## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Joint Australian-Taiwanese Workshop on Analysis and Applications. Tim Cranny and Bevan Thompson, eds. Proceedings of the Centre for Mathematics and its Applications, v. 37. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1999), 15 - 25

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

Xuan Thinh Duong and Alan McIntosh

#### 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 $\Omega$.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416323120

**Zentralblatt MATH identifier**

1193.42089

#### Citation

Duong, Xuan Thinh; McIntosh, Alan. The $L^p$ boundedness of Riesz transforms associated with divergence form operators. Joint Australian-Taiwanese Workshop on Analysis and Applications, 15--25, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1999. https://projecteuclid.org/euclid.pcma/1416323120