论文标题
Hölder条件和$τ$ - 分析Lipschitz功能
Hölder conditions and $τ$-spikes for analytic Lipschitz functions
论文作者
论文摘要
让$ u $为$ \ mathbb {c} $的开放子集，边界点$ x_0 $，让$a_α（u）$是属于lip $α（u）$的$ u $的函数分析的空间，是“小lipschitz class”。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，$ t $ $ t $是非简化的，$ 0 <λ<λ<λ<λ<λ $ 1+α$ dimensional hausdorff content，$ a_n = \ {z：2^{ - n-1} <| z-x_0 | <2^{ - n} \} $。这类似于$ x_0 $的$a_α（u）$上的有限点派生的必要条件。 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.
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.
