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

DOI:
https://doi.org/10.1090/S0002-9939-03-06925-9

Published electronically:
February 14, 2003

MathSciNet review:
1990635

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Abstract: Let be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space . We prove that if the dyadic martingale transforms are uniformly bounded on for each dyadic grid in , then the Hilbert transform is bounded on as well, thus providing an analogue of Burkholder's theorem for operator-weighted -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

Email:
stefanie@math.msu.edu

**S. Pott**

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

Email:
sp23@york.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-03-06925-9

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