Non-tangential limits for analytic Lipschitz functions

Abstract

Let $U$ be a bounded open subset of the complex plane. Let $0<\alpha <1$ and let $A_{\alpha }(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $\alpha$ on the complex plane, are analytic on $U$ and are such that for each $\epsilon >0$, there exists $\delta >0$ such that for all $z$, $w\in U$, $|f(z)-f(w)|\le \epsilon |z-w{|}^{\alpha }$ whenever $|z-w|<\delta$. We show that if a boundary point $x_0$ for $U$ admits a bounded point derivation for $A_{\alpha }(U)$ and $U$ has an interior cone at $x_0$ then one can evaluate the bounded point derivation by taking a limit of a difference quotient over a non-tangential ray to $x_0$. Notably our proofs are constructive in the sense that they make explicit use of the Cauchy integral formula.

arXiv comment9 pages, 1 figure, to appear in the Conference Proceedings of International Conference on Complex Analysis, Potential Theory, and Applications