Vol. 3, No. 1, 2010

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Bilinear forms on the Dirichlet space

Nicola Arcozzi, Richard Rochberg, Eric Sawyer and Brett D. Wick

Vol. 3 (2010), No. 1, 21–47

We show that the bilinear form Bb(f,g) = fg,b is bounded on the Dirichlet space of holomorphic functions on the unit disk if and only if |b|2dxdy is a Carleson measure for the Dirichlet space. This is completely analogous to the results for boundedness of Hankel forms on the Hardy and Bergman spaces, but the proof is quite different, relying heavily on potential-theoretic constructions.

Dirichlet space, Hankel form
Mathematical Subject Classification 2000
Primary: 30C85, 47B35, 31C25
Received: 8 July 2009
Revised: 99 2999
Accepted: 5 August 2009
Published: 4 March 2010
Nicola Arcozzi
Dipartimento di Matematica
Università di Bologna
40127 Bologna
Richard Rochberg
Department of Mathematics
Washington University
St. Louis, MO 63130
United States
Eric Sawyer
Department of Mathematics and Statistics
McMaster University
Hamilton, ON  L8S 4K1
Brett D. Wick
Department of Mathematics
LeConte College
1523 Greene Street
University of South Carolina
Columbia, SC 29208
United States