Bilinear forms on the Dirichlet space

Abstract

We show that the bilinear form $B_b\left(f,g\right)=\langle fg,b\rangle$ is bounded on the Dirichlet space of holomorphic functions on the unit disk if and only if $|b^{\prime }{|}^{2}dxdy$ 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.