Hardy spaces for quasiregular mappings and composition operators

TL;DR

This paper extends Hardy space theory to certain quasiregular mappings in the plane, showing classical properties hold and exploring composition operators and Carleson measures within this context.

Contribution

It introduces Hardy spaces for quasiregular mappings, characterizes a class of these mappings via composition operators, and relates Carleson measures to Hardy spaces.

Findings

Classical Hardy space properties extend to a specific class of quasiregular mappings.

Characterization of quasiregular mappings through composition operators with quasiconformal symbols.

Establishment of relations between Carleson measures and Hardy spaces for quasiregular mappings.

Abstract

We define Hardy spaces $\mathcal{H}^p$ for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $\mathcal{H}^p$-theory for Quasiconformal Mappings.