Derivatives at the Boundary for Analytic Lipschitz Functions

Abstract

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha <1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of classical difference quotients, for approach from a set having full area density at $b$.

20 pages. This complements the investigation reported in arXiv:1402.4307, and assumes background, notation and lemmas from that paper

References in corpus (1)

• Boundary smoothness of analytic functions