Bilinear Forms on the Dirichlet Space

Original · EN

Let D be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function b we define the associated Hankel type bilinear form, initially for polynomials f and g, by Tb(f,g):= < fg,b >D, where we are looking at the inner product in the space D. We let the norm of Tb denotes its norm as a bilinear map from D to the complex numbers. We say a function b is in the space X if the measure dμb:=| b′(z)| ²dA is a Carleson measure for D and norm X by bₓ:=| b(0)| + | b′(z)| ²dA(D)¹/². Our main result is Tb is bounded if and only if b and Tb× D≈ bₓ.