The classical embedding theorem of Carleson deals with finite positive Borel measures µ on the closed unit disk for which there exists a positive constant c such that f L 2 (µ) ≤ c f H 2 for all f ∈ H 2 , the Hardy space of the unit disk. Lefèvre et al. examined measures µ for which there exists a positive constant c such that f L 2 (µ) ≥ c f H 2 for all f ∈ H 2 . The first type of inequality above was explored with H 2 replaced by one of the model spaces (ΘH 2 ) ⊥ by Aleksandrov, Baranov, Cohn, Treil, and Volberg. In this paper we discuss the second type of inequality in (ΘH 2 ) ⊥ .
In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces H (b), mainly when b is a non-extreme point of the unit ball of H ∞ .Here b belongs to H ∞ 1 , the unit ball in H ∞ , and H ∞ is the Banach algebra of bounded analytic functions on D normed with the supremum norm · ∞ . The space H (b) is often known as the de Branges-Rovnyak space and we will review the basics of this space in a moment. For now, note that when b ∞ < 1, then H (b) is just the classical Hardy space H 2 [16,18] with an equivalent norm while if b is an inner function, meaning |b| = 1 almost everywhere on T = ∂D, then H (b) is the classical model space (bH 2 ) ⊥ = H 2 ⊖ bH 2 . For any b, the space H (b) is contractively contained in H 2 . As is often the case, the properties of H (b) spaces, including direct and reverse Carleson measures, depend on whether or not b is an extreme point ofwhere m is normalized Lebesgue measure on the unit circle T.Let M + (D − ) denote the positive finite Borel measures on the closed unit disk D − . By a reverse Carleson measure for H (b) we mean a 2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.
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