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.

Keywords: Weighted Hilbert transform, vector Hankel operators

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