Hardy spaces and quasiregular mappings: averaged derivatives and the $\mathbb{BMO}$ case

TL;DR

This paper investigates Hardy spaces of quasiregular mappings in the unit ball, focusing on averaged derivatives, boundary behavior, and their connections to BMO spaces and PDEs, extending prior results to this setting.

Contribution

It introduces new characterizations of Hardy spaces for quasiregular maps using averaged derivatives and explores their relations with BMO, Carleson measures, and elliptic PDEs.

Findings

Characterization of $ ext{H}^p$ spaces via averaged derivatives.

Establishment of boundary limit and maximal function properties.

Extension of previous results to quasiregular mappings.

Abstract

We study the Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ of quasiregular mappings on the unit ball $\mathbb{B}^n$ in ${\mathbb{R}}^n$ under the appropriate growth and multiplicity conditions. Our focus is on the averaged derivatives of maps and their Harnack and quantitative Harnack estimates. The averaged derivatives are employed to study the non-tangential limit functions and non-tangential maximal functions of quasiregular mappings and to characterize $\mathcal{H}^p$ in the case of finite multiplicity of $f$. Moreover, we study relations between quasiregular mappings, averaged derivatives, BMO spaces and Carleson measures on $\mathbb{B}^n$ and the role of the multiplicity of a map. We also apply our results to the second order elliptic PDEs and $\mathcal{A}$-harmonic equations. Our paper extends results by Astala and Koskela [AK] and Nolder [No1] to the setting of quasiregular maps.