Weighted Bernstein-type inequalities, and embedding theorems for the model subspaces

Weighted estimates are obtained for the derivatives in the model (shift-coinvariant) subspaces $K^p_{\Theta}$, generated by meromorphic inner functions $\Theta$ of the Hardy class $H^p(\mathbb{C} ^+)$. It is shown that the differentiation operator acts from $K^p_{\Theta}$ to a space $L^p(w)$, where the weight $w$ depends on the function $\vert\Theta'\vert$, the rate of growth of the argument of $\Theta$ along the real line.

As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces $K^p_{\Theta}$ in $L^p(\mu)$ are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures $\mu$ for which the norms $\Vert\cdot\Vert _{L^p(\mu)}$ and $\Vert\cdot\Vert _p$ are equivalent on a given model subspace $K^p_{\Theta}$, are established.