Let X be a compact subset of the complex plane with the property that every relatively open subset of X has positive area and let \(A_0(X)\) denote the space of VMO functions that are analytic on X. \(A_0(X)\) is said to admit a bounded point derivation of order t at a point \(x_0 \in \partial X\) if there exists a constant C such that \(|f^{(t)}(x_0)|\le C \Vert f\Vert _{{\text {BMO}}}\) for all functions in \({\text {VMO}}(X)\) that are analytic on X. In this paper, we give necessary and sufficient conditions in terms of lower 1-dimensional Hausdorff content for \(A_0(X)\) to admit a bounded point derivation at \(x_0\). These conditions are similar to conditions for the existence of bounded point derivations on other functions spaces.

令X为复平面的紧子集，其性质为X的每个相对开放子集均具有正面积，令\（A_0（X）\）表示对X进行分析的VMO函数的空间。\（A_0（X）\）被认为承认的顺序的有界点推导吨在点\（X_0 \在\局部X \）如果存在一个常数C ^使得\（| F ^ {（T）} （X_0）| \文件ç\ Vert的˚F\ Vert的_ {{\文本{BMO}}} \）用于所有功能\（{\文本{VMO}}（X）\）是上解析X。在本文中，我们针对\（A_0（X）\）的较低一维Hausdorff含量给出了充要条件，以允许在\（x_0 \）处有界点导数。这些条件类似于其他函数空间上存在有界点推导的条件。