COMMUTATOR LIPSCHITZ FUNCTIONS AND ANALYTIC CONTINUATION

摘要

Let (F)_0 and (F) be perfect subsets of the complex plane C. Assume that (F)_0 is contained in (F) and the set Ω (def=) (F)(F)_0 is open. We say that a continuous function f : (F) → C is an analytic continuation of a function f_0 : (F)_0 → C if f is analytic on Ω and f|(F)_0 = f_0. In the paper, it is proved that if F is bounded, then the commutator Lipschitz seminorm of the analytic continuation f coincides with the commutator Lipschitz seminorm of f_0. The same is true for unbounded (F) if some natural restrictions concerning the behavior of f at infinity are imposed. Bibliography: 14 titles.