Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings

We study weighted norm inequalities for the derivatives (Bernstein-type inequalities) in the shift-coinvariant subspaces KΘp of the Hardy class Hp in the upper half-plane. It is shown that the differentiation operator acts from KΘp to certain spaces of the form Lp(w), where the weight w(x) depends on the density of the spectrum of Θ near the point x of the real line. We discuss an application of the Bernstein-type inequalities to the problems of the description of measures μ, for which KΘp⊂Lp(μ), and of compactness of such embeddings. New versions of Carleson-type embedding theorems are obtained generalizing the theorems due to W.S. Cohn and A.L. Volberg-S.R. Treil.