Derivatives at the Boundary for Analytic Lipschitz Functions

TL;DR

This paper investigates the boundary behavior of Lipschitz continuous holomorphic functions, establishing conditions under which boundary derivatives can be obtained via classical difference quotients approaching a boundary point with full area density.

Contribution

It links the abstract concept of bounded point derivations to classical derivatives for Lipschitz holomorphic functions at boundary points.

Findings

Bounded point derivations can be evaluated by classical difference quotients.

Approach from sets with full area density at boundary points is sufficient.

Results connect functional analysis with classical complex analysis techniques.

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 $b$.