Reverse Carleson embeddings for model spaces (Q2854249)

Let \(H^2\) be the classical Hardy space of the open unit disk \(\mathbb{D}\) with norm \(\|\cdot\|_2\). Moreover, let \(M_+(\mathbb{D})\) denote the finite positive Borel measures on \(\mathbb{D}\), and, for \(\mu \in M_+(\mathbb{D})\), let \(\|\cdot\|_{\mu}\) be the norm in \(L^2(\mu)\). By Carleson, \(H^2\) can be continuously embedded into \(L^2(\mu)\), that is, NEWLINE\[NEWLINE \exists C >0\; \text{ such that } \; \| f\|_{\mu} \leq C \| f\|_2\; \text{ for every }\; f \in H^2 NEWLINE\]NEWLINE if and only if \(\sup_{I} \frac{\mu(S(I))}{m(I)} < \infty\), where the supremum is taken over all arcs \(I\) of the unit circle \(\mathbb{T}\), \(m:= \frac{d \Theta}{2 \pi}\) is the normalized Lebesgue measure on \(\mathbb{T}\) and \(S(I)\) is the Carleson window NEWLINE\[NEWLINE S(I):= \left \{ |z| \leq 1; \; \frac{z}{|z|} \in I, \; 1-|z| \leq \frac{m(I)}{2} \right \}. NEWLINE\]NEWLINE \textit{P. Lefèvre} et al. [Stud. Math. 202, No. 2, 123--144 (2011; Zbl 1232.46024)] studied the reverse embedding problem. More precisely, they investigated when the embedding is injective with closed range. They showed that if \(\mu\) is a Carleson measure, then the reverse embedding happens if and only if the following inequality holds NEWLINE\[NEWLINE \inf_{I} \frac{\mu(S(I))}{m(I)} > 0. NEWLINE\]NEWLINE This paper deals with reverse embeddings for model spaces \((\Theta H^2)^{\perp}= H^2 \ominus \Theta H^2\), where \(\Theta\) is a non-constant inner function.