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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Nonsmooth calculus


Author: Juha Heinonen
Journal: Bull. Amer. Math. Soc. 44 (2007), 163-232
MSC (2000): Primary 28A75, 49J52, 53C23, 51-02
DOI: https://doi.org/10.1090/S0273-0979-07-01140-8
Published electronically: January 24, 2007
MathSciNet review: 2291675
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Abstract: We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts.


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  • 1. Geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 70, Springer-Verlag, Berlin, 1993. Nonregular Riemannian geometry, A translation of Geometry, 4 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. Translation by E. Primrose. MR 1263963 (94i:53038)
  • 2. Problems in low-dimensional topology, Geometric Topology (Athens, GA, 1993) (Rob Kirby, ed.), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35-473. MR 1470751
  • 3. David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441 (97j:46024)
  • 4. Robert A. Adams, Sobolev spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, Vol. 65. MR 0450957 (56:9247)
  • 5. L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101-129. MR 0036841 (12:171c)
  • 6. A. D. Aleksandrov and V. A. Zalgaller, Intrinsic geometry of surfaces, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15, American Mathematical Society, Providence, RI, 1967. MR 0216434 (35:7267)
  • 7. Luigi Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 3, 439-478. MR 1079985 (92d:26022)
  • 8. -, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), no. 1, 51-67. MR 1823840 (2002b:31002)
  • 9. Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498 (2006k:49001)
  • 10. Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1-80. MR 1794185 (2001k:49095)
  • 11. L. Ambrosio, M. Miranda, Jr., and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat., vol. 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, pp. 1-45. MR 2118414 (2005j:49036)
  • 12. Luigi Ambrosio and Francesco Serra Cassano (eds.), Lectures notes on analysis in metric spaces, Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses Given by Teachers at the School], Scuola Normale Superiore, Pisa, 2000. Papers from the International Summer School held in Trento, May 1999. MR 2023120 (2004h:00022)
  • 13. Luigi Ambrosio and Paolo Tilli, Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and Its Applications, vol. 25, Oxford University Press, Oxford, 2004. MR 2039660 (2004k:28001)
  • 14. P. Assouad, Plongements lipschitziens dans $ {\bf R}^n$, Bull. Soc. Math. France 111 (1983), 429-448. MR 0763553 (86f:54050)
  • 15. Kari Astala, Mario Bonk, and Juha Heinonen, Quasiconformal mappings with Sobolev boundary values, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 3, 687-731. MR 1990676 (2004k:30045)
  • 16. Pascal Auscher, Thierry Coulhon, and Alexander Grigor'yan (eds.), Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemporary Mathematics, vol. 338, American Mathematical Society, Providence, RI, 2003. Lecture notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs held in Paris, April 16-July 13, 2002. MR 2041910 (2004k:00018)
  • 17. Z. Balogh and P. Koskela, Quasiconformality, quasisymmetry and removability in Loewner spaces, Duke Math. J. 101 (2000), no. 3, 554-577. With an appendix by J. Väisälä. MR 1740689 (2001d:30029)
  • 18. Z. M. Balogh and S. M. Buckley, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003), no. 2, 261-301. MR 1992014 (2004i:30042)
  • 19. Z. M. Balogh, P. Koskela, and S. Rogovin, Absolute continuity of quasiconformal mappings on curves, Geometric and Functional Analysis (to appear).
  • 20. Z. M. Balogh, K. Rogovin, and T. Zürcher, The Stepanov differentiability theorem in metric measure spaces, J. Geom. Anal. 14 (2004), no. 3, 405-422. MR 2077159 (2005d:28008)
  • 21. D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675 (2001g:53130)
  • 22. A. Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 1-78. MR 1421822 (98a:53108)
  • 23. Arne Beurling, The collected works of Arne Beurling. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston Inc., Boston, MA, 1989. Complex Analysis, edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR 1057613 (92k:01046a)
  • 24. Anders Björn, Jana Björn, and Nageswari Shanmugalingam, The Dirichlet problem for $ p$-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173-203. MR 1971145 (2004e:31007)
  • 25. -, The Perron method for $ p$-harmonic functions in metric spaces, J. Differential Equations 195 (2003), no. 2, 398-429. MR 2016818 (2004i:31012)
  • 26. M. Bonk, Quasiconformal geometry of fractals, ICM Proceedings, Madrid (2006).
  • 27. M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001). MR 1829896 (2003b:30024)
  • 28. Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127-183. MR 1930885 (2004k:53057)
  • 29. -, Rigidity for quasi-Möbius group actions, J. Differential Geom. 61 (2002), no. 1, 81-106. MR 1949785 (2004b:53059)
  • 30. -, Rigidity for quasi-Fuchsian actions on negatively curved spaces, Int. Math. Res. Not. (2004), no. 61, 3309-3316. MR 2096259 (2006b:53051)
  • 31. -, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005), 219-246 (electronic). MR 2116315 (2005k:20102)
  • 32. Mario Bonk and Urs Lang, Bi-Lipschitz parameterization of surfaces, Math. Ann. 327 (2003), no. 1, 135-169. MR 2006006
  • 33. Nicolas Bourbaki, Elements of the history of mathematics, Springer-Verlag, Berlin, 1994, Translated from the 1984 French original by John Meldrum. MR 1290116 (95c:01001)
  • 34. M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), 245-268. MR 1445387 (98c:20056)
  • 35. M. Bourdon and H. Pajot, Poincaré inequalities and quasiconformal structure on the boundaries of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315-2324. MR 1610912 (99j:30024)
  • 36. -, Cohomologie $ l\sb p$ et espaces de Besov, J. Reine Angew. Math. 558 (2003), 85-108. MR 1979183 (2004e:20073)
  • 37. Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • 38. A. Bruckner, J. Bruckner, and B. Thomson, Real analysis, Prentice-Hall, NJ, 1997.
  • 39. J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. of Math. (2) 143 (1996), no. 3, 435-467. MR 1394965 (97b:57017)
  • 40. Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418 (2002e:53053)
  • 41. P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), 213-230. MR 0683635 (84e:58076)
  • 42. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. MR 1708448 (2000g:53043)
  • 43. -, Degeneration of Riemannian metrics under Ricci curvature bounds, Scuola Normale Superiore, 2001. Lezioni Fermiane, Pisa, 2001. MR 006642 (2004j:53049)
  • 44. -, Integral bounds on curvature elliptic estimates and rectifiability of singular sets, Geom. Funct. Anal. 13 (2003), no. 1, 20-72. MR 1978491 (2004i:53041)
  • 45. Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406-480. MR 1484888 (98k:53044)
  • 46. -, On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), no. 1, 13-35. MR 1815410 (2003a:53043)
  • 47. -, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37-74. MR 1815411 (2003a:53044)
  • 48. J. Cheeger, T. H. Colding, and G. Tian, On the singularities of spaces with bounded Ricci curvature, Geom. Funct. Anal. 12 (2002), no. 5, 873-914. MR 1937830 (2003m:53053)
  • 49. Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128. MR 0303460 (46:2597)
  • 50. R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, no. 242, Springer-Verlag, Berlin, 1971. MR 0499948 (58:17690)
  • 51. Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569-645. MR 0447954 (56:6264)
  • 52. Tobias H. Colding and William P. Minicozzi II, Harmonic functions on manifolds, Ann. of Math. (2) 146 (1997), no. 3, 725-747. MR 1491451 (98m:53052)
  • 53. Alain Connes, Dennis Sullivan, and Nicolas Teleman, Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Topology 33 (1994), no. 4, 663-681. MR 1293305 (95g:58232)
  • 54. Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press Inc., Orlando, FL, 1986. MR 872468 (88a:57001)
  • 55. G. David and S. Semmes, Strong $ {A}\sb\infty$ weights, Sobolev inequalities and quasiconformal mappings, Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., vol. 122, Marcel Dekker, 1990, pp. 101-111. MR 1044784 (91c:30037)
  • 56. -, Fractured fractals and broken dreams: self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, vol. 7, Clarendon Press, Oxford University Press, 1997. MR 1616732 (99h:28018)
  • 57. S. K. Donaldson and D. P. Sullivan, Quasiconformal $ 4$-manifolds, Acta Math. 163 (1989), 181-252. MR 1032074 (91d:57012)
  • 58. J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften, vol. 262, Springer-Verlag, New York, 1984. MR 0731258 (85k:31001)
  • 59. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. General Theory, with the assistance of William G. Bade and Robert G. Bartle, reprint of the 1958 original, a Wiley-Interscience Publication. MR 1009162 (90g:47001a)
  • 60. Robert D. Edwards, The topology of manifolds and cell-like maps, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 111-127. MR 0562601 (81g:57010)
  • 61. J. Eells and B. Fuglede, Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics, vol. 142, Cambridge University Press, Cambridge, 2001. With a preface by M. Gromov. MR 1848068 (2002h:58017)
  • 62. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1992. MR 1158660 (93f:28001)
  • 63. E. B. Fabes, C. E. Kenig, and R. Serapioni, The local regularity of solutions to degenerate elliptic equations, Comm. PDE 7 (1982), 77-116. MR 0643158 (84i:35070)
  • 64. H. Federer, Geometric measure theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 153, Springer-Verlag, New York, 1969. MR 0257325 (41:1976)
  • 65. B. Franchi, P. Haj\lasz, and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1903-1924. MR 1738070 (2001a:46033)
  • 66. Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453. MR 0679066 (84b:57006)
  • 67. M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • 68. Joseph H. G. Fu, Bi-Lipschitz rough normal coordinates for surfaces with an $ L\sp 1$ curvature bound, Indiana Univ. Math. J. 47 (1998), no. 2, 439-453. MR 1647908 (99k:49090)
  • 69. B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-219. MR 0097720 (20:4187)
  • 70. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 0737190 (86c:35035)
  • 71. M. Gromov, Hyperbolic groups, Essays in Group Theory, S. Gersten, editor, MSRI Publications, Springer-Verlag, 1987, pp. 75-265. MR 0919826 (88e:20004)
  • 72. -, Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Progress in Mathematics, vol. 144, Birkhäuser, Basel, 1996, pp. 79-323. MR 1421823 (2000f:53034)
  • 73. -, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates. MR 1699320 (2000d:53065)
  • 74. M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Geometric Topology: Recent Developments (Montecatini Terme, 1990), Lecture Notes in Math., vol. 1504, Springer, Berlin, 1991, pp. 39-137. MR 1168043 (93f:53036)
  • 75. M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1-12. MR 0892185 (88e:53058)
  • 76. Peter Haïssinsky, Rigidity and expansion for rational maps, J. London Math. Soc. (2) 63 (2001), no. 1, 128-140. MR 1802762 (2001m:37085)
  • 77. P. Haj\lasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. MR 1401074 (97f:46050)
  • 78. -, Sobolev spaces on metric-measure spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 173-218. MR 2039955 (2005c:46039)
  • 79. P. Haj\lasz and P. Koskela, Sobolev met Poincaré, Memoirs Amer. Math. Soc. 145 (2000), no. 688. MR 1683160 (2000j:46063)
  • 80. P. Haj\lasz and O. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, J. Funct. Anal. 143 (1997), 221-246. MR 1428124 (98d:46034)
  • 81. J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • 82. -, The branch set of a quasiregular mapping, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 691-700. MR 1957076 (2003k:30034)
  • 83. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory of degenerate elliptic equations, The Clarendon Press, Oxford University Press, New York, 1993, Oxford Science Publications. MR 1207810 (94e:31003)
  • 84. J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61-79. MR 1323982 (96e:30051)
  • 85. -, From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 554-556. MR 1372507 (96m:30034)
  • 86. -, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 1654771 (99j:30025)
  • 87. -, A note on Lipschitz functions, upper gradients, and the Poincaré inequality, New Zealand J. Math. 28 (1999), 37-42. MR 1691958 (2000d:46041)
  • 88. J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87-139. MR 1869604 (2002k:46090)
  • 89. Juha Heinonen and Seppo Rickman, Geometric branched covers between generalized manifolds, Duke Math. J. 113 (2002), no. 3, 465-529. MR 1909607 (2003h:57003)
  • 90. Juha Heinonen and Dennis Sullivan, On the locally branched Euclidean metric gauge, Duke Math. J. 114 (2002), no. 1, 15-41. MR 1915034 (2004b:30044)
  • 91. Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34. MR 0353210 (50:5695)
  • 92. David A. Herron, Conformal deformations of uniform Loewner spaces, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2, 325-360. MR 2040578 (2005i:30029)
  • 93. Francis Hirsch and Gilles Lacombe, Elements of functional analysis, Graduate Texts in Mathematics, vol. 192, Springer-Verlag, New York, 1999. Translated from the 1997 French original by Silvio Levy. MR 1678925 (99j:46001)
  • 94. Tadeusz Iwaniec, The Gehring lemma, Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 181-204. MR 1488451 (99e:30012)
  • 95. D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. MR 0850547 (87i:35027)
  • 96. S. Kallunki and P. Koskela, Exceptional sets for the definition of quasiconformality, Amer. J. Math. 122 (2000), no. 4, 735-743. MR 1771571 (2001h:37095)
  • 97. S. Kallunki and N. Shanmugalingam, Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 455-464. MR 1833251 (2002c:31008)
  • 98. Michael Kapovich and Bruce Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 647-669. MR 1834498 (2002j:20077)
  • 99. Stephen Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), no. 2, 255-292. MR 2013501 (2004k:31019)
  • 100. -, A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), no. 2, 271-315. MR 2041901 (2005g:46070)
  • 101. -, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J. 53 (2004), no. 4, 1127-1150. MR 2095451 (2005g:53068)
  • 102. S. Keith and T. Laakso, Conformal Assouad dimension and modules, Geom. Funct. Anal. 14 (2004), no. 6, 1278-1321. MR 2135168 (2006g:30027)
  • 103. Stephen Keith and Kai Rajala, A remark on Poincaré inequalities on metric measure spaces, Math. Scand. 95 (2004), no. 2, 299-304. MR 2098359 (2005f:26057)
  • 104. S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Annals of Mathematics (to appear).
  • 105. Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • 106. Tero Kilpeläinen, A remark on the uniqueness of quasi continuous functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 261-262. MR 1601887 (99j:31016)
  • 107. Juha Kinnunen and Olli Martio, Potential theory of quasiminimizers, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 459-490. MR 1996447 (2004b:31010)
  • 108. -, Sobolev space properties of superharmonic functions on metric spaces, Results Math. 44 (2003), no. 1-2, 114-129. MR 2011911 (2004i:31013)
  • 109. Juha Kinnunen and Nageswari Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), no. 3, 401-423. MR 1856619 (2002i:35054)
  • 110. -, Polar sets on metric spaces, Trans. Amer. Math. Soc. 358 (2006), no. 1, 11-37. MR 2171221
  • 111. B. Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geome- trization and rigidity, ICM Proceedings, Madrid (2006).
  • 112. N. J. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659. MR 1266480 (95b:58043)
  • 113. P. Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291-304. MR 1714767 (2001g:46077)
  • 114. -, Upper gradients and Poincaré inequalities, Lecture Notes on Analysis in Metric Spaces (Trento, 1999), Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, 2000, pp. 55-69. MR 2023123 (2004i:46048)
  • 115. -, Sobolev spaces and quasiconformal mappings on metric spaces, European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 457-467. MR 1905335 (2003m:30049)
  • 116. P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1-17. MR 1628655 (99e:46042)
  • 117. P. Koskela, Kai Rajala, and Nageswari Shanmugalingam, Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces, J. Funct. Anal. 202 (2003), no. 1, 147-173. MR 1994768 (2004e:31006)
  • 118. P. Koskela, N. Shanmugalingam, and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces, Indiana Univ. Math. J. 49 (2000), no. 1, 333-352. MR 1777027 (2001g:46076)
  • 119. P. Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Dirichlet forms, Poincaré inequalities, and the Sobolev spaces of Korevaar and Schoen, Potential Anal. 21 (2004), no. 3, 241-262. MR 2075670 (2005f:31015)
  • 120. Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316. MR 1865418 (2002m:58052)
  • 121. T. Laakso, Ahlfors $ {Q}$-regular spaces with arbitrary $ {Q}$ admitting weak Poincaré inequalities, Geom. Funct. Anal. 10 (2000), 111-123. MR 1748917 (2001m:30027)
  • 122. Jean-François Lafont, Rigidity result for certain three-dimensional singular spaces and their fundamental groups, Geom. Dedicata 109 (2004), 197-219. MR 2114076 (2006g:57032)
  • 123. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der Mathematischen Wissenschaften, Band 180. MR 0350027 (50:2520)
  • 124. D. G. Larman, A new theory of dimension, Proc. London Math. Soc. 17 (1967), 178-192. MR 0203691 (34:3540)
  • 125. James R. Lee, Manor Mendel, and Assaf Naor, Metric structures in $ L\sb 1$: dimension, snowflakes, and average distortion, European J. Combin. 26 (2005), no. 8, 1180-1190. MR 2163751 (2006g:46012)
  • 126. B. Levi, Sul principio di Dirichlet, Rend. Circ. Mat. Palermo 22 (1906), 293-359.
  • 127. Genadi Levin and Sebastian van Strien, Bounds for maps of an interval with one critical point of inflection type. II, Invent. Math. 141 (2000), no. 2, 399-465. MR 1775218 (2001i:37061)
  • 128. J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Mathematics (to appear).
  • 129. -, Weak curvature conditions and functional inequalities, Journal of Functional Analysis (to appear).
  • 130. J. Luukkainen and E. Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), 531-534. MR 1443161 (99c:28009)
  • 131. J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), 85-122. MR 0515647 (80b:57015)
  • 132. Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542 (98h:35080)
  • 133. P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)
  • 134. Vladimir G. Maz´ja, Sobolev spaces, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 0817985 (87g:46056)
  • 135. J. Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35-45. MR 0806700 (87d:53086)
  • 136. C. B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. MR 0202511 (34:2380)
  • 137. G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, NJ, 1973, Annals of Mathematics Studies, No. 78. MR 0385004 (52:5874)
  • 138. -, A remark on quasiconformal mappings on Carnot groups, Michigan Math. J. 41 (1994), no. 1, 31-37. MR 1260606 (95c:22017)
  • 139. F. Nazarov, S. Treil, and A. Volberg, The $ Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151-239. MR 1998349 (2005d:30053)
  • 140. Shin-ichi Ohta, Cheeger type Sobolev spaces for metric space targets, Potential Anal. 20 (2004), no. 2, 149-175. MR 2032946 (2005h:58017)
  • 141. -, Harmonic maps and totally geodesic maps between metric spaces, Tohoku Mathematical Publications, vol. 28, Tohoku University Mathematical Institute, Sendai, 2004. Dissertation, Tohoku University, Sendai, 2004. MR 2051353 (2005e:58023)
  • 142. P. Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. MR 1024425 (90k:53079)
  • 143. -, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60. MR 0979599 (90e:53058)
  • 144. Conrad Plaut, Metric spaces of curvature $ \geq k$, Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, pp. 819-898. MR 1886682 (2002m:53063)
  • 145. Hans Rademacher, Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Math. Ann. 79 (1919), no. 4, 340-359. MR 1511935
  • 146. K. Rajala, Alexandrov-avaruudet, ylägradientit ja Poincarén epäyhtälö (in Finnish), Pro gradu -tutkielma. Jyväskylän yliopisto (2000), 1-41.
  • 147. Yu. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), 657-675. MR 1457485 (98h:46031)
  • 148. -, On the conformal representation of Alexandrov surfaces, Papers on Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 287-304. MR 1886629 (2003b:53075)
  • 149. Juha Rissanen, Wavelets on self-similar sets and the structure of the spaces $ M\sp {1,p}(E,\mu)$, Ann. Acad. Sci. Fenn. Math. Diss. (2002), no. 125, 46. Dissertation, University of Jyväskylä, Jyväskylä, 2002. MR 1880640 (2002k:42081)
  • 150. W. Rudin, Functional analysis, second ed., McGraw-Hill Inc., New York, 1991. MR 1157815 (92k:46001)
  • 151. Stanis\law Saks, Theory of the integral, Second revised edition. English translation by L. C. Young, with two additional notes by Stefan Banach, Dover Publications Inc., New York, 1964. MR 0167578 (29:4850)
  • 152. L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417-450. MR 1180389 (93m:58122)
  • 153. L. Schwartz, Théorie des distributions. Tome I, Actualités Sci. Ind., no. 1091 = Publ. Inst. Math. Univ. Strasbourg 9, Hermann & Cie., Paris, 1950. MR 0035918 (12:31d)
  • 154. S. Semmes, Finding structure in sets with little smoothness, Proc. of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 875-885. MR 1403987 (97j:28010)
  • 155. -, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. 2 (1996), 155-295. MR 1414889 (97j:46033)
  • 156. -, Good metric spaces without good parameterizations, Rev. Mat. Iberoamericana 12 (1996), 187-275. MR 1387590 (97e:57025)
  • 157. -, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $ {A}\sb\infty$-weights, Rev. Mat. Iberoamericana 12 (1996), 337-410. MR 1402671 (97e:30040)
  • 158. -, Mappings and spaces, Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 347-368. MR 1488459 (99c:30045)
  • 159. -, Metric spaces and mappings seen at many scales (appendix), Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original, translated from the French by Sean Michael Bates. MR 1699320 (2000d:53065)
  • 160. -, Some novel types of fractal geometry, Oxford Mathematical Monographs, Oxford University Press, New York, 2001. MR 1815356 (2002h:53073)
  • 161. -, Some topics concerning homeomorphic parameterizations, Publ. Mat. 45 (2001), no. 1, 3-67. MR 1829576 (2002c:57039)
  • 162. N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Ph.D. thesis, University of Michigan, 1999.
  • 163. -, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243-279. MR 1809341 (2002b:46059)
  • 164. -, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), no. 3, 1021-1050. MR 1879250 (2003c:31010)
  • 165. -, Some convergence results for $ p$-harmonic functions on metric measure spaces, Proc. London Math. Soc. (3) 87 (2003), no. 1, 226-246. MR 1978575 (2005f:31010)
  • 166. L. Siebenmann and D. Sullivan, On complexes that are Lipschitz manifolds, Geometric Topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York, 1979, pp. 503-525. MR 0537747 (80h:57027)
  • 167. S. L. Sobolev, On some estimates relating to families of functions having derivatives that are square integrable, Dokl. Akad. Nauk SSSR 1 (1936), 267-270 (in Russian).
  • 168. -, On a theorem in functional analysis, Math. Sb. 4 (1938), 471-497 (in Russian).
  • 169. -, Some applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, vol. 90, American Mathematical Society, Providence, RI, 1991. Translated from the third Russian edition by Harold H. McFaden, with comments by V. P. Palamodov. MR 1125990 (92e:46067)
  • 170. Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, second ed., Publish or Perish Inc., Wilmington, Del., 1979. MR 0532831 (82g:53003b)
  • 171. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970, Princeton Mathematical Series, No. 30. MR 0290095 (44:7280)
  • 172. -, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993, Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
  • 173. K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Mathematica 196 (2006), no. 1, 65-131. MR 2237206
  • 174. -, On the geometry of metric measure spaces. II, Acta Mathematica 196 (2006), no. 1, 133-177. MR 2237207
  • 175. D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric Topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York, 1979, pp. 543-555. MR 0537749 (81m:57012)
  • 176. -, Exterior $ d$, the local degree, and smoothability, Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 328-338. MR 1368667 (97d:57034)
  • 177. L. Tonelli, Sulla quadratura delle superficie, Atti Reale Accad. Lincei 3 (1926), 633-638.
  • 178. Tatiana Toro, Surfaces with generalized second fundamental form in $ L\sp 2$ are Lipschitz manifolds, J. Differential Geom. 39 (1994), no. 1, 65-101. MR 1258915 (95b:49066)
  • 179. -, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995), no. 1, 193-227. MR 1317632 (96b:28006)
  • 180. Hans Triebel, The structure of functions, Monographs in Mathematics, vol. 97, Birkhäuser Verlag, Basel, 2001. MR 1851996 (2002k:46087)
  • 181. P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303-342 (1982). MR 0658932 (84a:57016)
  • 182. J. T. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525-548. MR 1642158 (99i:30038)
  • 183. -, Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conf. Geom. Dynam. 5 (2001), 21-73. MR 1872156 (2002m:30026)
  • 184. N. Th. Varopoulos, Fonctions harmoniques sur les groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 519-521. MR 0892879 (88f:22026)
  • 185. A. L. Vol´berg and S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 666-675. English translation: Math. USSR-Izv., 30:629-638, 1988. MR 0903629 (88i:28006)
  • 186. M.-K. von Renesse, On local Poincaré via transportation, Mathematische Zeitschrift (to appear).
  • 187. Nik Weaver, Lipschitz algebras, World Scientific Publishing Co. Inc., River Edge, NJ, 1999. MR 1832645 (2002g:46002)
  • 188. -, Lipschitz algebras and derivations. II. Exterior differentiation, J. Funct. Anal. 178 (2000), no. 1, 64-112. MR 1800791 (2002g:46040a)
  • 189. Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, NJ, 1957. MR 0087148 (19:309c)
  • 190. J-M. Wu, Hausdorff dimension and doubling measures on metric spaces, Proc. Amer. Math. Soc. 126 (1998), 1453-1459. MR 1443418 (99h:28016)
  • 191. K. Yosida, Functional analysis, Springer-Verlag, New York, 1980. MR 0617913 (82i:46002)
  • 192. W. P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. MR 1014685 (91e:46046)

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Additional Information

Juha Heinonen
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: juha@umich.edu

DOI: https://doi.org/10.1090/S0273-0979-07-01140-8
Received by editor(s): November 7, 2005
Received by editor(s) in revised form: June 16, 2006
Published electronically: January 24, 2007
Additional Notes: This paper constitutes an expanded version of the AMS invited address given by the author in Boulder, Colorado, in October 2003
The author is grateful for the support and hospitality of MSRI and UC Berkeley, where the bulk of this paper was prepared during a visit in 2002-2003. Supported also by NSF grants DMS 0353549 and DMS 0244421.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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