| Title:
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$BV$ spaces and rectifiability for Carnot-Carathéodory metrics: an introduction (English) |
| Author:
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Franchi, Bruno |
| Language:
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English |
| Journal:
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Nonlinear Analysis, Function Spaces and Applications |
| Volume:
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Vol. 7 |
| Issue:
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2002 |
| Year:
|
|
| Pages:
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73-132 |
| . |
| Category:
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math |
| . |
| Summary:
|
This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned with topics of Geometric Measure Theory in Carnot groups and in particular with rectifiability theory in this setting. Thus, the core of the paper consists of Section 3 (dedicated to the study of BV functions with respect to Carnot-Carathéodory metrics), of Section 4 (dedicated more specifically to the theory of Carnot groups and, in particular, to the calculus associated with their differential structure as differential bundles) and of Section 5 (dedicated to the theory of intrinsic hypersurfaces and to rectifiability theory in Carnot groups). These sections rely basically on a group of results obtained in several papers in collaboration with R. Serapioni and F. Serra Cassano, starting from 1996. On the other hand, Section 2 and 6 are dedicated to the notion of Carnot-Carathéodory metric, to the properties of related Sobolev spaces and to Poincaré inequality associated with a family of Lipschitz continuous vector fields. In particular, relying on a group of joint papers with R. L. Wheeden, S. Gallot, C. Gutiérrez, P. Hajłasz, P. Koskela, G. Lu and C. Pérez, deep relationships between Poincaré inequality and the geometry of Carnot-Carathéodory spaces are studied. (English) |
| Keyword:
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Carnot-Carathéodory metrics |
| Keyword:
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Carnot groups |
| Keyword:
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Poincaré inequality |
| Keyword:
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hypersurfaces |
| Keyword:
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rectifiability |
| Keyword:
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Sobolev spaces |
| Keyword:
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BV spaces |
| MSC:
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22E25 |
| MSC:
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28A75 |
| MSC:
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28A78 |
| MSC:
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42B25 |
| MSC:
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46E35 |
| . |
| Date available:
|
2009-10-08T09:49:49Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702484 |
| . |
| Reference:
|
[1] Ambrosio L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces.Adv. Math. 159 (2001), 51–67. Zbl 1002.28004, MR 2002b: 31002. Zbl 1002.28004, MR 1823840 |
| Reference:
|
[2] Ambrosio L.: Fine properties of sets of finite perimeter in doubling metric measure spaces.In: Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10 (2002), 111–128. Zbl 1037.28002, MR 1926376 |
| Reference:
|
[3] Ambrosio L., Kirchheim B.: Rectifiable sets in metric and Banach spaces.Math. Ann. 318 (2000), 527–555. Zbl 0966.28002, MR 2003a:28009. Zbl 0966.28002, MR 1800768 |
| Reference:
|
[4] Ambrosio L., Kirchheim B.: Currents in metric spaces.Acta Math. 185 (2000), 1–80. Zbl 0984.49025, MR 2001k:49095. Zbl 0984.49025, MR 1794185 |
| Reference:
|
[5] Ambrosio L., Magnani V.: Some fine properties of $BV$ functions on sub-Riemannian groups.Preprint. Scuola Normale Superiore, 2002. |
| Reference:
|
[6] Anzellotti G., Giaquinta M.: $BV$ functions and traces.Rend. Sem. Mat. Univ. Padova 60 (1978), 1–21. Zbl 0432.46031, MR 82e:46046. MR 0555952 |
| Reference:
|
[7] Auscher P., Qafsaoui M.: Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form.J. Funct. Anal. 177 (2000), 310–364. Zbl 0979.35044, MR 2001j:35057. Zbl 0979.35044, MR 1795955 |
| Reference:
|
[8] Bakry D., Coulhon T., Ledoux M., Coste L. Saloff,- : Sobolev inequalities in disguise.Indiana Univ. Math. J. 44 (1995), 1033–1074. Zbl 0857.26006, MR 97c:46039. MR 1386760 |
| Reference:
|
[9] Balogh Z.: Size of characteristic sets and functions with prescribed gradient.Preprint, 2000. Zbl 1051.53024, MR 2021034 |
| Reference:
|
[10] Balogh Z., Rickly M., Cassano F. Serra: Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric.Preprint, 2001. MR 1970902 |
| Reference:
|
[11] Bellaïche A.: The tangent space in subriemannian geometry.In: Subriemannian Geometry (A. Bellaïche and J. Risler, eds.). Progress in Mathematics 144. Birkhäuser, Basel, 1996, pp. 1–78. Zbl 0862.53031, MR 98a:53108. |
| Reference:
|
[12] Biroli M., Mosco U.: 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. Zbl 0837.31006, MR 96i:46034a. Zbl 0837.31006, MR 1340280 |
| Reference:
|
[13] Bonfiglioli A.: Carnot groups related to sets of vector fields.Boll. Un. Mat. Ital. (to appear). Zbl 1178.35140 |
| Reference:
|
[14] Bonfiglioli A., Uguzzoni F.: A note on lifting of Carnot groups.Preprint, 2002. Zbl 1100.35029 |
| Reference:
|
[15] Buckley S., Koskela P., Lu G.: Boman equals John.In: 16th Rolf Nevanlinna Colloquium. Proceedings of the international conference held in Joensuu, Finland, August 1–5, 1995. (I. Laine et al., eds.). De Gruyter, Berlin, 1996, pp. 91–99. Zbl 0861.43007, MR 98m:43013. MR 1427074 |
| Reference:
|
[16] Busemann H.: The geometry of geodesics.Academic Press, New York, N. Y., 1955. Zbl 0112.37002, MR 17,779a. Zbl 0112.37002, MR 0075623 |
| Reference:
|
[17] Buttazzo G.: Semicontinuity, relaxation and integral rapresentation in the calculus of variations.Pitman Research Notes in Mathematics Series 207. Longman Scientific & Technical, Longman, Harlow, 1989. Zbl 0669.49005, MR 91c:49002. MR 1020296 |
| Reference:
|
[18] Cancelier C., Franchi B., Serra E.: Agmon metric for sum-of-squares operators.J. Anal. Math. 83 (2001), 89–107. Zbl pre01640251, MR 2002e:35059. Zbl 1200.35061, MR 1828487 |
| Reference:
|
[19] Capogna L., Danielli D., Garofalo N.: The geometric Sobolev embedding for vector fields and the isoperimetric inequality.Comm. Anal. Geom. 2 (1994), 203–215. Zbl 0864.46018, MR 96d:46032. Zbl 0864.46018, MR 1312686 |
| Reference:
|
[20] Capogna L., Danielli D., Garofalo N.: Subelliptic mollifiers and a basic pointwise estimate of Poincaré type.Math. Z. 226 (1997), 147–154. Zbl 0893.35023, MR 98i:35025. Zbl 0893.35023, MR 1472145 |
| Reference:
|
[21] Cohn W., Lu G., Lu S.: Higher order Poincaré inequalities associated with linear operators on stratified groups and applications.Math. Z. (to appear). Zbl 1023.46032, MR 1992541 |
| Reference:
|
[22] Coifman R. R., Weiss G.: Analyse harmonique non-commutative sur certains espaces homogenes. Etude de certaines intégrales singuliéres.Lecture Notes in Math. 242. Springer-Verlag, Berlin–Heidelberg–New York , 1971. Zbl 0224.43006, MR 58 #17690. Zbl 0224.43006, MR 0499948 |
| Reference:
|
[23] Coulhon T.: Inégalités de Gagliardo-Nirenberg pour les semi-groupes d’opérateurs et applications.Potential Anal. 1 (1992), 343–353. Zbl 0768.47018, MR 94k:47064. Zbl 0768.47018, MR 1245890 |
| Reference:
|
[24] Coulhon T., Russ E., Nachef V. Tardivel,- : Sobolev algebras on Lie groups and Riemannian manifolds.Amer. J. Math. 123 (2001), 283–342. Zbl 0990.43003, MR 2002g:43003. MR 1828225 |
| Reference:
|
[25] Coulhon T., Coste L. Saloff,- : Semi-groupes d’opérateurs et espaces fonctionnels sur les groupes de Lie.J. Approximation Theory 65 (1991), 176–199. Zbl 0745.47030, MR 92c:47049. MR 1104158 |
| Reference:
|
[26] Coulhon T., Coste L. Saloff,- : Isopérimétrie pour les groupes et les variétés.Revista Mat. Iberoamericana 9 (1993), 293–314. Zbl 0782.53066, MR 94g:58263. MR 1232845 |
| Reference:
|
[27] Maso G. Dal: An introduction to $\Gamma $-convergence.Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Basel, 1993. Zbl 0816.49001, MR 94a:49001. MR 1201152 |
| Reference:
|
[28] Danielli D., Garofalo N., Nhieu D. M.: Traces inequalities for Carnot-Carathéodory spaces and applications.Ann. Scuola Norm. Sup. Pisa Cl. Sci., IV. Ser. 27 (1998), 195–252. Zbl 0938.46036, MR 2000d:46039. MR 1664688 |
| Reference:
|
[29] David G., Semmes S.: Fractured fractals and broken dreams. Self-similargeometry through metric and measure.Oxford Lecture Series in Mathematics andits Applications 7. Clarendon Press, Oxford University Press, Oxford, 1997.Zbl 0887.54001, MR 99h:28018. MR 1616732 |
| Reference:
|
[30] Giorgi E. De: Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni.Ann. Mat. Pura Appl., IV. Ser. 36 (1954), 191–213.Zbl 0055.28504, MR 15,945d. Zbl 0055.28504, MR 0062214 |
| Reference:
|
[31] Giorgi E. De: Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni.Ricerche Mat. 4 (1955), 95–113. Zbl 0066.29903, MR 17,596a. Zbl 0066.29903, MR 0074499 |
| Reference:
|
[32] Delladio S.: Lower semicontinuity and continuity of function measures with respect to the strict convergence.Proc. Royal Soc. Edinburgh 119A (1991), 265–278. Zbl 0747.28007, MR 92i:28012. MR 1135973 |
| Reference:
|
[33] Derridj M.: Sur un théorème de trace.Ann. Inst. Fourier (Grenoble) 22 (1972), 72–83. Zbl 0226.46041, MR 49 #7755. MR 0343011 |
| Reference:
|
[34] Derridj M., Dias J.-P.: Le problème de Dirichlet pour une classe d’opérateurs non linéaires.J. Math. Pures Appl., IX. Sér. 51 (1972), 219–230. Zbl 0229.35038, MR 54 #8015. Zbl 0229.35038, MR 0419998 |
| Reference:
|
[35] Derridj M., Zuily C.: Régularité $C_\infty $ à la frontière d’opérateurs dégénérés.C. R. Acad. Sci. Paris, Sér. A 271 (1970), 786–788. Zbl 0206.44702, MR 43 #2347. MR 0276603 |
| Reference:
|
[36] Federer H.: Geometric measure theory.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153. Springer, Berlin, 1969. Zbl 0176.00801, MR 41 #1976. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[37] Fefferman C., Phong D. H.: Subelliptic eigenvalue problems.In: Harmonic analysis. Conference in honor of A. Zygmund, Chicago 1981, Vol. I, II. Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590–606. Zbl 0503.35071, MR 86c:35112. MR 0730094 |
| Reference:
|
[38] Folland G. B.: Subelliptic estimates and function spaces on nilpotent Lie groups.Ark. Mat. 13 (1975), 161–207. Zbl 0312.35026, MR 58 #13215. Zbl 0312.35026, MR 0494315 |
| Reference:
|
[39] Folland G. B., Stein E. M.: Hardy spaces on homogeneous groups.Mathematical Notes 28. Princeton University Press, Princeton, N.J., 1982. Zbl 0508.42025, MR 84h:43027. Zbl 0508.42025, MR 0657581 |
| Reference:
|
[40] Franchi B.: Stime subellittiche e metriche Riemanniane singolari II.Seminario di Analisi Matematica, Università di Bologna, 1983, VIII-1–VIII-17. |
| Reference:
|
[41] Franchi B.: Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations.Trans. Amer. Math. Soc. 327 (1991),125–158. Zbl 0751.46023, MR 91m:35095. Zbl 0751.46023, MR 1040042 |
| Reference:
|
[42] Franchi B., Gallot S., Wheeden R. L.: Sobolev and isoperimetric inequalities for degenerate metrics.Math. Ann. 300 (1994), 557–571. Zbl 0830.46027, MR 96a:46066. MR 1314734 |
| Reference:
|
[43] Franchi B., Gutiérrez C. E., Wheeden R. L.: Weighted Sobolev-Poincaré inequalities for Grushin type operators.Comm. Partial Differ. Equations 19 (1994), 523–604. Zbl 0822.46032, MR 96h:26019. MR 1265808 |
| Reference:
|
[44] Franchi B., Hajłasz P., Koskela P.: Definitions of Sobolev classes on metric spaces.Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924. Zbl 0938.46037, MR 2001a:46033. Zbl 0938.46037, MR 1738070 |
| Reference:
|
[45] Franchi B., Lanconelli E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients.Ann. Scuola Norm. Super. Pisa, Cl. Sci., IV. Ser. 10 (1983), 523–541. Zbl 0552.35032, MR 85k:35094. Zbl 0552.35032, MR 0753153 |
| Reference:
|
[46] Franchi B., Lu G., Wheeden R. L.: Representation formulas and weighted Poincaré inequalities for Hörmander vector fields.Ann. Inst. Fourier 45 (1995), 577–604. Zbl 0820.46026, MR 96i:46037. Zbl 0820.46026, MR 1343563 |
| Reference:
|
[47] Franchi B., Lu G., Wheeden R. L.: A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type.Internat. Math. Res. Notices (1996), 1–14. Zbl 0856.43006, MR 97k:26012. Zbl 0856.43006, MR 1383947 |
| Reference:
|
[48] Franchi B., Pérez C., Wheeden R. L.: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type.J. Funct. Anal. 153 (1998), 108–146. Zbl 0892.43005, MR 99d:42042. Zbl 0892.43005, MR 1609261 |
| Reference:
|
[49] Franchi B., Pérez C., Wheeden R. L.: A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces.J. Fourier Anal. Appl. (to appear). Zbl 1074.46022, MR 2027891 |
| Reference:
|
[50] Franchi B., Serapioni R., Cassano F. Serra: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields.Houston J. Math. 22 (1996), 859–890. Zbl 0876.49014, MR 98c:49037. MR 1437714 |
| Reference:
|
[51] Franchi B., Serapioni R., Cassano F. Serra: Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields.Boll. Unione Mat. Ital., VII. Ser. 11-B (1997), 83–117. Zbl 0952.49010, MR 98c:46062. MR 1448000 |
| Reference:
|
[52] Franchi B., Serapioni R., Cassano F. Serra: Sur les ensembles des périmètre fini dans le groupe de Heisenberg.C. R. Acad. Sci. Paris, Sér. I, Math. 329 (1999), 183–188. Zbl pre01340017, MR 2000e:49008. MR 1711057 |
| Reference:
|
[53] Franchi B., Serapioni R., Cassano F. Serra: Rectifiability and perimeter in the Heisenberg group.Math. Ann. 321 (2001), 479–531. Zbl pre01695175,MR 1 871 966. MR 1871966 |
| Reference:
|
[54] Franchi B., Serapioni R., Cassano F. Serra: Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups.Comm. Anal. Geom. (to appear). MR 2032504 |
| Reference:
|
[55] Franchi B., Serapioni R., Cassano F. Serra: Rectifiability and perimeter in step 2 groups.Proceedings of Equadiff 10, Prague, August 2001. Math. Bohem. 127 (2002), 219–228. Zbl pre01842893. MR 1981527 |
| Reference:
|
[56] Franchi B., Serapioni R., Cassano F. Serra: On the structure of finite perimeter sets in step 2 Carnot groups.J. Geometric Anal. (to appear). MR 1984849 |
| Reference:
|
[57] Franchi B., Wheeden R. L.: Compensation couples and isoperimetric estimates for vector fields.Colloq. Math. 74 (1997), 9–27. Zbl 0915.46028, MR 98g:46042. Zbl 0915.46028, MR 1455453 |
| Reference:
|
[58] Franchi B., Wheeden R. L.: Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type.J. Inequal. Appl. 3 (1999), 65–89. Zbl 0934.46037, MR 2001b:53027. Zbl 0934.46037, MR 1731670 |
| Reference:
|
[59] Friedrichs K. O.: The identity of weak and strong extensions of differential operators.Trans. Amer. Math. Soc. 55 (1944), 132–151. Zbl 0061.26201, MR 5,188b. Zbl 0061.26201, MR 0009701 |
| Reference:
|
[60] Garofalo N., Nhieu D. M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces.Commun. Pure Appl. Math. 49 (1996), 1081–1144. Zbl 0880.35032, MR 97i:58032. MR 1404326 |
| Reference:
|
[61] Garofalo N., Nhieu D. M.: Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot Carathéodory spaces.J. Anal. Math. 74 (1998), 67–97. Zbl 0906.46026, MR 2000i:46025. Zbl 0906.46026, MR 1631642 |
| Reference:
|
[62] Garofalo N., Pauls S. D.: The Bernstein problem in the Heisenberg group.(to appear). Zbl 1161.53024 |
| Reference:
|
[63] Gromov M.: Carnot-Carathéodory spaces seen from within.In: Sub-Riemannian Geometry. Proceedings of the satellite meeting of the 1st European congress of mathematics “Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique”, Paris, June 30–July 1, 1992 (A. Bellaïche and J. Risler, eds.). Progress in Mathematics 144. Birkhäuser, Basel, 1996, pp. 79–323. Zbl 0864.53025, MR 2000f:53034. MR 1421823 |
| Reference:
|
[64] Gromov M.: Metric structures for Riemannian and non-Riemannian spaces.Progress in Mathematics 152. Birkhäuser, Boston, MA, 1999. Zbl 0953.53002,MR 2000d:53065. Zbl 0953.53002, MR 1699320 |
| Reference:
|
[65] Hajłasz P.: Sobolev spaces on an arbitrary metric space.Potential Anal. 5 (1996), 403–415. Zbl 0859.46022, MR 97f:46050. MR 1401074 |
| Reference:
|
[66] Hajłasz P.: Geometric approach to Sobolev spaces and badly degenerated elliptic equations.In: Proceedings of the Banach Center minisemester on nonlinear analysis and applications, Warsaw, Poland, November 14–December 15, 1994. (N. Kenmochi et al., eds.). Int. Ser., Math. Sci. Appl. 7. Gakkotosho Co., Ltd. GAKUTO, Tokyo, 1995, pp. 141–168. Zbl 0877.46024, MR 97m:46051. MR 1422932 |
| Reference:
|
[67] Hajłasz P., Koskela P.: Sobolev met Poincaré.Memoirs Amer. Math. Soc. 688 (2000). Zbl 0954.46022, MR 2000j:46063. |
| Reference:
|
[68] Heinonen J.: Calculus on Carnot groups.In: Fall School in Analysis, Jyväskylä, Finland, October 3–7, 1994 (T. Kilpeläinen, ed.). Univ. Jyväskylä Math. Inst. Report 68. Univ. of Jyväskylä, 1995, pp. 1–31. Zbl 0863.22009, MR 96j:22015. MR 1351042 |
| Reference:
|
[69] Heinonen J.: Lectures on analysis on metric spaces.Universitext. Springer-Verlag, New York, 2001. Zbl 0985.46008. Zbl 0985.46008, MR 1800917 |
| Reference:
|
[70] Jerison D.: The Poincaré inequality for vector fields satisfying Hörmander condition.Duke Math. J. 53 (1986), 503–523. Zbl 0614.35066, MR 87i:35027. MR 0850547 |
| Reference:
|
[71] Kirchheim B., Cassano F. Serra: Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group.Preprint, 2002. MR 2124590 |
| Reference:
|
[72] Korányi A., Reimann H. M.: Foundation for the theory of quasiconformal mappings on the Heisenberg group.Adv. Math. 111 (1995), 1–87. Zbl 0876.30019, MR 96c:30021. MR 1317384 |
| Reference:
|
[73] Lanconelli E., Morbidelli D.: On the Poincaré inequality for vector fields.Ark. Mat. 38 (2000), 327–342. MR 2002a:46037. Zbl 1131.46304, MR 1785405 |
| Reference:
|
[74] Leonardi G., Rigot S.: Isoperimetric sets in the Heisenberg groups.Preprint, 2001. |
| Reference:
|
[75] Long R., Nie F.: Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators.In: Harmonic analysis. Proceedings of the special program at the Nankai Institute of Mathematics, Tianjin, PR China, March–July, 1988(M.-T. Cheng et al., eds.). Lect. Notes in Math. 1494. Springer, Berlin, 1991, pp. 131–141. Zbl 0786.46034, MR 94c:46066. MR 1187073 |
| Reference:
|
[76] Lu G.: The sharp Poincaré inequality for free vector fields: an endpoint result.Rev. Mat. Iberoamericana 10 (1994), 453–466. Zbl 0860.35006, MR 96g:26023. Zbl 0860.35006, MR 1286482 |
| Reference:
|
[77] Lu G.: Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups.Acta Math. Sin. (Engl. Ser.) 16 (2000), 405–444. Zbl 0973.46020, MR 2001k:46054. Zbl 0973.46020, MR 1787096 |
| Reference:
|
[78] Lu G., Liu Y., Wheeden R. L.: Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces.Math. Ann. 323 (2002), 157–174. Zbl pre01801594, MR 1 906 913. Zbl 1007.46034, MR 1906913 |
| Reference:
|
[79] Lu G., Liu Y., Wheeden R. L.: High order representation formulas and embedding theorems on stratified groups and generalizations.Studia Math. 142 (2000), 101–133. MR 2001k:46055. MR 1792599 |
| Reference:
|
[80] Lu G., Wheeden R. L.: An optimal representation formula for Carnot-Carathéodory vector fields.Bull. London Math. Soc. 30 (1998), 578–584. Zbl 0931.31003, MR 2000a:31005. Zbl 0931.31003, MR 1642822 |
| Reference:
|
[81] Lu G., Wheeden R. L.: Simultaneous representation and approximation formulas and higher order Sobolev embedding theorems on stratified groups.Preprint, 2002. MR 2078090 |
| Reference:
|
[82] Luckhaus S., Modica L.: The Gibbs-Thompson relation within the gradient theory of phase transition.Arch. Rat. Mech. Anal. 107 (1989), 71–83. Zbl 0681.49012, MR 90k:49041. MR 1000224 |
| Reference:
|
[83] Magnani V.: On a general coarea inequality and applications.Ann. Acad. Sci. Fenn. Math. 27 (2002), 121–140. MR 2002k:49080. Zbl 1064.49034, MR 1884354 |
| Reference:
|
[84] Magnani V.: A blow-up theorem for regular hypersurfaces in nilpotent groups.Manuscripta Math. (to appear). MR 1951800 |
| Reference:
|
[85] Magnani V.: An area formula in metric spaces.Preprint, 2002. Zbl 1231.28007, MR 2842953 |
| Reference:
|
[86] Magnani V.: Perimeter measure and higher codimension rectifiability in homogeneous groups.Preprint, 2001. |
| Reference:
|
[87] Magnani V.: Elements of geometric measure theory on sub-Riemannian groups.PhD Thesis, Scuola Normale Superiore, Pisa, 2002 (to appear in the series Tesi of SNS Pisa). Zbl 1064.28007, MR 2115223 |
| Reference:
|
[88] Maheux P., Coste L. Saloff,- : Analyse sur les boules d’un opérateur sous-elliptique.Math. Ann. 303 (1995), 713–740. Zbl 0836.35106, MR 96m:35049. MR 1359957 |
| Reference:
|
[89] Martio O., Sarvas J.: Injectivity theorems in plane and space.Ann. Acad. Sci. Fenn., Ser. A I Math. 4 (1979), 383–401. Zbl 0406.30013, MR 81i:30039. Zbl 0406.30013, MR 0565886 |
| Reference:
|
[90] Mattila P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability.Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995. Zbl 0819.28004, MR 96h:28006. Zbl 0819.28004, MR 1333890 |
| Reference:
|
[91] Jr M. Miranda ,. : Functions of bounded variation on “good” metric spaces.J. Math. Pures Appl. (to appear). Zbl 1109.46030, MR 2005202 |
| Reference:
|
[92] Mitchell J.: On Carnot-Carathèodory metrics.J. Differ. Geom. 21 (1985), 35–45. Zbl 0554.53023, MR 87d:53086. Zbl 0554.53023, MR 0806700 |
| Reference:
|
[93] Montanari A., Morbidelli D.: Balls defined by nonsmooth vector fields and the Poincaré inequality.Preprint, 2003. Zbl 1069.46504, MR 2073841 |
| Reference:
|
[94] Montefalcone F.: Sets of finite perimeter associated with vector fields and polyhedral approximation.Preprint, 2001. Zbl 1072.49031, MR 2104216 |
| Reference:
|
[95] Montgomery R.: A tour of subriemannian geometries, their geodesics and applications.Mathematical Surveys and Monographs 91. American Mathematical Society, Providence, R.I., 2002. Zbl pre01731778, MR 2002m:53045. Zbl 1044.53022, MR 1867362 |
| Reference:
|
[96] Monti R.: Distances, boundaries and surface measures in Carnot-Carathéodory spaces.PhD Thesis, University of Trento, 2001. |
| Reference:
|
[97] Monti R., Morbidelli D.: Regular domains in homogeneous groups.Preprint, 2001. Zbl 1067.43003, MR 2135732 |
| Reference:
|
[98] Monti R., Morbidelli D.: Domains with the cone property in Carnot-Carathéodory spaces.Preprint, 2002. |
| Reference:
|
[99] Monti R., Cassano F. Serra: Surface measures in Carnot-Carathéodory spaces.Calc. Var. Partial Differ. Equations 13 (2001), 339–376. Zbl pre01703143, MR 2002j:49052. MR 1865002 |
| Reference:
|
[100] Morbidelli D.: Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields.Studia Math. 139 (2000), 213–244. Zbl 0981.46034, MR 2002a:46039. Zbl 0981.46034, MR 1762582 |
| Reference:
|
[101] Nagel A., Stein E. M., Wainger S.: Balls and metrics defined by vector fields I: Basic properties.Acta Math. 155 (1985), 103–147. Zbl 0578.32044, MR 86k:46049. Zbl 0578.32044, MR 0793239 |
| Reference:
|
[102] Pansu P.: Une inégalité isopérimétrique sur le groupe de Heisenberg.C. R. Acad. Sci. Paris, Sér. I 295 (1982), 127–130. Zbl 0502.53039, MR 85b:53044. Zbl 0502.53039, MR 0676380 |
| Reference:
|
[103] Pansu P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. Math. 129 (1989), 1–60. Zbl 0678.53042, MR 90e:53058. Zbl 0678.53042, MR 0979599 |
| Reference:
|
[104] Pansu P.: Geometrie du group d’Heisenberg.These pour le titre de Docteur 3ème cycle, Université Paris VII, Paris, 1982. |
| Reference:
|
[105] Pauls S. D.: The large scale geometry of nilpotent Lie groups.Commun. Anal. Geom. 9 (2001), 951–982. Zbl pre01751515. MR 1883722 |
| Reference:
|
[106] Pauls S. D.: A notion of rectifiability modelled on Carnot groups.(to appear). MR 2048183 |
| Reference:
|
[107] Reschetnyak, Yu. G.: Weak convergence of completely additive vector functions on a set.Sibirsk. Mat. Z. 9 (1968), 1386–1394. Zbl 0176.44402. MR 0240274 |
| Reference:
|
[108] Rothschild L., Stein E. M.: Hypoelliptic differential operators and nilpotent groups.Acta Math. 137 (1976), 247–320. Zbl 0346.35030, MR 55 #9171. MR 0436223 |
| Reference:
|
[109] Coste L. Saloff,- : A note on Poincaré, Sobolev and Harnack inequalities.Internat. Math. Res. Notices (1992), 27–38. Zbl 0769.58054, MR 93d:58158. MR 1150597 |
| Reference:
|
[110] Coste L. Saloff,- : Aspects of Sobolev-type inequalities.London Mathematical Society Lecture Note Series 289. University Press, Cambridge, 2002. Zbl 0991.35002, MR 2003c:46048. MR 1872526 |
| Reference:
|
[111] Sawyer E., Wheeden R. L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces.Amer. J. Math. 114 (1992), 813–874.Zbl 0783.42011, MR 94i:42024. Zbl 0783.42011, MR 1175693 |
| Reference:
|
[112] Semmes S.: On the non existence of bilipschitz parametrization and geometric problems about $A_\infty $ weights.Revista Mat. Iberoamericana 12 (1996), 337–410. Zbl 0858.46017, MR 97e:30040. MR 1402671 |
| Reference:
|
[113] Serrin J.: On the definition and properties of certain variational integrals.Trans. Amer. Math. Soc. 101 (1961), 139–167. Zbl 0102.04601, MR 25 #1466. Zbl 0102.04601, MR 0138018 |
| Reference:
|
[114] Stein E. M.: Harmonic analysis: Real variable methods, orthogonality, and oscillatory integrals.Princeton Mathematical Series 43. Princeton University Press, Princeton, N.J., 1993. Zbl 0821.42001, MR 95c:42002. Zbl 0821.42001, MR 1232192 |
| Reference:
|
[115] Varopoulos N. Th.: Analysis on Lie groups.J. Funct. Anal. 76 (1988), 346–410. Zbl 0634.22008, MR 89i:22018. Zbl 0634.22008, MR 0924464 |
| Reference:
|
[116] Varopoulos N. Th., Coste L. Saloff,- Coulhon T.: Analysis and geometry on groups.Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge, 1992. Zbl 0813.22003, MR 95f:43008. MR 1218884 |
| Reference:
|
[117] Vodop’yanov S. K. : $\Cal P$-differentiability on Carnot groups in different topologies and related topics.In: Proceedings on analysis and geometry. International conference in honor of the 70th birthday of Professor Yu. G. Reshetnyak, Novosibirsk, Russia, August 30–September 3, 1999 (S. K. Vodop’yanov, ed.). Izdatel’stvo Instituta Matematiki Im. S. L. Soboleva SO RAN, Novosibirsk, 2000, pp. 603–670. Zbl 0992.58005. MR 1847541 |
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