Bilinear forms on vector hardy spaces

Let Phi:(H) over tilde(2)Hx (H) over tilde(2)H-->C be a bilinear form on vector Hardy space, Introduce the symbol phi of Phi by (phi(z(1), z(2)), a x b) = Phi(k(z1) x a, k(z2) x b), where k(w) is the reproducing kernel for w is an element of D. We show that Phi extends to a bounded bilinear form on (H) over tilde(1)Hx (H) over tilde(1)H provided that the gradient \\partial derivative(1) partial derivative(2) phi\\(Bi(H,H))A(dz(1))A(dz(2)) defines a Carleson measure in the bidisc D-2. We obtain a sufficient condition for Phi to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.