PreprintCC BY 4.0arXiv
Characterizations for quasiconformal weighted Bergman spaces
Abstract
We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces Aαp(𝔻). For p>0 and α>−1, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for −1<α≤ 0, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power α is sharp: for every α>0, we construct a bounded Kα-quasiconformal mapping for which the weighted level-set length integral with exponent α+2 diverges.
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BibTeX
@misc{koskela2026characterizations,
title = {{Characterizations for quasiconformal weighted Bergman spaces}},
author = {Pekka Koskela and Chengxi Li and Abhishek Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14267},
eprint = {2610.14267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14267},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterizations for quasiconformal weighted Bergman spaces
AU - Pekka Koskela
AU - Chengxi Li
AU - Abhishek Pandey
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14267
UR - https://arxiv.org/abs/2610.14267
AB - We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces A_α^p(𝔻). For p>0 and α>−1, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for −1<α≤ 0, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power α is sharp: for every α>0, we construct a bounded K_α-quasiconformal mapping for which the weighted level-set length integral with exponent α+2 diverges.
ER -