Some remarks on Riesz transform on exterior Lipschitz domains

TL;DR

This paper investigates the boundedness properties of the Riesz transform on exterior Lipschitz domains for elliptic operators with specific coefficient regularity, establishing new equivalences in Sobolev spaces and applications to heat kernel estimates.

Contribution

It proves a new equivalence involving the Riesz transform and Sobolev norms on exterior Lipschitz domains with VMO or CMO coefficients, extending previous unboundedness results.

Findings

Established equivalence of Sobolev norms involving the Riesz transform

Identified the kernel of the square root of the elliptic operator as a one-dimensional space

Provided a uniform $L^p$ bound for the gradient of the heat semigroup for $p eq 2$

Abstract

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 $p>2$.