H\"older conditions and $\tau$-spikes for analytic Lipschitz functions

Abstract

Let $U$ be an open subset of $\mathbb{C}$ with boundary point $x_0$ and let $A_{\alpha}(U)$ be the space of functions analytic on $U$ that belong to lip$\alpha(U)$, the "little Lipschitz class". We consider the condition $S= \displaystyle \sum_{n=1}^{\infty}2^{(t+\lambda+1)n}M_*^{1+\alpha}(A_n \setminus U)< \infty,$ where $t$ is a non-negative integer, $0<\lambda<1$, $M_*^{1+\alpha}$ is the lower $1+\alpha$ dimensional Hausdorff content, and $A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}\}$. This is similar to a necessary and sufficient condition for bounded point derivations on $A_{\alpha}(U)$ at $x_0$. We show that $S= \infty$ implies that $x_0$ is a $(t+\lambda)$-spike for $A_{\alpha}(U)$ and that if $S<\infty$ and $U$ satisfies a cone condition, then the $t$-th derivatives of functions in $A_{\alpha}(U)$ satisfy a H\"older condition at $x_0$ for a non-tangential approach.