Intrinsic Hardy-Orlicz spaces of conformal mappings

TL;DR

This paper introduces a new class of Hardy-Orlicz spaces for conformal mappings based on intrinsic path distances, establishing conditions under which they coincide with classical spaces and providing counterexamples.

Contribution

It defines intrinsic Hardy-Orlicz spaces for conformal maps and compares them to classical spaces, revealing when they are equivalent or distinct.

Findings

Intrinsic Hardy-Orlicz spaces coincide with classical spaces if the Orlicz function is doubling.

An example shows the intrinsic space can be strictly smaller than the classical space.

The paper characterizes conditions for space equivalence and difference.

Abstract

We define a new type of Hardy-Orlicz spaces of conformal mappings on the unit disk where in place of the value |f(x)| we consider the intrinsic path distance between f(x) and f(0) in the image domain. We show that if the Orlicz function is doubling then these two spaces are actually the same, and we give an example when the intrinsic Hardy-Orlicz space is strictly smaller.