Reverse Carleson Embeddings for Model Spaces

TL;DR

This paper investigates reverse Carleson embedding inequalities for model spaces, extending classical results by exploring lower bounds of measures on these spaces, which has implications for understanding their structure and measures.

Contribution

It introduces and analyzes reverse Carleson embeddings for model spaces, providing new insights into measure inequalities in these functional spaces.

Findings

Established conditions for reverse Carleson embeddings in model spaces

Extended classical embedding theorems to a new inequality type

Provided theoretical framework for measure inequalities in Hardy space subspaces

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 $(\Theta H^2)^{\perp}$.