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On the Norm of the Beurling–Ahlfors Operator in Several Dimensions

Published online by Cambridge University Press:  20 November 2018

Tuomas P. Hytönen
Tuomas P. Hytönen*
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finlande-mail: [email protected]
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Abstract

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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

$$||S||L\left( {{L}^{p}}\left( {{\mathbb{R}}^{n}};\Lambda \right) \right)\le \left( n/2+1 \right)\left( {{p}^{*}}-1 \right),\,\,\,{{p}^{*}}:=\,\max \{p,\,{{p}^{'}}\}.$$
.

This improves on earlier results in all dimensions $n\,\ge \,3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

Keywords

42B2060G46
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 113 - 125
Copyright
Copyright © Canadian Mathematical Society 2011

References

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