Bounded point derivations on Campanato spaces

TL;DR

This paper characterizes when bounded point derivations exist on vanishing Campanato spaces near a point, using Hausdorff contents, generalizing previous results for other function spaces.

Contribution

It provides a necessary and sufficient condition based on Hausdorff contents for bounded point derivations on Campanato spaces, extending known criteria.

Findings

Characterization of bounded point derivations using Hausdorff contents

Generalization of conditions from other function spaces

Necessary and sufficient criteria established

Abstract

Let $X$ be a compact subset of the complex plane and $x \in X$. A necessary and sufficient condition is given in terms of Hausdorff contents for the existence of a bounded point derivation at $x$ on the space of vanishing Campanato functions that are analytic in a neighborhood of $X$. This generalizes many known conditions for the existence of bounded point derivations on other function spaces.