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

Abstract:

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 $|\Theta ’|$, 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 $\|\cdot \|_{L^p(\mu )}$ and $\|\cdot \|_p$ are equivalent on a given model subspace $K^p_{\Theta }$, are established.