On Embedding Theorems for Coinvariant Subspaces of the Shift Operator. I

Abstract

Weighted estimates are obtained for the derivatives in the model (shift-coinvariant) subspaces K p Θ , generated by meromorphic inner functions Θ of the Hardy class H p (C +). It is shown that the differentiation operator acts from K p Θ to a space L p (w), where the weight w depends on the function |Θ |, the rate of growth of the argument of Θ along the real line. As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces K p Θ in L p (µ) are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures µ for which the norms · L p (µ) and · p are equivalent on a given model subspace K p Θ , are established.

Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

“…We finish by showing that (3) implies (1). By (ii), the operator T : l 2 −→ Y , (µ n ) −→ n µ n y n , already introduced above, is bounded and has dense range since by construction the canonical system is dense in l 2 and (y n ) n is dense in Y .…”

“…For one component inner functions we have already pointed out the following estimate by Aleksandrov (see [1])…”

“…In the special situation when I is one-component, which means that the level set L(I, ε) = {z ∈ D : |I(z)| < ε} of I is connected for some ε > 0, then Aleksandrov shows the following estimate (see [1])…”

“…We will write S ∈ (C) for short when S satisfies (1). Obviously in this situation we also have the embedding H ∞ |S ⊂ l ∞ , so that S ∈ (C) is equivalent to H ∞ |S = l ∞ .…”

One-component inner functions

“…It was shown that [for instance, [10], p. 355] arclength on {z ∈ D : |u(z)| = ε} is such a measure whenever is connected and η < ε < 1.A thorough study of the class I c was given by Aleksandrov [1] who showed the interesting result that u ∈ I c if and only if there is a constant C = C(u) such that for all a ∈ D Many operator-theoretic applications are given in [1][2][3]7]. In our paper here, we are interested in explicit examples, which are somewhat lacking in the literature.…”