Averaged derivatives and reverse inequalities in quasiregular Bergman spaces
Abstract
We study weighted Bergman spaces of quasiregular mappings on the unit ball in ℝn, with weight (1−|x|)α, α>−1. For every fixed distortion bound K≥1 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 Möbius class K=1 in dimensions n≥3. At the critical exponent p=n+α, 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 González~\cite{AG21,AG25} and Adamowicz and Caamaño~\cite{AC} on quasiregular Hardy spaces.
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BibTeX
@misc{das2026averaged,
title = {{Averaged derivatives and reverse inequalities in quasiregular Bergman spaces}},
author = {Suman Das},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14837},
eprint = {2610.14837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14837},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Averaged derivatives and reverse inequalities in quasiregular Bergman spaces
AU - Suman Das
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14837
UR - https://arxiv.org/abs/2610.14837
AB - We study weighted Bergman spaces of quasiregular mappings on the unit ball in ℝ^n, with weight (1−|x|)^α, α>−1. For every fixed distortion bound K≥1 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 Möbius class K=1 in dimensions n≥3. At the critical exponent p=n+α, 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 González~\cite{AG21,AG25} and Adamowicz and Caamaño~\cite{AC} on quasiregular Hardy spaces.
ER -