Characterizing regularity of domains via the Riesz transforms on their boundaries

Abstract

Under mild geometric measure-theoretic assumptions on an open subset $\Omega$ of ${ℝ}^n$, we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space ${\mathcal{C}}^{\alpha }\left(\partial \Omega \right)$ if and only if $\Omega$ is a Lyapunov domain of order $\alpha$ (i.e., a domain of class ${\mathcal{C}}^{1+\alpha }$). In the category of Lyapunov domains we also establish the boundedness on Hölder spaces of singular integral operators with kernels of the form $P\left(x-y\right)∕|x-y{|}^{n-1+l}$, where $P$ is any odd homogeneous polynomial of degree $l$ in ${ℝ}^n$. This family of singular integral operators, which may be thought of as generalized Riesz transforms, includes the boundary layer potentials associated with basic PDEs of mathematical physics, such as the Laplacian, the Lamé system, and the Stokes system. We also consider the limiting case $\alpha =0$ (with $VMO\left(\partial \Omega \right)$ as the natural replacement of ${\mathcal{C}}^{\alpha }\left(\partial \Omega \right)$), and discuss an extension to the scale of Besov spaces.