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A version of Burkholder's theorem for operator-weighted spaces

Authors: S. Petermichl and S. Pott
Journal: Proc. Amer. Math. Soc. 131 (2003), 3457-3461
MSC (2000): Primary 42A50, 47B37; Secondary 42A61
Published electronically: February 14, 2003
MathSciNet review: 1990635
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Abstract: Let $W$ be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space $\mathcal{H}$. We prove that if the dyadic martingale transforms are uniformly bounded on $L^2_{\mathbb{R} }(W)$ for each dyadic grid in $\mathbb{R} $, then the Hilbert transform is bounded on $L^2_{\mathbb{R} }(W)$ as well, thus providing an analogue of Burkholder's theorem for operator-weighted $L^2$-spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts''.

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Additional Information

S. Petermichl
Affiliation: School of Mathematics, Institute of Advanced Studies, Einstein Drive, Princeton, New Jersey 08540

S. Pott
Affiliation: Department of Mathematics, University of York, York YO10 5DD, United Kingdom

Keywords: Operator-weighted inequalities, Hilbert transform, martingale transforms, UMD spaces
Received by editor(s): August 19, 2001
Received by editor(s) in revised form: May 28, 2002
Published electronically: February 14, 2003
Additional Notes: The second author gratefully acknowledges support by EPSRC
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society