Reverse Carleson Embeddings for Model Spaces

Abstract

The classical embedding theorem of Carleson deals with finite positive Borel measures $\mu$ on the closed unit disk for which there exists a positive constant $c$ such that $|f|_{L^2(\mu)} \leq c |f|_{H^2}$ for all $f \in H^2$, the Hardy space of the unit disk. Lef\'evre et al. examined measures $\mu$ for which there exists a positive constant $c$ such that $\|f\|_{L^2(\mu)} \geq c |f|_{H^2}$ for all $f \in H^2$. The first type of inequality above was explored with $H^2$ replaced by one of the model spaces $(\Theta H^2)^{\perp}$ by Aleksandrov, Baranov, Cohn, Treil, and Volberg. In this paper we discuss the second type of inequality in $(\Theta H^2)^{\perp}$.