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VOL. 37 | 1999 The $L^p$ boundedness of Riesz transforms associated with divergence form operators
Xuan Thinh Duong, Alan McIntosh

Editor(s) Tim Cranny, Bevan Thompson

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 $\Omega$.

Information

Published: 1 January 1999
First available in Project Euclid: 18 November 2014

zbMATH: 1193.42089

Rights: Copyright © 1999, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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