Lipschitz surfaces, perimeter and trace theorems for BV functions in Carnot-Carathéodory spaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 939-998.

We introduce intrinsic Lipschitz hypersurfaces in Carnot-Carathéodory spaces and prove that intrinsic Lipschitz domains have locally finite perimeter. We also show the existence of a boundary trace operator for functions with bounded variation on Lipschitz domains and obtain extension results for such functions. In particular, we characterize their trace space.

Published online:
Classification: 53C17,  46E35
Vittone, Davide 1

1 Dipartimento di Matematica Università di Padova Via Trieste, 63 35121 Padova, Italia
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Vittone, Davide. Lipschitz surfaces, perimeter and trace theorems for BV functions in Carnot-Carathéodory spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 939-998. http://www.numdam.org/item/ASNSP_2012_5_11_4_939_0/

[1] L. Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), 51–67. | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.. | MR | Zbl

[3] L. Ambrosio, F. Serra Cassano and D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal. 16 (2006), 187–232. | MR | Zbl

[4] G. Anzellotti and M. Giaquinta, BV functions and traces (Italian), Rend. Sem. Mat. Univ. Padova 60 (1978), 1–21. | EuDML | Numdam | MR | Zbl

[5] G. Arena and R. Serapioni, Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs, Calc. Var. Partial Differential Equations 35 (2009), 517–536. | MR | Zbl

[6] H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace theorem in Sobolev spaces associated with Hörmander’s vector fields, In: “Partial Differential Equations and their Applications” (Wuhan, 1999), 1–14, World Sci. Publ., River Edge, NJ, 1999. | MR | Zbl

[7] H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace and trace lifting theorems in weighted Sobolev spaces, J. Inst. Math. Jussieu 4 (2005), 509–552. | MR | Zbl

[8] H. Bahouri, J.-Y. Chemin and C.-J. Xu, Trace theorem on the Heisenberg group, Ann. Inst. Fourier (Grenoble) 59 (2009), 491–514. | EuDML | Numdam | MR | Zbl

[9] A. Baldi and F. Montefalcone, A note on the extension of BV functions in metric measure spaces, J. Math. Anal. Appl. 340 (2008), 197–208. | MR | Zbl

[10] A. Bellaïche and J.-J. Risler (editors), “Sub-Riemannian Geometry”, Progress in Mathematics, Vol. 144, Birkhäuser Verlag, Basel, 1996. | MR

[11] S. Berhanu and I. Pesenson, The trace problem for vector fields satisfying Hörmander’s condition, Math. Z. 231 (1999), 103–122. | MR | Zbl

[12] F. Bigolin and F. Serra Cassano, Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs, Adv. Calc. Var. 3 (2010), 69–97. | MR | Zbl

[13] F. Bigolin and F. Serra Cassano, Distributional solutions of Burgers’ equation and intrinsic regular graphs in Heisenberg groups, J. Math. Anal. Appl. 366 (2010), 561–568. | MR | Zbl

[14] M. Biroli and U. Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser. Rend. Lincei, Mat. Appl., 6 (1995), 37–44. | EuDML | MR | Zbl

[15] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, “Stratified Lie Groups and Potential Theory for their sub-Laplacians”, Springer Monographs in Mathematics. Springer, Berlin, 2007. | MR | Zbl

[16] L. Capogna, D. Danielli and N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203–215. | MR | Zbl

[17] L. Capogna and N. Garofalo, Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces, J. Geom. Anal. 16 (2006), 455–497. | MR | Zbl

[18] L. Capogna, N. Garofalo and D.-M. Nhieu, A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett. 5 (1998), 541–549. | MR | Zbl

[19] L. Capogna, N. Garofalo and D.-M. Nhieu, Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type, Amer. J. Math. 124 (2002), 273–306. | MR | Zbl

[20] L. Capogna, N. Garofalo & D.-M. Nhieu, Mutual absolute continuity of harmonic and surface measures for Hörmander type operators, In: “Perspectives in Partial Differential Equations, Harmonic Analysis and Applications”, 49–100, Proc. Sympos. Pure Math., 79, Amer. Math. Soc., Providence, RI, 2008. | MR | Zbl

[21] G. Citti and M. Manfredini, Blow-up in non homogeneous Lie groups and rectifiability, Houston J. Math. 31 (2005), 333–353. | MR | Zbl

[22] G. Citti and M. Manfredini, Implicit function theorem in Carnot-Carathéodory spaces, Commun. Contemp. Math. 8 (2006), 657–680. | MR | Zbl

[23] D. Danielli, A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Anal. 11 (1999), 387–413. | MR | Zbl

[24] D. Danielli, N. Garofalo and D.-M. Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27 (1998), 195–252. | EuDML | Numdam | MR | Zbl

[25] D. Danielli, N. Garofalo and D.-M. Nhieu, Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds, In: “Harmonic Analysis and Boundary Value Problems” (Fayetteville, AR, 2000), 19–37, Contemp. Math. 277, Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[26] D. Danielli, N. Garofalo and D.-M. Nhieu, “Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces”, Mem. Amer. Math. Soc. 182 (2006). | MR | Zbl

[27] H. Federer, “Geometric Measure Theory”, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York, 1969. | MR | Zbl

[28] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207. | MR | Zbl

[29] G. B. Folland and E. M. Stein, “Hardy Spaces on Homogeneous Groups”, Princeton University Press, 1982. | MR | Zbl

[30] B. Franchi, S. Gallot and R. L. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557–571. | EuDML | MR | Zbl

[31] B. Franchi, R. Serapioni and F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending vector fields, Houston J. Math. 22 (1996), 859–889. | MR | Zbl

[32] B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7) 11 (1997), 83–117. | MR | Zbl

[33] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (2001), 479–531. | MR | Zbl

[34] B. Franchi, R. Serapioni and F. Serra Cassano, On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003), 421–466. | MR | Zbl

[35] B. Franchi, R. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom. 11 (2003), 909–944. | MR | Zbl

[36] B. Franchi, R. Serapioni and F. Serra Cassano, Intrinsic Lipschitz graphs in Heisenberg groups, J. Nonlinear Convex Anal. 7 (2006), 423–441. | MR | Zbl

[37] B. Franchi, R. Serapioni and F. Serra Cassano, Regular submanifolds, graphs and area formula in Heisenberg groups, Adv. Math. 211 (2007), 152–203. | MR | Zbl

[38] B. Franchi, R. Serapioni and F. Serra Cassano, Differentiability of intrinsic Lipschitz functions within Heisenberg groups, J. Geom. Anal. 21 (2011), 1044–1084. | MR | Zbl

[39] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math. 49 (1996), 1081–1144. | MR | Zbl

[40] N. Garofalo and D.-M. Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67–97. | MR | Zbl

[41] E. Giusti, “Minimal Surfaces and Functions of Bounded Variation”, Monographs in Mathematics, 80. Birkhäser Verlag, Basel, 1984. | MR | Zbl

[42] B. Kirchheim and F. Serra Cassano, Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (2004), 871–896. | EuDML | Numdam | MR | Zbl

[43] V. Magnani, “Elements of Geometric Measure Theory on Sub-Riemannian Groups”, PhD thesis, Scuola Normale Superiore, Pisa, 2002. | MR | Zbl

[44] V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc. (JEMS) 8 (2006), 585–609. | EuDML | MR | Zbl

[45] V. Magnani and D. Vittone, An intrinsic measure for submanifolds in stratified groups, J. Reine Angew. Math. 619 (2008), 203–232. | MR | Zbl

[46] N. G. Meyers and W. P. Ziemer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math. 99 (1977), 1345–1360. | MR | Zbl

[47] M. Miranda, Comportamento delle successioni convergenti di frontiere minimali (Italian), Rend. Sem. Mat. Univ. Padova 38 (1967), 238–257. | EuDML | Numdam | MR | Zbl

[48] J. Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35–45. | MR | Zbl

[49] R. Monti and D. Morbidelli, Trace theorems for vector fields, Math. Z. 239 (2002), 747–776. | MR | Zbl

[50] R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 339–376. | MR | Zbl

[51] D. Morbidelli, Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields, Studia Math. 139 (2000), 213–244. | EuDML | MR | Zbl

[52] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103–147. | MR | Zbl

[53] L. Rothschild and E. M. Stein, Hypoelliptic differential operatorsand nilpotent groups, Acta Math. 137 (1976), 247–320. | MR | Zbl

[54] C. Selby, An extension and trace theorem for functions of H-bounded variation in Carnot groups of step 2, Houston J. Math. 33 (2007), 593–616 (electronic). | MR | Zbl

[55] L. Simon, Lectures on geometric measure theory, In: “Proceedings of the Centre for Mathematical Analysis, Australian National University”, Vol. 3, Centre for Mathematical Analysis, Canberra, 1983. | MR | Zbl

[56] E. Stein, “Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals”, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. | MR | Zbl

[57] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis and Geometry on Groups”, Cambridge Tracts in Mathematics, Vol. 100, Cambridge University Press, Cambridge, 1992. | MR | Zbl

[58] F. Vigneron, The trace problem for Sobolev spaces over the Heisenberg group, J. Anal. Math. 103 (2007), 279–306. | MR | Zbl

[59] D. Vittone, “Submanifolds in Carnot Groups”, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 2008. | MR | Zbl