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TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

Part of: Functions of several variables Lie groups

Published online by Cambridge University Press:  17 June 2013

VALENTINO MAGNANI
VALENTINO MAGNANI*
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127, Pisa email magnani@dm.unipi.it
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Abstract

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We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

Keywords

stratified groupsimplicit function theoremmean value inequalityrank theoremintrinsic submanifolds

MSC classification

Secondary: 22E30: Analysis on real and complex Lie groups 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B12: Calculus of vector functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 1 , August 2013 , pp. 76 - 128
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Ambrosio, L. and Tilli, P., Selected Topics on Analysis in Metric Spaces (Oxford University Press, 2003).CrossRefGoogle Scholar
Capogna, L. and Cowling, M., ‘Conformality and Q-Harmonicity in Carnot groups’, Duke Math. J. 135 (3) (2006), 455–479.CrossRefGoogle Scholar
Federer, H., Geometric Measure Theory (Springer, 1969).Google Scholar
Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups (Princeton University Press, 1982).Google Scholar
Franchi, B., Serapioni, R. and Serra Cassano, F., ‘Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups’, Comm. Anal. Geom. 11 (5) (2003), 909–944.Google Scholar
Franchi, B., Serapioni, R. and Serra Cassano, F., ‘On the structure of finite perimeter sets in step 2 Carnot groups’, J. Geom. Anal. 13 (3) (2003), 421–466.CrossRefGoogle Scholar
Franchi, B., Serapioni, R. and Serra Cassano, F., ‘Regular submanifolds, graphs and area formula in Heisenberg groups’, Adv. Math. 211 (1) (2007), 152–203.CrossRefGoogle Scholar
Gromov, M., ‘Carnot–Carathéodory spaces seen from within’, in: Subriemannian Geometry, Progress in Mathematics, 144 (eds. Bellaiche, A. and Risler, J.) (Birkhäuser, Basel, 1996).Google Scholar
Hajlasz, P. and Koskela, P., ‘Sobolev met Poincaré’, Mem. Amer. Math. Soc. 145 (2000).Google Scholar
Hebisch, W. and Sikora, A., ‘A smooth subadditive homogeneous norm on a homogeneous group’, Studia Math. 96 (3) (1990), 231–236.CrossRefGoogle Scholar
Herstein, I. N., Algebra, Editori riuniti (1994), translation from Topics in Algebra.Google Scholar
Kaplan, A., ‘Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms’, Trans. Amer. Math. Soc. 258 (1) (1980), 147–153.CrossRefGoogle Scholar
Kirchheim, B. and Serra Cassano, F., ‘Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (4) (2004), 871–896.Google Scholar
LLoyd, N. G., Degree Theory (Cambridge University Press, 1978).Google Scholar
Magnani, V., ‘Differentiability and area formula on stratified Lie groups’, Houston J. Math. 27 (2) (2001), 297–323.Google Scholar
Magnani, V., ‘Elements of geometric measure theory on sub-Riemannian groups’, PhD Theses series of Scuola Normale Superiore, 2002.Google Scholar
Magnani, V., ‘On a general coarea inequality and applications’, Ann. Acad. Sci. Fenn. Math. 27 (2002), 121–140.Google Scholar
Magnani, V., ‘The coarea formula for real-valued Lipschitz maps on stratified groups’, Math. Nachr. 278 (14) (2005), 1–17.CrossRefGoogle Scholar
Magnani, V., ‘Characteristic points, rectifiability and perimeter measure on stratified groups’, J. Eur. Math. Soc. 8 (4) (2006), 585–609.CrossRefGoogle Scholar
Magnani, V., ‘Pansu differentiability and intrinsic submanifolds in stratified groups’, 2007, arXiv:math/0701322v1.Google Scholar
Magnani, V., ‘Contact equations, Lipschitz extensions and isoperimetric inequalities’, Calc. Var. Partial Differential Equations 39 (1–2) (2010), 233–271.CrossRefGoogle Scholar
McDuff, D. and Salamon, D., Introduction to Symplectic Topology (Oxford University Press, 1995).CrossRefGoogle Scholar
Pansu, P., ‘Croissance des boules et des géodésiques fermeés dans les nilvariété’, Ergod. Theory Dynam. Sys. 3 (1983), 415–445.CrossRefGoogle Scholar
Pansu, P., ‘Métriques de Carnot–Carathéodory quasiisométries des espaces symétriques de rang un’, Ann. Math. 129 (1989), 1–60.CrossRefGoogle Scholar
Reimann, H. M. and Ricci, F., ‘The complexified Heisenberg group’, in: Proceedings on Analysis and Geometry (in Russian) (Novosibirsk Akademgorodok, 1999) (Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat, Novosibirsk, 2000), 465–480.Google Scholar
Schwartz, J. T., ‘Nonlinear functional analysis’, in: Courant Institute of Mathematical Sciences (New York University, 1965).Google Scholar
Stein, E. M., Harmonic Analysis (Princeton University Press, 1993).Google Scholar
Varadarajan, V. S., Lie groups, Lie Algebras and their Representation (Springer, New York, 1984).CrossRefGoogle Scholar
Warhurst, B., ‘Contact and Pansu differentiable maps on Carnot groups’, Bull. Aust. Math. Soc. 77 (3) (2008), 495–507.CrossRefGoogle Scholar
Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups (Foresman and Company, London, 1971).Google Scholar