Hölder 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 $S=\sum _{n=1}^{\infty }{2}^{(t+\lambda +1)n}{M}_{\ast }^{1+\alpha }(A_n\setminus U)<\infty ,$ where $t$ is a non-negative integer, $0<\lambda <1$, $M_{\ast }^{1+\alpha }$ 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 Hölder condition at $x_0$ for a non-tangential approach.

arXiv comment16 pages, 1 figure. arXiv admin note: text overlap with arXiv:1811.11370