, Volume 48, Issue 6, pp 984-997

Local stability of mappings with bounded distortion on Heisenberg groups

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Abstract

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.

Original Russian Text Copyright © 2007 Isangulova D. V.
The author was partially supported by the Russian Foundation for Basic Research (Grant 06-01-00735), the State Maintenance Program for the Junior Scientists and the Leading Scientific Schools of the Russian Federation (Grant NSh-8526.2006.1), and INTAS (Grant YSF 03-55-905).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1228–1245, November–December, 2007.