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Smooth functions in the range of a Hankel operator. (English) Zbl 0821.30026
Of concern are Hankel operators of the form \(H_{\overline{\theta}}\), where the symbol \(\overline{\theta}\) is the complex conjugate of an inner function \(\theta\). We are interested in conditions on a function \(f \in H^ 2\) under which the image \(H_{\overline{\theta}} f\) becomes smooth, in a sense, on the unit circle. For various smoothness classes \(X\) (such as the Gevrey, Sobolev and Besov spaces), we find sharp quantitative conditions that ensure either the inclusion \(H_{\overline{\theta}} f \in X\) or the totality of inclusions \(H_{\overline {\theta}^ k} f \in X\) for all \(k \in \mathbb{N}\), accompanied with natural norm estimates. Also, the paper contains a number of differential (Bernstein-type) inequalities for pseudocontinuable functions and a result in approximation theory.

MSC:
30D55 \(H^p\)-classes (MSC2000)
30D50 Blaschke products, etc. (MSC2000)
30D45 Normal functions of one complex variable, normal families
47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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