Imbedding theorems for invariant subspaces of backward shift operator.

A. L. Vol'berg ; S. R. Treil'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 38-51

Résumé

For subspaces $K_\theta^p=H^p\cap\theta\bar H^p_0$, $\theta$ being an inner function in the unit disc $\mathbb D$, we find conditions on a measure in $\operatorname{clos}\mathbb D$ ensuring the imbedding $K_\theta^p\subset L^p(\mu)$, $0$. The main result claims that $K_\theta^p\subset L^p(\mu)$ if there are positive constants $\varepsilon$ and $c$ such that $\mu(\Delta)\leqslant c\cdot r_\Delta$ for every disc $\Delta$ of radius $r_\Delta$ centered on $\mathbb T$ and such that $|\theta(z)|\varepsilon$ for some $z\in\Delta$. Cohn's criterion for the imbedding $K_\theta^2\subset L^2(\mu)$ is obtained as a corollary. It is also shown that a necessary and sufficient condition for $K_\theta^p\subset L^p(\mu)$ must depend on $p$.