Let H∞ denote the space of bounded analytic functions on the upper half-plane ℂ+. We prove that each function in the model space H∞ ∩ Θ(Equation Presented) is an operator Lipschitz function on ℝ if and only if the inner function Θ is a usual Lipschitz function, i.e., Θ′ ∈ H∞. Let (OL)′(ℝ) denote the set of all functions f ∈ L∞ whose antiderivative is operator Lipschitz on the real line ℝ. We prove that H∞ ∩ Θ(Equation Presented) ⊂ (OL)′(ℝ) if Θ is a Blaschke product with zeros satisfying the uniform Frostman condition. We also deal with the following questions. When does an inner function Θ belong to (OL)′(ℝ)? When does each divisor of an inner function Θ belong to (OL)′(ℝ)? As an application, we deduce that (OL)′(ℝ) is not a subalgebra of L∞(ℝ). Another application is related to a description of the sets of discontinuity points for the derivatives of operator Lipschitz functions. We prove that a set ℰ,ℰ ⊂ ℝ, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if ℰ is an Fσ set of first category. A considerable proportion of the results of the paper is based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton, and Friedman. We also give a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition. Bibliography: 27 titles.
Original language | English |
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Pages (from-to) | 485-518 |
Number of pages | 34 |
Journal | Journal of Mathematical Sciences (United States) |
Volume | 202 |
Issue number | 4 |
DOIs | |
State | Published - Oct 2014 |
Externally published | Yes |
ID: 87317335