Hardy spaces and quasiregular mappings

TL;DR

This paper extends classical Hardy space results to quasiregular mappings in higher dimensions, characterizing these spaces and measures, and exploring their relations to Bergman and harmonic functions under growth and multiplicity conditions.

Contribution

It introduces Hardy spaces for quasiregular mappings with growth and multiplicity conditions, generalizing known results for quasiconformal mappings and analytic functions.

Findings

Characterization of Hardy spaces via non-tangential limits and maximal functions.

Quasiregular maps belong to Hardy spaces for some p depending on dimension and distortion.

Characterization of Carleson measures through integral inequalities for quasiregular mappings.

Abstract

We study Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ for quasiregular mappings on the unit ball $B$ in $\mathbb{R}^n$ which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in $\mathcal{H}^p$. In particular, we characterize $\mathcal{H}^p$ in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to $\mathcal{H}^p$ for some $p=p(n,K)$. Moreover, we provide characterization of Carleson measures on $B$ via integral inequalities for quasiregular mappings on $B$. We also discuss the Bergman spaces of quasiregular mappings and their relations to $\mathcal{H}^p$ spaces and analyze correspondence between results for $\mathcal{H}^p$ spaces and $\mathcal{A}$-harmonic functions. A key difference between the previously known results for quasiconformal mappings in $\mathbb{R}^n$ and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.