Riesz transform on exterior Lipschitz domains and applications

TL;DR

This paper proves the boundedness of Riesz transforms associated with elliptic operators on exterior Lipschitz domains, extending known results and applying to boundary value problems with new estimates even for the Laplacian.

Contribution

It establishes the boundedness of Riesz transforms for elliptic operators on exterior Lipschitz domains, including VMO coefficient cases, and derives new inequalities for these operators.

Findings

Boundedness of Riesz transforms in $L^p$ spaces for exterior Lipschitz domains.

Reverse inequality for the square root of the elliptic operator.

Applicability to inhomogeneous boundary value problems.

Abstract

Let ${\mathscr{L}}=-\text{div}A\nabla$ be a uniformly elliptic operator on $\mathbb{R}^n$, $n\ge 2$. Let $\Omega$ be an exterior Lipschitz domain, and let ${\mathscr{L}}_D$ and ${\mathscr{L}}_N$ be the operator ${\mathscr{L}}$ on $\Omega$ subject to the Dirichlet and Neumann boundary values, respectively. We establish the boundedness of the Riesz transforms $\nabla{\mathscr{L}}_D^{-1/2}$, $\nabla {\mathscr{L}}_N^{-1/2}$ in $L^p$ spaces. As a byproduct, we show the reverse inequality $\|{\mathscr{L}}_D^{1/2}f\|_{L^p(\Omega)}\le C\|\nabla f\|_{L^p(\Omega)}$ holds for any $1<p<\infty$. The proof can be generalized to show the boundedness of the Riesz transforms, for operators with VMO coefficients on exterior Lipschitz or $C^1$ domains. The estimates can be also applied to the inhomogeneous Dirichlet and Neumann problems. These results are new even for the Dirichlet and Neumann of the Laplacian operator on the exterior Lipschitz and $C^1$ domains.