Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 103-112.

Logarithmic growth for weighted Hilbert transforms and vector Hankel operators

Authors: T.A. Gillespie, (1) S. Pott, (2) S. Treil (3) and A. Volberg (4)
Author institution: (1) Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ Scotland, UK
(2) Department of Mathematics, University of York, York Y010 5DD, UK
(3) Brown University, Department of Mathematics, Providence, RI 02912, USA
(4) Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Summary: We give an example of an operator weight $W$ satisfying the operator Hunt-Muckenhoupt-Wheeden $\mathbb{A}_2$ condition, but for which the Hilbert transform on $L^2(W)$ is unbounded. The construction relates weighted boundedness with the boundedness of vector Hankel operators. We establish a relationship between the norm of a vector Hankel operator and a certain natural (but not Nehari-Page) $\bmo$ norm of its symbol, which is logarithmic in dimension.

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