Abstract
We prove a global Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non singular and satisfies some growth conditions at infinity. An estimate for the Lipschitz constant (with respect to the Carnot–Carathéodory distance in \(\mathbb{H}\)) of a continuously Pansu differentiable map is included. This gives a characterization of (continuously Pansu differentiable) globally biLipscitz deformations of \(\mathbb{H}\) in term of a pointwise estimate of their differential.
Keywords: Inverse problem, Heisenberg group
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1. Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford (2004)
2. Arcozzi, N., Morbidelli, D.: Stability of biLipschiz maps in the Heisenberg group. Preprint 2005, available at http://arxiv.org/abs/math/0508474
3. Balogh, Z.: Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group. J. Anal. Math. 83, 289–312 (2001)
4. Capogna, L., Tang, P.: Uniform domains and quasiconformal mappings on the Heisenberg group. Manuscripta Math. 86, 267–281 (1995)
5. Balogh, Z., Hoeffer-Isenegger, R., Tyson, J.: Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group. Erg. Theory and Dynam. Systems (to appear)
6. Capogna, L.: Regularity of quasi-linear equations in the Heisenberg group. Comm. Pure Appl. Math. 50, 867–889 (1997)
7. John, F.: Rotation and strain. Comm. Pure Appl. Math. 14, 391–413 (1961)
8. Hadamard, J.: Sur les transformations ponctuelles. Bull. Soc. Mat. France 34, 71–84 (1906)
9. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 688, (2000)
10. Korányi, A., Reimann, H.M.: Quasiconformal mappings on the Heisenberg group. Invent. Math. 80, 309–338 (1985)
11. Lévy, P.: Sur les fonctions de lignes implicites. Bull. Soc. Mat. France 48, 13–27 (1920)
12. Magnani, V.: Elements of geometric measure theory on sub-Riemannian groups. Scuola Normale Superiore, Pisa (2002)
13. Pansu, P.: Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. 129, 1–60 (1989)
14. Schwartz, J.: Nonlinear functional analysis. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. Notes on Mathematics and its Applications. Gordon and Breach Science Publishers, New York-London-Paris (1969)
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Arcozzi, N., Morbidelli, D. A global Inverse Map Theorem and biLipschitz maps in the Heisenberg group . Ann. Univ. Ferrara 52, 189–197 (2006). https://doi.org/10.1007/s11565-006-0015-4
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DOI: https://doi.org/10.1007/s11565-006-0015-4