## A version of Burkholder’s theorem for operator-weighted spaces

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- by S. Petermichl and S. Pott
- Proc. Amer. Math. Soc.
**131**(2003), 3457-3461 - DOI: https://doi.org/10.1090/S0002-9939-03-06925-9
- Published electronically: February 14, 2003
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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”.## References

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## Bibliographic Information

**S. Petermichl**- Affiliation: School of Mathematics, Institute of Advanced Studies, Einstein Drive, Princeton, New Jersey 08540
- MR Author ID: 662756
- Email: stefanie@math.msu.edu
**S. Pott**- Affiliation: Department of Mathematics, University of York, York YO10 5DD, United Kingdom
- Email: sp23@york.ac.uk
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 3457-3461 - MSC (2000): Primary 42A50, 47B37; Secondary 42A61
- DOI: https://doi.org/10.1090/S0002-9939-03-06925-9
- MathSciNet review: 1990635