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 \(S= \sum _{n=1}^{\infty }2^{(t+\lambda +1)n}M_*^{1+\alpha }(A_n \setminus U)< \infty ,\) 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 Hölder condition at \(x_0\) for a non-tangential approach.