Geodesic
The geometry of Carnot-Carathéodory spaces, quasiconformal analysis, and geometric measure theory
Vladikavkazskij matematičeskij žurnal, Tome 5 (2003) no. 1, pp. 14-34 Cet article a éte moissonné depuis la source Math-Net.Ru

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S. K. Vodop'yanov. The geometry of Carnot-Carathéodory spaces, quasiconformal analysis, and geometric measure theory. Vladikavkazskij matematičeskij žurnal, Tome 5 (2003) no. 1, pp. 14-34. http://geodesic.mathdoc.fr/item/VMJ_2003_5_1_a3/

[1] Vodopyanov S. K., “$L_p$-teoriya potentsiala i kvazikonformnye otobrazheniya na odnorodnykh gruppakh”, Sovremennye problemy geometrii i analiza, Nauka, Novosibirsk, 1989, 45–89 | MR

[2] Vodopyanov S. K., “Monotonnye funktsii i kvazikonformnye otobrazheniya na gruppakh Karno”, Sib. mat. zhurn., 37:6 (1996), 1269–1295 | MR

[3] Vodopyanov S. K., “O zamknutosti klassov otobrazhenii s ogranichennym iskazheniem”, Matematicheskie trudy, 5:2 (2002), 91–136 | MR

[4] Vodopyanov S. K., Greshnov A. V., “O differentsiruemosti otobrazhenii prostranstv Karno — Karateodori”, Dokl. RAN, 389:5 (2003), 1–5 | MR

[5] Vodopyanov S. K., Greshnov A. V., “Analiticheskie svoistva kvazikonformnykh otobrazhenii na gruppakh Karno”, Sib. mat. zhurn., 36:6 (1995), 1317–1327 | MR

[6] Vodopyanov S. K., Greshnov A. V., “O prodolzhenii funktsii ogranichennoi srednei ostsillyatsii na prostranstvakh odnorodnogo tipa s vnutrennei metrikoi”, Sib. mat. zhurn., 36:5 (1995), 1015–1048 | MR

[7] Vodopyanov S. K., Latfullin T. G., “Vesovye prostranstva Soboleva i kvazigiperbolicheskie otobrazheniya na gruppakh Karno”, Kompleksnyi analiz v sovremennoi matematike, K 80-letiyu B. V. Shabata, FAZIS, M., 2001, 81–98 | MR

[8] Vodopyanov S. K., Ukhlov A. D., “Approksimativno differentsiruemye preobrazovaniya i zamena peremennykh na nilpotentnykh gruppakh”, Sib. mat. zhurn., 37:1 (1996), 70–89 | MR

[9] Vodopyanov S. K., Ukhlov A. D., “Operatory superpozitsii v prostranstvakh Lebega i differentsiruemost kvaziadditivnykh funktsii mnozhestva”, Vladikavk. mat. zhurn., 4:1 (2002), 11–33 | MR | Zbl

[10] Postnikov M. M., Gruppy i algebry Li, Nauka, M., 1982, 447 pp. | MR | Zbl

[11] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982, 278 pp. | MR | Zbl

[12] Reshetnyak Yu. G., “Sobolevskie klassy funktsii so znacheniyami v metricheskom prostranstve”, Sib. mat. zhurn., 38:3 (1997), 657–675 | MR | Zbl

[13] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, v. I, Mir, M., 1964, 533 pp.

[14] Chernikov V. M., Vodop'yanov S. K., “Sobolev Spaces and Hypoelliptic Equations, I”, Siberian Advances in Mathematics, 6:3 (1996), 27–67 | MR

[15] Chernikov V. M., Vodop'yanov S. K., “Sobolev Spaces and Hypoelliptic Equations, II”, Siberian Advances in Mathematics, 6:4 (1996), 64–96 | MR

[16] Cheeger J., “Differentiability of Lipschitz functions on metric measure spaces”, Geometric and Functional Analysis, 9 (1999), 428–517 | DOI | MR | Zbl

[17] Folland G. B., Stein I. M., Hardy spaces on homogeneous groups, Math. Notes, 28, Princeton Univ. Press, Princeton, 1982, 274 pp. | MR | Zbl

[18] Fanghua L., Xiaoping Y., Geometric measure theory, Science Press, Beijing–New York, 2002, 237 pp. | MR | Zbl

[19] Federer H., Geometric measure theory, Springer, Berlin, 1969, 676 pp. | MR | Zbl

[20] Gromov M., “Carnot–Caratheodory spaces seen from within”, Sub-Reimannian geometry, Birkhäuser, Basel, 1996, 79–323 | MR | Zbl

[21] Hajlasz P., “Sobolev spaces on an arbitrary metric space”, Potential Analysis, 6 (1996), 403–415 | MR

[22] Heinonen J., Lectures on analysis on metric spaces, Springer, NY, 2001, 140 pp. | MR

[23] Hörmander L., “Hypoelliptic second order differential equations”, Acta Math., 119 (1967), 147–171 | DOI | MR | Zbl

[24] Jerison D., “The Poincaré inequality for vector fields satisfying Hörmander's condition”, Duke Math. J., 53:2 (1986), 503–523 | DOI | MR | Zbl

[25] Koranyi A., Reimann H. M., “Foundations for the theory of quasiconformal mappings on the Heisenberg group”, Adv. Math., 111 (1995), 1–87 | DOI | MR | Zbl

[26] Lu G., “Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications”, Rev. Mat. Iberoamericana, 8:3 (1992), 367–439 | MR | Zbl

[27] Lu G., “The sharp Poincaré inequality for free vector fields: An endpoint result”, Rev. Mat. Iberoamericana, 10:3 (1994), 453–466 | MR | Zbl

[28] Magnani V., “Differentiability and area formula on stratified Lie groups”, Houston Journal of Math., 27:2 (2001), 297–323 | MR | Zbl

[29] Margulis G. A., Mostow G. D., “The differential of quasi-conformal mapping of a Carnot–Carathéodory spaces”, Geometric and Functional Analysis, 5:2 (1995), 402–433 | DOI | MR | Zbl

[30] Martio O., Malý J., “Luzin's condition $(N)$ and mappings of the class $W^{1,n}$”, J. Reine Angew. Math., 485 (1995), 19–36 | MR

[31] Mitchell J., “On Carnot–Carathéodory metrics”, J. Differential Geometry, 21 (1985), 35–45 | MR | Zbl

[32] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, American Mathematical Society, Providence, 2002, 260 pp. | MR

[33] Mostow G. D., Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies, 78, Princeton University Press, Princeton, N.J.; Tokyo Univ. Press, Tokyo, 1973, 195 pp. | MR | Zbl

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

[35] Pansu P., “Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un”, Ann. of Math., 119 (1989), 1–60 | DOI | MR

[36] Reshetnyak Yu. G., “Some geometric properties of functions and mappings with generalized derivatives”, Sib. Mat. Zh., 7:4 (1966), 886–919 | MR | Zbl

[37] Rademakher H., “Ueber partielle und totale Differenzierbarkeit, I”, Math. Ann., 79 (1919), 340–359 | DOI | MR

[38] Pauls S. D., A notion of rectifiability modeled on Carnot groups, Preprint, Dartmooth College, Hanover, 2001, 21 pp. http://hilbert.dartmooth.edu/~pauls/preprin4.html

[39] Stein E. M., Harmonic Analysis: Real-Variables Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. press, Princeton, 1993, 695 pp. | MR | Zbl

[40] Stepanoff W., “Ueber totale Differenzierbarkeit”, Math. Ann., 90 (1923), 318–320 | DOI | MR | Zbl

[41] Vodop'yanov S. K., “$\mathcal P$-differentiability on Carnot groups in different topologies and related topics”, Trudy po analizu i geometrii, Redaktor-sostavitel S. K. Vodopyanov, Izd-vo IM SO RAN, Novosibirsk, 2000, 603–670 | MR

[42] Vodop'yanov S. K., “Sobolev classes and quasiconformal mappings on Carnot-Caratheodory spaces”, Geometry, Topology and Physics, Proceedings of the First Brazil-USA Workshop (Campinas, Brazil, June 30–July 7, 1996), eds. B. N. Apanasov, S. B. Bradlow, W. A. Rodrigues, K. K. Uhlenbeck, Walter de Gruyter Co., Berlin–New York, 1997, 301–316 | MR

[43] Vodop'yanov S. K., Greshnov A. V., “Quasiconformal mappings and $BMO$-spaces on metric structures”, Siberian Advances in Mathematics, 8:3 (1998), 132–150 | MR