Hardy spaces and quasiconformal maps in the Heisenberg group
Adamowicz, T., & Fässler, K. (2023). Hardy spaces and quasiconformal maps in the Heisenberg group. Journal of Functional Analysis, 284(6), Article 109832. https://doi.org/10.1016/j.jfa.2022.109832
Published in
Journal of Functional AnalysisDate
2023Copyright
© 2022 The Author(s). Published by Elsevier Inc.
We define Hardy spaces Hp, 0 < p < ∞, for quasiconformal mappings on the Korányi unit ball B in the first Heisenberg group H1. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of p0 (K) > 0 such that every K-quasiconformal map f : B → f (B) ⊂ H1 belongs to Hp for all 0 < p < p0(K). Second, we give two equivalent conditions for the Hp membership of a quasiconformal map f , one in terms of the radial limits of f , and one using a nontangential maximal function of f . As an application, we characterize Carleson measures on B via integral inequalities for quasiconformal mappings on B and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from Rn to H1. A crucial difference between the proofs in Rn
Publisher
ElsevierISSN Search the Publication Forum
0022-1236Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/164888980
Metadata
Show full item recordCollections
Related funder(s)
Research Council of FinlandFunding program(s)
Research costs of Academy Research Fellow, AoF; Academy Research Fellow, AoFLicense
Related items
Showing items with similar title or keywords.
-
Weighted Hardy Spaces of Quasiconformal Mappings
Benedict, Sita; Koskela, Pekka; Li, Xining (Springer, 2022)We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the n-dimensional unit ball using the weight (1−r)n−2+α. We extend the known results for univalent functions on the unit disk. ... -
Uniformization with Infinitesimally Metric Measures
Rajala, Kai; Rasimus, Martti; Romney, Matthew (Springer, 2021)We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to R2R2. Given a measure μμ on such a space, we introduce μμ-quasiconformal maps f:X→R2f:X→R2, ... -
Bi-Sobolev Extensions
Koski, Aleksis; Onninen, Jani (Springer, 2023)We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension ... -
A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group
Adamowicz, Tomasz; Fässler, Katrin; Warhurst, Ben (Springer, 2020)We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group H1. Several auxiliary properties of quasiconformal mappings between subdomains ... -
Singular quasisymmetric mappings in dimensions two and greater
Romney, Matthew (Academic Press, 2019)For all n ≥2, we construct a metric space (X, d)and a quasisymmetric mapping f:[0, 1]n→X with the property that f−1 is not absolutely continuous with respect to the Hausdorff n-measure on X. That is, there exists a Borel ...