On the uniqueness question for Hahn-Banach extensions from the space of $\mathcal {L}^1$ analytic functions

Abstract:

Let $\Omega$ be a region of the complex plane, $\mathcal {B}\left ( \Omega \right )$ the space of analytic ${\mathcal {L}^1}$ functions over $\Omega$, and $\kappa \in {\mathcal {L}^\infty }\left ( \Omega \right )$. An evident necessary condition for the linear functional ${\Lambda _\kappa }\left [ \varphi \right ] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal {B}(\Omega ))$ to have a unique Hahn-Banach extension from $\mathcal {B}(\Omega )$ to ${\mathcal {L}^1}(\Omega )$ is that $|| {{\Lambda _{\kappa |G}}} || = {|| {\kappa |G} ||_\infty }$ for every restriction $\kappa |G$ of $\kappa$ to a subregion $G$ of $\Omega$. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether $\left | {\kappa (z)} \right |$ is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.