# Bounded point derivations and functions of bounded mean oscillation

## TL;DR

This paper characterizes when the space of VMO functions analytic on a subset of the complex plane admits bounded point derivations at boundary points, using conditions based on Hausdorff content.

## Contribution

It provides necessary and sufficient conditions for bounded point derivations in terms of Hausdorff content, extending understanding of derivations in analytic VMO spaces.

## Key findings

- Conditions for bounded point derivations are characterized by lower 1-dimensional Hausdorff content.
- Results are analogous to those for other function spaces.
- Provides a geometric criterion for the existence of derivations.

## Abstract

Let $X$ be a subset of the complex plane 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 $|f^{(t)}(x_0)|\leq C ||f||_{BMO}$ for all functions in $VMO(X)$ that are analytic on $X \cup \{x_0\}$. 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.