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# On the Norm of the Beurling-Ahlfors Operator in Several Dimensions

Published:2010-08-26
Printed: Mar 2011
• Tuomas P. Hytönen,
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland
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## Abstract

The generalized Beurling-Ahlfors operator $S$ on $L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate $$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad p^*:=\max\{p,p'\}$$ This improves on earlier results in all dimensions $n\geq 3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.
 MSC Classifications: 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.) 60G46 - Martingales and classical analysis