On harmonic quasiregular mappings in Bergman spaces

TL;DR

This paper extends classical harmonic analysis results to harmonic quasiregular and quasiconformal mappings in Bergman spaces, establishing new conditions under which harmonic functions belong to these spaces.

Contribution

It proves the Hardy-Littlewood theorem for harmonic quasiregular functions and identifies a range of p for which univalent harmonic functions are in Bergman spaces.

Findings

Hardy-Littlewood theorem holds for harmonic quasiregular functions.

Univalent harmonic functions belong to Bergman spaces for certain p.

Results extend to harmonic quasiconformal mappings.

Abstract

A classical result of Hardy and Littlewood says that if $f=u+iv$ is analytic in the unit disk $\mathbb{D}$ and $u$ is in the harmonic Bergman space $a^p$ ($0<p<\infty$), then $v$ is also in $a^p$. This complements a celebrated result of M. Riesz on Hardy spaces, which only holds for $1<p<\infty$. These results do not extend directly to complex-valued harmonic functions. We prove that the Hardy-Littlewood theorem holds for a harmonic function $f=u+iv$ if we place the assumption that $f$ is quasiregular in $\mathbb{D}$. This makes further progress on the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Then we consider univalent harmonic mappings in $\mathbb{D}$ and study their membership in Bergman spaces. In particular, we produce a non-trivial range of $p>0$ such that every univalent harmonic function $f$ (and the partial derivatives $f_\theta,\, rf_r$) is of class $a^p$. This result extends nicely to harmonic quasiconformal mappings in $\mathbb{D}$.