解析Lipschitz函数的赫尔德条件和$τ$ -尖峰
Hölder conditions and $τ$-spikes for analytic Lipschitz functions
摘要：
设$U$是$\mathbb{C}$的一个开子集，其边界点为$x_0$，$A_α(U)$是$U$上解析的属于LipPDMATHABMQ（小Lipschitz类）的函数空间.我们考虑条件$S= \displaystyle \sum_{n=1}^{\infty}2^{(t+λ+1)n}M_*^{1+α}(A_n \setminus U)< \infty,$，其中$t$是非负整数，$0<λ<1$，$M_*^{1+α}$是较低的$1+α$维Hausdorff含量，以及$A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}\}$ .这类似于有界点导子在$A_α(U)$上在$x_0$的一个充分必要条件。我们表明，$S= \infty$意味着$x_0$是$(t+λ)$ -穗$A_α(U)$和，如果$S<\infty$和$U$满足锥条件，那么$t$ -次导数的功能在$A_α(U)$满足Hölder条件$x_0$的非切向的方法。
英文摘要：
Let $U$ be an open subset of $\mathbb{C}$ with boundary point $x_0$ and let $A_α(U)$ be the space of functions analytic on $U$ that belong to lip$α(U)$, the "little Lipschitz class". We consider the condition $S= \displaystyle \sum_{n=1}^{\infty}2^{(t+λ+1)n}M_*^{1+α}(A_n \setminus U)< \infty,$ where $t$ is a non-negative integer, $0<λ<1$, $M_*^{1+α}$ is the lower $1+α$ 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 $A_α(U)$ at $x_0$. We show that $S= \infty$ implies that $x_0$ is a $(t+λ)$-spike for $A_α(U)$ and that if $S<\infty$ and $U$ satisfies a cone condition, then the $t$-th derivatives of functions in $A_α(U)$ satisfy a Hölder condition at $x_0$ for a non-tangential approach.
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