, Volume 48, Issue 6, pp 984997
First online:
Local stability of mappings with bounded distortion on Heisenberg groups
 D. V. IsangulovaAffiliated withSobolev Institute of Mathematics Email author
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This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K  1} \) in the uniform norm and order of closeness K − 1 in the Sobolev norm L _{p} ^{1} for all \(p < \tfrac{C}{{K  1}}\).
In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness \(\sqrt {K  1} \) in the uniform norm and order of closeness K − 1 in the Sobolev norm L _{p} ^{1} for all \(p < \tfrac{C}{{K  1}}\). We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm.
Keywords
Heisenberg group mapping with bounded distortion coercive estimate stability Title
 Local stability of mappings with bounded distortion on Heisenberg groups
 Journal

Siberian Mathematical Journal
Volume 48, Issue 6 , pp 984997
 Cover Date
 200711
 DOI
 10.1007/s1120200701016
 Print ISSN
 00374466
 Online ISSN
 15739260
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Heisenberg group
 mapping with bounded distortion
 coercive estimate
 stability
 Authors

 D. V. Isangulova ^{(1)}
 Author Affiliations

 1. Sobolev Institute of Mathematics, Novosibirsk, Russia