A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meant to stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensional intrinsic Lipschitz graphs are sets with locally finite G-perimeter. From this a Rademacher’s type theorem for one codimensional graphs in a general class of groups is proved.
@article{bwmeta1.element.doi-10_2478_agms-2014-0010,
author = {Bruno Franchi and Marco Marchi and Raul Paolo Serapioni},
title = {Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem},
journal = {Analysis and Geometry in Metric Spaces},
volume = {2},
year = {2014},
zbl = {1307.22007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0010}
}
Bruno Franchi; Marco Marchi; Raul Paolo Serapioni. Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem. Analysis and Geometry in Metric Spaces, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2014-0010/
[1] L.Ambrosio, N.Fusco, D.Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, (2000). | Zbl 0957.49001
[2] L.Ambrosio, B.Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Annalen, 318 (2000), 527–555. | Zbl 0966.28002
[3] L.Ambrosio, B.Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1–80. | Zbl 0984.49025
[4] L.Ambrosio, B.Kirchheim, E. Le Donne, Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal., 19, (2009), no. 3, 509–540. | Zbl 1187.28008
[5] L.Ambrosio, F. Serra Cassano, D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16, (2006), 187–232. | Zbl 1085.49045
[6] G.Arena, Intrinsic Graphs, Convexity and Calculus on Carnot Groups, PhD Thesis, University of Trento, (2008).
[7] G.Arena, R.Serapioni, Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs, Calc. Var. Partial Differential Equations 35 (2009), no. 4, 517–536. | Zbl 1225.53031
[8] F.Bigolin, D.Vittone, Some remarks about parametrizations of intrinsic regular surfaces in the Heisenberg group, Publ. Mat. 54 (2010), no. 1, 159–172. | Zbl 1188.53028
[9] F. Bigolin, F. Serra Cassano, Distributional solutions of Burgers’ equation and intrinsic regular graphs in Heisenberg groups, J. Math. Anal. Appl. 366 (2010), no. 2, 561–568. | Zbl 1186.35030
[10] F. Bigolin, F. Serra Cassano, Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs, Adv. Calc. Var. 3 (2010), no. 1, 69–97. | Zbl 1188.53027
[11] A.Bonfiglioli, E.Lanconelli, F.Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub–Laplacians, Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, New York (2007). | Zbl 1128.43001
[12] G.Citti, M.Manfredini, Implicit function theorem in Carnot-Carathéodory spaces. Commun. Contemp. Math. 8 (2006), no. 5, 657–680. | Zbl 1160.53017
[13] G.Citti, M.Manfredini, Blow-up in non homogeneous Lie groups and rectifiability. Houston J.Math. 31 (2005), no. 2, 333–353. | Zbl 1078.49030
[14] L.C.Evans, R.F.Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, BocaRaton, (1992). | Zbl 0804.28001
[15] H.Federer, Geometric Measure Theory, Springer, (1969). | Zbl 0176.00801
[16] G.B.Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161–207. | Zbl 0312.35026
[17] B.Franchi, R.Serapioni, Intrinsic Lipschitz graphs within Carnot groups, Preprint 2014 | Zbl 1307.22007
[18] B.Franchi, R.Serapioni, F.Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479– 531. | Zbl 1057.49032
[19] B.Franchi, R.Serapioni, F.Serra Cassano, Regular hypersurfaces, Intrinsic Perimeter and Implicit Function Theoremin Carnot Groups, Comm. Anal. Geom., 11 (2003), no 5, 909–944. | Zbl 1077.22008
[20] B.Franchi, R.Serapioni, F.Serra Cassano, On the Structure of Finite Perimeter Sets in Step 2 Carnot Groups, The Journal of Geometric Analysis. 13 (2003), no. 3, 421–466. | Zbl 1064.49033
[21] B.Franchi, R.Serapioni, F.Serra Cassano, Intrinsic Lipschitz graphs in Heisenberg groups. J. Nonlinear Convex Anal. 7 (2006), no. 3, 423–441. | Zbl 1151.58005
[22] B.Franchi, R.Serapioni, F.Serra Cassano, Regular submanifolds, graphs and area formula in Heisenberg Groups. Advances in Math. 211 (2007), no.1, 152–203. | Zbl 1125.28002
[23] B.Franchi, R.Serapioni, F.Serra Cassano, Rademacher theoremfor intrinsic Lipschitz continuous functions, J. Geom. Anal. 21 (2011), 1044–1084. | Zbl 1234.22002
[24] N.Garofalo, D.M.Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math., 49 (1996), 1081–1144. | Zbl 0880.35032
[25] M.Gromov, Metric structures for Riemannian and Non Riemannian Spaces, Progress inMathematics, 152, Birkhauser Verlag, Boston, (1999).
[26] B.Kirchheim, F.Serra Cassano, Rectifiability and parameterizations of intrinsically regular surfaces in the Heisenberg group, Ann. Scuola Norm. Sup. Pisa, Cl.Sc. (5) III, (2005), 871–896. | Zbl 1170.28300
[27] R. Korte, P. Lahti, N. Shanmugalingam, Semmes family of curves a a characterization of functions of bounded variation in terms of curves, http://cvgmt.sns.it/media/doc/paper/2229/lineBV-CalcVarPDE-Sept2013.pdf (2013). | Zbl 1327.31026
[28] V.Magnani, Elements of Geometric Measure Theory on Sub-Riemannian Groups, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, (2003). | Zbl 1064.28007
[29] V.Magnani, Towards differential calculus in stratified groups, Journal of Australian Mathematical Society, 95 no.1, (2013), 76–128. | Zbl 1279.22011
[30] M.Marchi, Regularity of sets with constant intrinsic normal in a class of Carnot groups to appear on Ann.Inst.Fourier (Grenoble).
[31] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, (1995).
[32] P.Mattila, R.Serapioni, F.Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 9 (2010), no. 4, 687–723. | Zbl 1229.28004
[33] J.Mitchell, On Carnot-Carathèodory metrics, J.Differ. Geom. 21 (1985), 35–45. | Zbl 0554.53023
[34] R.Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI (2002). | Zbl 1044.53022
[35] S. Nicolussi Golo, The tangent space in Sub-Riemannian Geometry, Doctoral Thesis, Università di Trento, (2012). | Zbl 1312.53004
[36] P.Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989), 1–60. | Zbl 0678.53042
[37] S.D.Pauls, A notion of rectifiability modelled on Carnot groups, Indiana Univ. Math. J. 53 (2004), no. 1, 49–81. | Zbl 1076.49025
[38] S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A–weights Rev. Mat. Iberoamericana 12 (1996), no. 2, 337–410. | Zbl 0858.46017
[39] E.M.Stein, Harmonic Analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton University Press, Princeton (1993). | Zbl 0821.42001
[40] N.Th.Varopoulos, L.Saloff-Coste, T.Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, (1992) | Zbl 0813.22003
[41] D.Vittone, Submanifolds in Carnot groups, Edizioni della Normale, Scuola Normale Superiore, Pisa, (2008). | Zbl 1142.53046
[42] D.Vittone, Lipschitz surfaces, Perimeter and trace Theorems for BV Functions in Carnot-Caratheéodory Spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 11(4), (2012), 939–998. | Zbl 1270.53068