Commutator Lipschitz Functions and Analytic Continuation

Abstract

Let \( \mathfrak{F} \)₀ and \( \mathfrak{F} \) be perfect subsets of the complex plane ℂ. Assume that \( \mathfrak{F} \)₀ ⊂ \( \mathfrak{F} \) and the set \( \Omega \overset{\mathrm{def}}{=}\mathfrak{F}\backslash {\mathfrak{F}}_0 \) is open. We say that a continuous function f : \( \mathfrak{F} \) → ℂ is an analytic continuation of a function f₀ : \( \mathfrak{F} \)₀ → ℂ if f is analytic on Ω and f|\( \mathfrak{F} \)₀ = f₀. In the paper, it is proved that if \( \mathfrak{F} \) is bounded, then the commutator Lipschitz seminorm of the analytic continuation f coincides with the commutator Lipschitz seminorm of f₀. The same is true for unbounded \( \mathfrak{F} \) if some natural restrictions concerning the behavior of f at infinity are imposed.