Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$

TL;DR

This paper characterizes when the integration operator is bounded on the space of bounded analytic functions in a simply connected domain, using new uniform approximation results for Bloch functions, especially for univalent symbols.

Contribution

It provides a necessary and sufficient condition for the boundedness of the integration operator on H-infinity, based on a novel uniform approximation of Bloch functions.

Findings

Characterization of boundedness of the integration operator on H-infinity

New uniform approximation results for Bloch functions

Full characterization for univalent symbols in Volterra operators

Abstract

We obtain a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.