On harmonic quasiregular mappings in Bergman spaces

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 $\mathbb{D}$.