}}_p({\mathcal{L}}_D^{1/2})}{\inf }{\Vert {\mathcal{L}}_D^{1/2}(f-\ddot{I}•)\Vert }_{L^p(\hat{I}\text{c◯})}$$ for $f\in {\dot{W}}_0^{1,p}(\hat{I}\text{c◯})$. Here ${\mathcal{K}}_p({\mathcal{L}}_D^{1/2})$ is the kernel of ${\mathcal{L}}_D^{1/2}$ in ${\dot{W}}_0^{1,p}(\hat{I}\text{c◯})$, which coincides with ${\tilde{\mathcal{A}}}_0^p(\hat{I}\text{c◯}):=\{f\in {\dot{W}}_0^{1,p}(\hat{I}\text{c◯}):{\mathcal{L}}_{D}f=0\}$ 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 $p>2$.

18pp, substancially revised, comments are welcome