Proceedings of the Centre for Mathematics and its Applications

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

Xuan Thinh Duong and Alan McIntosh

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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

Source
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

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


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