We show that the bilinear form is bounded on the Dirichlet space of holomorphic functions on the unit disk if and only if 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.
"Bilinear forms on the Dirichlet space." Anal. PDE 3 (1) 21 - 47, 2010. https://doi.org/10.2140/apde.2010.3.21