Bounded Point Derivations and Functions of Bounded Mean Oscillation

Abstract

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.