Bellman function , two weighted Hilbert transform , and embeddings of the model spaces

This paper is devoted to embedding theorems for the space Kθ, where θ is an inner function in the unit disc D. As we will see soon, the question of embedding of Kθ into L2(μ) is virtually equivalent to the boundedness of two-weight Hilbert transform. This makes the embedding question quite difficult (the general bondedness criteria of Hunt-Muckenhoupt-Wheeden type for two weight Hilbert transform is yet to be found). Here we are not interested in sufficient conditions for the embedding of Kθ into L2(μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that certain natural set of conditions is not sufficient for the embedding of Kθ into L2(μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding of Kθ into L2(μ). Our technique naturally leads to the conclusion that there can be a uniform embedding of all the reproducing kernels of Kθ but the embedding of the whole Kθ into L2(μ) may fail. Moreover, it may happen that the embedding into a potentially larger space L2(|1− θ|2 dμ) fails too.