Bilinear Forms on the Dirichlet Space

Hankel operators on the Hardy space of the disk, H (D) , can be studied as linear operators from H (D) to its dual space, as conjugate linear operators from H (D) to itself, or, in the viewpoint we will take here, as bilinear functionals on H (D) × H (D) . In that formulation, given a holomorphic symbol function b we consider the bilinear Hankel form, defined initially for f, g in P (D) , the space of polynomials, by Sb (f, g) := 〈fg, b〉H2 . The norm of Sb is ‖Sb‖H2×H2 = sup {|Sb (f, g)| : ‖f‖H2 = ‖g‖H2 = 1} . Nehari’s classical criterion for the boundedness of Sb can be cast in modern language using Fefferman’s duality theorem. We say a positive measure μ on the disk is a Carleson measure for H if