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Transactions of the American Mathematical Society

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On singular integral and martingale transforms

Authors: Stefan Geiss, Stephen Montgomery-Smith and Eero Saksman
Journal: Trans. Amer. Math. Soc. 362 (2010), 553-575
MSC (2000): Primary 60G46, 42B15; Secondary 42B20, 46B09, 46B20
Published electronically: September 11, 2009
MathSciNet review: 2551497
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Abstract: Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space $ X$ equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on $ L^p_X(\mathbf{R}^2)$ with $ p\in (1,\infty).$ Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals $ p^*-1$ with $ p^*:= \max \{p, (p/(p-1))\}$, where the novelty is the lower bound.

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Additional Information

Stefan Geiss
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland

Stephen Montgomery-Smith
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Eero Saksman
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 Helsinki, Finland

Keywords: UMD property, singular integrals, martingale transforms
Received by editor(s): January 29, 2007
Published electronically: September 11, 2009
Additional Notes: The first and the last author are supported by Project #110599 of the Academy of Finland.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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