Averaged derivatives and reverse inequalities in quasiregular Bergman spaces

Abstract

We study weighted Bergman spaces of quasiregular mappings on the unit ball in $\mathbb{R}^n$, with weight $(1-|x|)^\alpha$, $\alpha>-1$. For every fixed distortion bound $K\ge1$ and every $p>0$, we characterize Bergman integrability by averaged derivatives. We also give geometric conditions under which integration over a subset, or against another measure, controls the full Bergman integral. When $K$ is sufficiently large, the dominating sets are precisely the relatively dense sets. For comparison, we determine the dominating sets for the M\"obius class $K=1$ in dimensions $n\ge3$. At the critical exponent $p=n+\alpha$, its domination criterion differs from that for every fixed $K>1$. Our results extend aspects of the Bergman theory developed by Nolder~\cite{Nolder90} and by Oleinik and Vuorinen~\cite{OV}, and complement the recent work of Adamowicz and Gonz\'alez~\cite{AG21,AG25} and Adamowicz and Caama\~no~\cite{AC} on quasiregular Hardy spaces.