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\( \overline{H^{\infty }} \) 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\( \overline{H^{\infty }} \) ⊂ (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 *F*
_{σ} 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.

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Translated from *Zapiski Nauchnykh Seminarov POMI*, Vol. 416, 2013, pp. 5–58.

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Aleksandrov, A.B. Operator Lipschitz Functions and Model Spaces.
*J Math Sci* **202**, 485–518 (2014). https://doi.org/10.1007/s10958-014-2057-y

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DOI: https://doi.org/10.1007/s10958-014-2057-y