Let be a -step Carnot group. The first aim of this paper is to show an interplay between volume and -perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for -regular submanifolds of codimension one. We then give some applications of this result: slicing of functions, integral geometric formulae for volume and -perimeter and, making use of a suitable notion of convexity, called -convexity, we state a Cauchy type formula for -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 on smooth domains.
@article{ASNSP_2005_5_4_1_79_0,
author = {Montefalcone, Francescopaolo},
title = {Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in {Carnot} groups},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {79--128},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 4},
number = {1},
year = {2005},
mrnumber = {2165404},
zbl = {1150.49022},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2005_5_4_1_79_0/}
}
TY - JOUR AU - Montefalcone, Francescopaolo TI - Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 79 EP - 128 VL - 4 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2005_5_4_1_79_0/ LA - en ID - ASNSP_2005_5_4_1_79_0 ER -
%0 Journal Article %A Montefalcone, Francescopaolo %T Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2005 %P 79-128 %V 4 %N 1 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2005_5_4_1_79_0/ %G en %F ASNSP_2005_5_4_1_79_0
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] , Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), 51-67. | MR | Zbl
[2] , and , “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. | MR | Zbl
[3] and , Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527-555. | MR | Zbl
[4] and , Currents in metric spaces, Acta Math. 185 (2000), 1-80. | MR | Zbl
[5] and , Some fine properties of BV functions on sub-Riemannian groups, Math. Z. 245 (2003). | MR
[6] and , “Selected topics on Analysis in Metric Spaces”, Quaderni della Scuola Normale Superiore, Pisa, 2000. | MR | Zbl
[7] , Size of characteristic sets and functions with prescribed gradients, J. Reine Angew. Math. 564 (2003), 63-83. | MR | Zbl
[8] and , Regularity of convex functions on Heisenberg groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003), 847-868. | EuDML | Numdam | MR | Zbl
[9] , and , Fundamental solution for the Q-Laplacian and sharp Moser-Trudinger inequality in Carnot groups, J. Funct. Anal. 204 (2003), 35-49. | MR | Zbl
[10] , 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] , “Manifolds all of whose Geodesics are Closed”, Springer Verlag, Berlin, 1978. | MR | Zbl
[12] and , Monotonicity and simmetry results for degenerate elliptic equations on nilpotent Lie groups, Pacific J. Math. 204 (2002), 1-17. | MR | Zbl
[13] and , “Geometric Inequalities”, Springer Verlag, Berlin, 1980. | MR | Zbl
[14] , and , The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203-215. | MR | Zbl
[15] , “Riemannian Geometry: a modern introduction”, Cambridge University Press, 1994. | MR | Zbl
[16] , “Isoperimetric Inequalities”, Cambridge University Press, 2001. | MR | Zbl
[17] , Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. | MR | Zbl
[18] , and , Minimum of the Mumford-Shah functional in a conctact manifold on the Heisenberg space, Preprint 2003.
[19] and , “Representations of nilpotent Lie groups and their applications”, Cambridge University Press, 1984. | Zbl
[20] , Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), 419-435. | Numdam | MR | Zbl
[21] and , A lower bound for on manifolds with boundary, Comment. Math. Helv. 62 (1987), 106-121. | MR | Zbl
[22] and , Isopérimétrie pour les groupes et les variétésf, Rev. Math. Iberoamericana 9 (1993), 293-314. | MR | Zbl
[23] , and , Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), 263-341. | MR | Zbl
[24] and , “Fractured Fractals and Broken Dreams. Self-Similar Geometry through Metric and Measure”, Oxford University Press, 1997. | MR | Zbl
[25] , “Heat Kernels and Spectral Theory”, Cambridge University Press, 1989. | MR | Zbl
[26] , 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] , 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] , Un progetto di teoria unitaria delle correnti, forme differenziali, varietà ambientate in spazi metrici, funzioni a variazione limitata, Manuscript (1995).
[29] , Problema di Plateau generale e funzionali geodetici, Atti Sem. Mat. Fis. Univ. Modena 43 (1995), 285-292. | MR | Zbl
[30] and , “Measure Theory and Fine Properties of functions”, CRC Press, Boca Raton, 1992. | MR | Zbl
[31] , “Geometric Measure Theory”, Springer Verlag, 1969. | MR | Zbl
[32] , Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. | MR | Zbl
[33] and , “Hardy spaces on homogeneous groups”, Princeton University Press, 1982. | MR | Zbl
[34] , and , Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571. | MR | Zbl
[35] and , Hlder 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] , and , Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier, Grenoble 45 (1995), 577-604. | Numdam | MR | Zbl
[37] , and , Meyers-Serrin type theorems and relaxationof variational integrals depending on vector fields, Houston J. Math. 22 (1996), 859-890. | MR | Zbl
[38] , and , 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] , and , Rectifiability and Perimeter in the Heisenberg Group, Math. Ann. 321 (2001), 479-531. | MR | Zbl
[40] , and , Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom. 11 (2003). | MR | Zbl
[41] , and , On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal. 13 (2003). | MR | Zbl
[42] and , 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] , Finite difference approximation of the Mumford Shah functional, Comm. Pure. Appl. Math. 51 (1998). | MR | Zbl
[44] , “Nilpotent Lie groups”, Springer Lecture notes in Mathematics, Vol. 562, 1976. | Zbl
[45] , 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] , “Metric Structures for Riemannian and Non Riemannian Spaces”, Progress in Mathematics 153, Birkhauser Verlag, Boston, 1999. | MR | Zbl
[47] and , 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] and , “Sobolev Met Poincaré”, Mem. Amer. Math. Soc. 688, Providence, RI, 2000. | MR | Zbl
[49] , Calculus on Carnot groups, In: “Fall School in Analysis” (Jyväskyla, 1994), Report, 68, Univ. Jyväskyla, Jyväskyla, 1995, 1-31. | MR | Zbl
[50] , “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, New York, 1978. | MR | Zbl
[51] , The Poincaré inequality for vector fields satisfying Hrmander condition, Duke Math. J. 53 (1986), 503-523. | MR | Zbl
[52] and , Foundation for the theory of quasiconformal mapping on the Heisenberg group, Adv. Math. 111 (1995), 1-85. | Zbl
[53] , “Differential and Riemannian Manifolds”, Springer Verlag, 1994. | MR | Zbl
[54] , “Introduction to Smooth Manifolds”, Springer Verlag, 2003. | MR | Zbl
[55] , and , Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), 1-22. | MR | Zbl
[56] , Differentiability and Area Formula on Statified Lie Groups, Houston J. Math. 27 (2001), 297-323. | MR | Zbl
[57] , Characteristic point, rectifiability and perimeter measure on stratified groups, Preprint 2003. | MR
[58] and , The differential of a quasi-conformal mapping of a Carnot-Carathéodory spaces, Geom. Functional Anal. 5 (1995), 402-433. | MR | Zbl
[59] , “Geometry of Sets and Measures in Euclidean Spaces”, Cambridge University Press, 1995. | MR | Zbl
[60] , On Carnot-Carathèodory metrics, J. Differential Geom. 21 (1985), 35-45. | MR | Zbl
[61] , “A Tour of Subriemannian Geometries, Their Geodesics and Applications”, American Mathematical Society, Math. Surveys and Monographs, vol. 91, 2002. | MR | Zbl
[62] , “Distances, Boundaries and surface measures in Carnot Carathéodory Spaces”, UTMPhDTS, 31, Ph. D. Thesis Series, Dip. Mat. Univ. Trento, Nov. 2001.
[63] and , Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), 339-376. | MR | Zbl
[64] , and , Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147. | MR | Zbl
[65] , “Geometrie du Group d'Heisenberg”, These pour le titre de Docteur, 3ème cycle, Universite Paris VII, 1982.
[66] , Métriques de Carnot Carathéodory et quasi-isométries des espaces symmétriques de rang un, Ann. of Math. 2 129 (1989), 1-60. | MR | Zbl
[67] and , On Besicovitch -problem, J. London Math. Soc. 45 (1992), 279-287. | Zbl
[68] and , Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. | MR | Zbl
[69] , “Integral Geometry and Geometric Probability”, Addison-Wesley, Reading, Mass., 1976. | MR | Zbl
[70] , “Harmonic Analysis”, Princeton University Press, 1993. | MR | Zbl
[71] , Sub-Riemannian geometry, J. Differential Geom. 24 (1986), 221-263. Corrections: J. Differential Geom. 30 (1989), 595-596. | MR | Zbl
[72] , 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] , “Lie Groups, Lie Algebras, and their Representations”, Springer, 1984. | MR | Zbl
[74] , Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410. | MR | Zbl
[75] , and , “Analysis and Geometry on Groups”, Cambridge University Press, 1992. | MR | Zbl
[76] , -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] , “Weakly Differentiable Functions”, Springer Verlag, 1989. | MR | Zbl
