Ahlfors-Beurling conformal invariant and relative capacity of compact sets

TL;DR

This paper introduces a new concept called relative capacity based on the Ahlfors-Beurling conformal invariant, explores its properties, and shows its equivalence to half-plane capacity in specific cases, with applications to bounded holomorphic functions.

Contribution

It defines the relative capacity using the Ahlfors-Beurling invariant, analyzes its asymptotic behavior, and establishes its properties and applications, extending classical capacity concepts.

Findings

Relative capacity matches half-plane capacity when domain is the upper half-plane.

Asymptotic expansion of the conformal invariant characterizes the relative capacity.

Properties under symmetrization and geometric transformations are established.

Abstract

For a given domain $D$ in the extended complex plane $\bar{\mathbb C}$ with an accessible boundary point $z_0 \in \partial D$ and for a subset $E \subset {D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\rc E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $r(D\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0.$ Here $r(G,z)$ denotes the inner radius at $z$ of the connected component of the set $G$ containing the point $z.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $D$ is the upper half plane and $z_0=\infty$ the capacity $\rc E$ coincides with the well-known half-plane capacity ${\hc} E.$ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.