The $L^p$ boundedness of Riesz transforms associated with divergence form operators

Abstract

Let $A$ be a divergence form elliptic operator associated with a quadratic form on $\Omega$ where $\Omega$ is the Euclidean space $\mathbb{R}^n$ or a domain of $\mathbb{R}^n$. Assume that $A$ generates an analytic semigroup $e^{-tA}$ on $L^2(\Omega)$ which has heat kernel bounds of Poisson type, and that the generalised Riesz transform $\nabla A^{-l/2}$ is bounded on $L^2(\Omega)$. We then prove that $\nabla A^{-1/2}$ is of weak type $(l,l)$, hence bounded on $L^p(\Omega)$ for $l \leq p \leq 2$. No specific assumptions are made concerning the Hölder continuity of the coefficients or the smoothness of the boundary of $\Omega$.