Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 79-128.

Let 𝔾 be a k-step Carnot group. The first aim of this paper is to show an interplay between volume and 𝔾-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for 𝔾-regular submanifolds of codimension one. We then give some applications of this result: slicing of BV 𝔾 functions, integral geometric formulae for volume and 𝔾-perimeter and, making use of a suitable notion of convexity, called 𝔾-convexity, we state a Cauchy type formula for 𝔾-convex sets. Finally, in the last section we prove a sub-riemannian Santaló formula showing some related applications. In particular we find two lower bounds for the first eigenvalue of the Dirichlet problem for the Carnot sub-laplacian Δ 𝔾 on smooth domains.

Classification : 49Q15, 46E35, 22E60
Montefalcone, Francescopaolo 1

1 Dipartimento di Matematica Università degli Studi di Bologna Piazza di P. ta S. Donato, 5 40126 Bologna, Italia
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Montefalcone, Francescopaolo. Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 79-128. http://www.numdam.org/item/ASNSP_2005_5_4_1_79_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 University Press, 2000. | MR | Zbl

[3] L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527-555. | MR | Zbl

[4] L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. | MR | Zbl

[5] L. Ambrosio and V. Magnani, Some fine properties of BV functions on sub-Riemannian groups, Math. Z. 245 (2003). | MR

[6] L. Ambrosio and P. Tilli, “Selected topics on Analysis in Metric Spaces”, Quaderni della Scuola Normale Superiore, Pisa, 2000. | MR | Zbl

[7] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradients, J. Reine Angew. Math. 564 (2003), 63-83. | MR | Zbl

[8] Z. M. Balogh and M. Rickly, Regularity of convex functions on Heisenberg groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003), 847-868. | EuDML | Numdam | MR | Zbl

[9] Z. M. Balogh, J. J. Manfredi and J. Tyson, Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups, J. Funct. Anal. 204 (2003), 35-49. | MR | Zbl

[10] A. Bellaïche, The tangent space in subriemannian geometry, In: “Subriemannian Geometry”, A. Bellaiche and J. Risler (eds.), Progress in Mathematics 144, Birkhauser Verlag, Basel, 1996. | MR | Zbl

[11] A. L. Besse, “Manifolds all of whose Geodesics are Closed”, Springer Verlag, Berlin, 1978. | MR | Zbl

[12] I. Birindelli and J. Prajapat, Monotonicity and simmetry results for degenerate elliptic equations on nilpotent Lie groups, Pacific J. Math. 204 (2002), 1-17. | MR | Zbl

[13] Yu. D. Burago and V. A. Zalgaller, “Geometric Inequalities”, Springer Verlag, Berlin, 1980. | MR | Zbl

[14] 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

[15] I. Chavel, “Riemannian Geometry: a modern introduction”, Cambridge University Press, 1994. | MR | Zbl

[16] I. Chavel, “Isoperimetric Inequalities”, Cambridge University Press, 2001. | MR | Zbl

[17] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. | MR | Zbl

[18] G. Citti, M. Manfredini and A. Sarti, Minimum of the Mumford-Shah functional in a conctact manifold on the Heisenberg space, Preprint 2003.

[19] L. J. Corvin and F. P. Greenleaf, “Representations of nilpotent Lie groups and their applications”, Cambridge University Press, 1984. | Zbl

[20] C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), 419-435. | Numdam | MR | Zbl

[21] C. B. Croke and A. Derdziński, A lower bound for λ 1 on manifolds with boundary, Comment. Math. Helv. 62 (1987), 106-121. | MR | Zbl

[22] T. Coulhon and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétésf, Rev. Math. Iberoamericana 9 (1993), 293-314. | MR | Zbl

[23] D. Danielli, N. Garofalo and D. M. Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263-341. | MR | Zbl

[24] G. David and S. Semmes, “Fractured Fractals and Broken Dreams. Self-Similar Geometry through Metric and Measure”, Oxford University Press, 1997. | MR | Zbl

[25] E. B. Davies, “Heat Kernels and Spectral Theory”, Cambridge University Press, 1989. | MR | Zbl

[26] E. De Giorgi, Sulla proprietà isoperimetrica dell'ipersfera, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 (1958), 33-44. | MR | Zbl

[27] E. De Giorgi, Un progetto di teoria delle correnti, forme differenziali e varietà non orientate in spazi metrici, In: “Variational Methods, Non Linear Analysys and Differential Equations in Honour of J. P. Cecconi”, M. Chicco et al. (eds.), ECIG, Genova, 1993, 67-71.

[28] E. De Giorgi, Un progetto di teoria unitaria delle correnti, forme differenziali, varietà ambientate in spazi metrici, funzioni a variazione limitata, Manuscript (1995).

[29] E. De Giorgi, Problema di Plateau generale e funzionali geodetici, Atti Sem. Mat. Fis. Univ. Modena 43 (1995), 285-292. | MR | Zbl

[30] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of functions”, CRC Press, Boca Raton, 1992. | MR | Zbl

[31] H. Federer, “Geometric Measure Theory”, Springer Verlag, 1969. | MR | Zbl

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

[33] G. B. Folland and E. M. Stein, “Hardy spaces on homogeneous groups”, Princeton University Press, 1982. | MR | Zbl

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

[35] B. Franchi and E. Lanconelli, Ho ¨lder regularity theorem for a class of non uniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983), 523-541. | Numdam | MR | Zbl

[36] B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier, Grenoble 45 (1995), 577-604. | Numdam | MR | Zbl

[37] B. Franchi, R. Serapioni and F. S. Cassano, Meyers-Serrin type theorems and relaxationof variational integrals depending on vector fields, Houston J. Math. 22 (1996), 859-890. | MR | Zbl

[38] B. Franchi, R. Serapioni and F. S. Cassano, Approximation and imbedding theorems for wheighted Sobolev spaces associated with Lipschitz continous vector fields, Boll. Unione Mat. Ital. 11-B 7 (1997). | MR | Zbl

[39] B. Franchi, R. Serapioni and F. S. Cassano, Rectifiability and Perimeter in the Heisenberg Group, Math. Ann. 321 (2001), 479-531. | MR | Zbl

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

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

[42] N. Garofalo and 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. | MR | Zbl

[43] M. Gobbino, Finite difference approximation of the Mumford Shah functional, Comm. Pure. Appl. Math. 51 (1998). | MR | Zbl

[44] N. Goodman, “Nilpotent Lie groups”, Springer Lecture notes in Mathematics, Vol. 562, 1976. | Zbl

[45] M. Gromov, Carnot-Carathéodory spaces seen from within, In: “Subriemannian Geometry”, Progress in Mathematics, 144, A. Bellaiche and J. Rislered (eds.), Birkhauser Verlag, Basel, 1996. | Zbl

[46] M. Gromov, “Metric Structures for Riemannian and Non Riemannian Spaces”, Progress in Mathematics 153, Birkhauser Verlag, Boston, 1999. | MR | Zbl

[47] C. E. Gutierrez and A. Montanari, On the second order derivatives of convex functions on the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), 349-366. | Numdam | MR | Zbl

[48] P. Hajłasz and P. Koskela, “Sobolev Met Poincaré”, Mem. Amer. Math. Soc. 688, Providence, RI, 2000. | MR | Zbl

[49] J. Heinonen, Calculus on Carnot groups, In: “Fall School in Analysis” (Jyväskyla, 1994), Report, 68, Univ. Jyväskyla, Jyväskyla, 1995, 1-31. | MR | Zbl

[50] S. Helgason, “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, New York, 1978. | MR | Zbl

[51] D. Jerison, The Poincaré inequality for vector fields satisfying Ho ¨rmander condition, Duke Math. J. 53 (1986), 503-523. | MR | Zbl

[52] A. Korányi and H. M. Reimann, Foundation for the theory of quasiconformal mapping on the Heisenberg group, Adv. Math. 111 (1995), 1-85. | Zbl

[53] S. Lang, “Differential and Riemannian Manifolds”, Springer Verlag, 1994. | MR | Zbl

[54] J. M. Lee, “Introduction to Smooth Manifolds”, Springer Verlag, 2003. | MR | Zbl

[55] G. Lu, J. J. Manfredi and B. Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), 1-22. | MR | Zbl

[56] V. Magnani, Differentiability and Area Formula on Statified Lie Groups, Houston J. Math. 27 (2001), 297-323. | MR | Zbl

[57] V. Magnani, Characteristic point, rectifiability and perimeter measure on stratified groups, Preprint 2003. | MR

[58] G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory spaces, Geom. Functional Anal. 5 (1995), 402-433. | MR | Zbl

[59] P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Cambridge University Press, 1995. | MR | Zbl

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

[61] R. Montgomery, “A Tour of Subriemannian Geometries, Their Geodesics and Applications”, American Mathematical Society, Math. Surveys and Monographs, vol. 91, 2002. | MR | Zbl

[62] R. Monti, “Distances, Boundaries and surface measures in Carnot Carathéodory Spaces”, UTMPhDTS, 31, Ph. D. Thesis Series, Dip. Mat. Univ. Trento, Nov. 2001.

[63] R. Monti and F. Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), 339-376. | MR | Zbl

[64] 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

[65] P. Pansu, “Geometrie du Group d'Heisenberg”, These pour le titre de Docteur, 3ème cycle, Universite Paris VII, 1982.

[66] P. Pansu, Métriques de Carnot Carathéodory et quasi-isométries des espaces symmétriques de rang un, Ann. of Math. 2 129 (1989), 1-60. | MR | Zbl

[67] D. Preiss and J. Tisêr, On Besicovitch 1/2-problem, J. London Math. Soc. 45 (1992), 279-287. | Zbl

[68] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. | MR | Zbl

[69] L. A. Santaló, “Integral Geometry and Geometric Probability”, Addison-Wesley, Reading, Mass., 1976. | MR | Zbl

[70] E. M. Stein, “Harmonic Analysis”, Princeton University Press, 1993. | MR | Zbl

[71] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263. Corrections: J. Differential Geom. 30 (1989), 595-596. | MR | Zbl

[72] G. Talenti, The standard isoperimetric theorem, In: “Handbook of Convexity”, Vol. A, P. M. Gruber and J. M. Wills (eds.), 73-123, Amsterdam, North Holland, 1993. | MR | Zbl

[73] V. S. Varadarajan, “Lie Groups, Lie Algebras, and their Representations”, Springer, 1984. | MR | Zbl

[74] N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410. | MR | Zbl

[75] N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, “Analysis and Geometry on Groups”, Cambridge University Press, 1992. | MR | Zbl

[76] S. K. Vodop'Yanov, 𝒫-differentiability on Carnot Groups in different topologies and related topics, Proc. on Analysis and Geometry, pp. 603-670, Sobolev Inst. Press, Novosibirsk, 2000. | MR | Zbl

[77] W. P. Ziemer, “Weakly Differentiable Functions”, Springer Verlag, 1989. | MR | Zbl