## NONSMOOTH CALCULUS

Citations: | 13 - 0 self |

### BibTeX

@MISC{Heinonen_nonsmoothcalculus,

author = {Juha Heinonen},

title = {NONSMOOTH CALCULUS},

year = {}

}

### OpenURL

### Abstract

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. 1.

### Citations

2094 |
Elliptic partial differential equation of second order, 2nd ed., Springer-Verlag
- Gilbarg, Trudinger
- 1983
(Show Context)
Citation Context ... using the space L ∞ (R n ) in place of L p (R n ). 6.1. Notes. For more information about Sobolev-Poincaré inequalities, potential estimates, and related issues, see, for example, [3], [172], [173], =-=[70]-=-, [62], [135], [192]. 7. Capacity and Modulus To reiterate, functions in W 1,p (R n ) are integrable beyond the initial requirement, and they have representatives that are absolutely continuous on lin... |

1338 |
Singular integrals and differentiability properties of functions
- Stein
- 1970
(Show Context)
Citation Context ...r W 1,p (R n ) using the space L ∞ (R n ) in place of L p (R n ). 6.1. Notes. For more information about Sobolev-Poincaré inequalities, potential estimates, and related issues, see, for example, [3], =-=[172]-=-, [173], [70], [62], [135], [192]. 7. Capacity and Modulus To reiterate, functions in W 1,p (R n ) are integrable beyond the initial requirement, and they have representatives that are absolutely cont... |

796 |
A Comprehensive Introduction to Differential Geometry. Publish or
- Spivak
- 1979
(Show Context)
Citation Context ...is given Hausdorff dimension that are suitable for calculus. These spaces cannot resemble the classical fractals; they must contain lots of rectifiable curves, for example. 8.12. Notes. Spivak’s book =-=[171]-=- contains a nice analysis of Riemann’s lecture. Of course, there is a huge literature on this topic. The most immediate generalization of a Riemannian metric leads to Finsler geometry, where one equip... |

756 |
Measure Theory and Fine Properties of Functions
- Evans, Gariepy
- 1992
(Show Context)
Citation Context ... thirties; see [168], [169]. Brief historical comments can be found for example in [137, Ch. 1.8, p. 19] and [135, p. 29]. Among the excellent modern sources for the material in this section are [4], =-=[62]-=-, [192]. For the theory of Sobolev spaces with weights, see [84]. For an example of a measure for which the gradient operator is not closable, see [63, pp. 91–92]. 6. Sobolev-Poincaré inequalities The... |

716 |
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43
- Stein
- 1995
(Show Context)
Citation Context ... (R n ) using the space L ∞ (R n ) in place of L p (R n ). 6.1. Notes. For more information about Sobolev-Poincaré inequalities, potential estimates, and related issues, see, for example, [3], [172], =-=[173]-=-, [70], [62], [135], [192]. 7. Capacity and Modulus To reiterate, functions in W 1,p (R n ) are integrable beyond the initial requirement, and they have representatives that are absolutely continuous ... |

561 | Metric Structures for Riemannian and Non-Riemannian Spaces with appendices by - Gromov - 1999 |

484 |
Geometry of sets and measures in Euclidean spaces: fractals and rectifiability
- Mattila
- 1995
(Show Context)
Citation Context ...7.3) Hn−p(E) < ∞ ⇒ cap p(E) = 0sNONSMOOTH CALCULUS 19 whenever 1 ≤ p ≤ n. 3 Moreover, singletons have positive p-capacity if p > n. Here and later Hs denotes the s-dimensional Hausdorff measure. (See =-=[134]-=- for the definition.) Theorem 7.1. Let 1 ≤ p ≤ n. Every function in W 1,p (R n ) has a representative that is p-quasicontinuous in the following sense: for every ɛ > 0 there is an open set U ⊂ R n suc... |

454 | Functional Analysis - Yosida - 1980 |

398 |
Metric Spaces of Non-Positive Curvature
- Bridson, Haefliger
- 1999
(Show Context)
Citation Context ...quips each tangent space of a smooth manifold with an arbitrary (smoothly varying) Banach norm; see, for example, [21]. Basic references to Lipschitz manifolds are [132], [175], [181]. The monographs =-=[37]-=-, [40] contain much information about spaces with bounded curvature in the sense of Alexandrov. See also the survey [145]. For the theory of surfaces of bounded curvature in the sense of Alexandrov, s... |

249 |
Hyperbolic groups
- Gromov
- 1987
(Show Context)
Citation Context ...gh the spaces by Semmes are singular as defined in this article, they still admit “branched Lipschitz parametrizations” by Euclidean space [90]. The fundamental source for Gromov hyperbolic spaces is =-=[71]-=-. See also [40], [37], and references there. For the boundaries of homogeneous spaces and buildings, see [138], [92], [144], [35], [36]. Further examples or references related to singular spaces can b... |

238 |
The topology of four-dimensional manifolds
- Freedman
- 1982
(Show Context)
Citation Context ...ask if a given four manifold can be metrized so that it has locally finite Hausdorff 4-measure. 25 There is a conjecture, attributed to Freedman, stating that every four manifold admits Hölder 24 See =-=[66]-=-, [67] for constructions of such manifolds. 25 Every noncompact four manifold is smoothable [67, p. 116], so the question is interesting only for compact manifolds.sNONSMOOTH CALCULUS 55 continuous ch... |

237 |
A Course in Metric Geometry
- Burago, Burago, et al.
- 2001
(Show Context)
Citation Context ...each tangent space of a smooth manifold with an arbitrary (smoothly varying) Banach norm; see, for example, [21]. Basic references to Lipschitz manifolds are [132], [175], [181]. The monographs [37], =-=[40]-=- contain much information about spaces with bounded curvature in the sense of Alexandrov. See also the survey [145]. For the theory of surfaces of bounded curvature in the sense of Alexandrov, see [6]... |

216 |
Multiple Integrals in the Calculus of Variations
- Morrey
- 1966
(Show Context)
Citation Context ...mpleteness. I apologize in advance to anyone who has been inadvertedly ignored or misrepresented. 1.1. Notes. Among the classic treatises on weakly differentiable functions and their applications are =-=[137]-=-, [170], [154]. For foundational treatments of analysis on spaces of homogeneous type, see [50], [51]. An abstract Poincaré inequality in a metric measure space as discussed in this article was formul... |

199 |
The topology of 4–manifolds
- Freedman, Quinn
- 1990
(Show Context)
Citation Context ... a given four manifold can be metrized so that it has locally finite Hausdorff 4-measure. 25 There is a conjecture, attributed to Freedman, stating that every four manifold admits Hölder 24 See [66], =-=[67]-=- for constructions of such manifolds. 25 Every noncompact four manifold is smoothable [67, p. 116], so the question is interesting only for compact manifolds.sNONSMOOTH CALCULUS 55 continuous charts m... |

198 |
Strong rigidity for locally symmetric spaces
- Mostow
- 1973
(Show Context)
Citation Context ...rtial differential equations and complex analysis, see [173] and the references there. Excellent sources for more geometric aspects of the theory are the articles [22], [72]. For Mostow rigidity, see =-=[138]-=-, [139]. The contemporary literature on harmonic analysis, partial differential equations, and geometric measure theory in sub-Riemannian contexts is large and growing. For a thorough discussion of th... |

195 |
Nonlinear potential theory of degenerate elliptic equations
- Heinonen, Kilpeläinen, et al.
- 1993
(Show Context)
Citation Context ...und for example in [137, Ch. 1.8, p. 19] and [135, p. 29]. Among the excellent modern sources for the material in this section are [4], [62], [192]. For the theory of Sobolev spaces with weights, see =-=[84]-=-. For an example of a measure for which the gradient operator is not closable, see [63, pp. 91–92]. 6. Sobolev-Poincaré inequalities The fundamental theorem of calculus gives global information about ... |

189 |
Carnot–Carathéodory spaces seen from within, in
- Gromov
- 1996
(Show Context)
Citation Context ...ory, related to hypoelliptic partial differential equations and complex analysis, see [173] and the references there. Excellent sources for more geometric aspects of the theory are the articles [22], =-=[72]-=-. For Mostow rigidity, see [138], [139]. The contemporary literature on harmonic analysis, partial differential equations, and geometric measure theory in sub-Riemannian contexts is large and growing.... |

182 |
Geometric integration theory
- Whitney
- 1957
(Show Context)
Citation Context ...s to satisfy the hypotheses in Theorem 11.2. It was known to Whitney already in the 1950s that Lipschitz charts on a manifold can be used to set up a measurable or L ∞ de Rhams54 JUHA HEINONEN theory =-=[189]-=-. Other similar analytic tools have been developed on Lipschitz manifolds. For example, using Lipschitz (or, more generally, quasiconformally) invariant notions of geometric analysis together with ide... |

179 |
An Introduction to Riemann-Finsler Geometry
- Bao, Chern, et al.
- 2000
(Show Context)
Citation Context ...immediate generalization of a Riemannian metric leads to Finsler geometry, where one equips each tangent space of a smooth manifold with an arbitrary (smoothly varying) Banach norm; see, for example, =-=[21]-=-. Basic references to Lipschitz manifolds are [132], [175], [181]. The monographs [37], [40] contain much information about spaces with bounded curvature in the sense of Alexandrov. See also the surve... |

175 |
Soboloev Spaces
- Maz’ja
- 1985
(Show Context)
Citation Context ...pace L ∞ (R n ) in place of L p (R n ). 6.1. Notes. For more information about Sobolev-Poincaré inequalities, potential estimates, and related issues, see, for example, [3], [172], [173], [70], [62], =-=[135]-=-, [192]. 7. Capacity and Modulus To reiterate, functions in W 1,p (R n ) are integrable beyond the initial requirement, and they have representatives that are absolutely continuous on lines. In additi... |

168 |
Foundations of modern potential theory
- Landkof
- 1972
(Show Context)
Citation Context ...or where such a theory is needed for external reasons. 7.3. Notes. The relationship between various capacities and pointwise behavior of functions was first studied in classical potential theory; see =-=[124]-=-, [58]. For general estimates between capacities and Hausdorff measures, see [124], [3], [135]. Elementary proofs for the particular estimates in (7.2) and (7.3) can be found in [84, Chapter 2]. There... |

167 |
Analyse harmonique non-commutative sur certains espaces homogenes. (French) Étude de certaines intégrales singulières
- Coifman, Weiss
- 1971
(Show Context)
Citation Context .... 1.1. Notes. Among the classic treatises on weakly differentiable functions and their applications are [137], [170], [154]. For foundational treatments of analysis on spaces of homogeneous type, see =-=[50]-=-, [51]. An abstract Poincaré inequality in a metric measure space as discussed in this article was formulated in [86], [87]. The work by Cheeger referred to in this introduction is [42]. For a subsequ... |

160 |
Extensions of Hardy spaces and their use in analysis
- Coifman, Weiss
- 1977
(Show Context)
Citation Context ... Notes. Among the classic treatises on weakly differentiable functions and their applications are [137], [170], [154]. For foundational treatments of analysis on spaces of homogeneous type, see [50], =-=[51]-=-. An abstract Poincaré inequality in a metric measure space as discussed in this article was formulated in [86], [87]. The work by Cheeger referred to in this introduction is [42]. For a subsequent wo... |

158 |
Lectures on Analysis on Metric Spaces
- Heinonen
- 2001
(Show Context)
Citation Context ...heeger referred to in this introduction is [42]. For a subsequent work by Keith that was also cited earlier, see [101]. Recent monographs and surveys on analysis on metric spaces include [12], [161], =-=[81]-=-, [13]. For the relevant literature on related but omitted topics, see [106], [180], [10], [9], and the numerous references in these papers. More references will be given in the course of the article.... |

157 |
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un
- Pansu
- 1989
(Show Context)
Citation Context ... Euclidean space [90]. The fundamental source for Gromov hyperbolic spaces is [71]. See also [40], [37], and references there. For the boundaries of homogeneous spaces and buildings, see [138], [92], =-=[144]-=-, [35], [36]. Further examples or references related to singular spaces can be found in [161], [81]. 9. Spaces of homogeneous type At this juncture, it is instructive to briefly discuss spaces of homo... |

146 |
Classical Potential Theory and its Probabilistic Counterpart
- Doob
- 1984
(Show Context)
Citation Context ...e such a theory is needed for external reasons. 7.3. Notes. The relationship between various capacities and pointwise behavior of functions was first studied in classical potential theory; see [124], =-=[58]-=-. For general estimates between capacities and Hausdorff measures, see [124], [3], [135]. Elementary proofs for the particular estimates in (7.2) and (7.3) can be found in [84, Chapter 2]. There is a ... |

137 |
The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, in
- Bellaïche
- 1996
(Show Context)
Citation Context ...he theory, related to hypoelliptic partial differential equations and complex analysis, see [173] and the references there. Excellent sources for more geometric aspects of the theory are the articles =-=[22]-=-, [72]. For Mostow rigidity, see [138], [139]. The contemporary literature on harmonic analysis, partial differential equations, and geometric measure theory in sub-Riemannian contexts is large and gr... |

131 |
Dierentiability of Lipschitz functions on metric measure spaces
- Cheeger
- 1999
(Show Context)
Citation Context ...eous type, see [50], [51]. An abstract Poincaré inequality in a metric measure space as discussed in this article was formulated in [86], [87]. The work by Cheeger referred to in this introduction is =-=[42]-=-. For a subsequent work by Keith that was also cited earlier, see [101]. Recent monographs and surveys on analysis on metric spaces include [12], [161], [81], [13]. For the relevant literature on rela... |

129 |
Quasiconformal maps in metric spaces with controlled geometry
- Heinonen, Koskela
- 1998
(Show Context)
Citation Context .... For foundational treatments of analysis on spaces of homogeneous type, see [50], [51]. An abstract Poincaré inequality in a metric measure space as discussed in this article was formulated in [86], =-=[87]-=-. The work by Cheeger referred to in this introduction is [42]. For a subsequent work by Keith that was also cited earlier, see [101]. Recent monographs and surveys on analysis on metric spaces includ... |

121 |
Sobolev met Poincaré
- Hajlasz, Koskela
(Show Context)
Citation Context ...quire only a Poincaré type inequality and a doubling measure. See, for example, [153], [52], [16], and the many references there. See also [116]. For examples of doubling p-Poincaré spaces, see [87], =-=[79]-=-, and references therein. Semmes’s article [156] introduced an important method to verify the validity of a Poincaré inequality; this method has been used later e.g. in [35], [122]. For an earlier art... |

100 | Problems in low-dimensional topology, Geometric topology - Kirby - 1997 |

99 |
Some Applications of Functional Analysis
- Sobolev
- 1988
(Show Context)
Citation Context ... ⊂ R n is an arbitrary open set. We also ignore the higher order Sobolev spaces in this article.s14 JUHA HEINONEN 5.2. Notes. The theory of Sobolev spaces dates from the nineteen thirties; see [168], =-=[169]-=-. Brief historical comments can be found for example in [137, Ch. 1.8, p. 19] and [135, p. 29]. Among the excellent modern sources for the material in this section are [4], [62], [192]. For the theory... |

96 |
Sobolev spaces and harmonic maps for metric space targets
- Korevaar, Schoen
- 1993
(Show Context)
Citation Context ...ach space. Ambrosio [7] was probably the first person to systematically study such mappings (in the context of mappings of bounded variation). For Riemannian domains, Sobolev mappings were studied in =-=[113]-=-, [148]. For the general theory, see [89]. These studies have applications to harmonic mappings with singular targets; see for example [113], [61], [120], [142], [141]. 15. Potential theory on singula... |

96 |
On the geometry of metric measure spaces
- Sturm
(Show Context)
Citation Context ...Similar questions for spaces with Ricci curvature bounds have attracted much attention recently. For an informative discussion of this problem, see [45, Appendix 2]. Lott and Villani [129], and Sturm =-=[174]-=- have independently proposed an approach for defining metric measure spaces with Ricci curvature bounded from below. Lott and Villani also prove a version of Federbush’s logarithmic Sobolev inequality... |

92 | T.H.Colding, On the structure of spaces with Ricci curvature bounded below
- Cheeger
(Show Context)
Citation Context ...survey of some of the applications of these ideas to Riemannian geometry, together with further references. Weak tangent spaces and tangent functions have been extensively used by Cheeger and Colding =-=[45]-=-, [46], [47], Cheeger [42], David and Semmes [56], Keith [101], Bonk and Kleiner [29], [30], and others. The concepts of a doubling metric space and Assouad dimension (under different name) can be fou... |

88 | Ricci curvature for metric-measure spaces via optimal transport
- Lott, Villani
(Show Context)
Citation Context ...ov in the 1950s. Similar questions for spaces with Ricci curvature bounds have attracted much attention recently. For an informative discussion of this problem, see [45, Appendix 2]. Lott and Villani =-=[129]-=-, and Sturm [174] have independently proposed an approach for defining metric measure spaces with Ricci curvature bounded from below. Lott and Villani also prove a version of Federbush’s logarithmic S... |

84 |
Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace
- Adams
- 1975
(Show Context)
Citation Context ...eteen thirties; see [168], [169]. Brief historical comments can be found for example in [137, Ch. 1.8, p. 19] and [135, p. 29]. Among the excellent modern sources for the material in this section are =-=[4]-=-, [62], [192]. For the theory of Sobolev spaces with weights, see [84]. For an example of a measure for which the gradient operator is not closable, see [63, pp. 91–92]. 6. Sobolev-Poincaré inequaliti... |

83 |
The splitting theorem for manifolds of non-negative Ricci curvature
- Cheeger, Gromoll
- 1971
(Show Context)
Citation Context ...t linear functions are differentiable or smooth, cf. [42, Remark 8.14]. A similar intrinsic characterization of linearity was used earlier by Cheeger and Gromoll in their celebrated splitting theorem =-=[49]-=-. In this connection, one should also mention works by Mostow [138], [139], and Pansu [144], where differentiability theorems in non-Riemannian settings were proved. As mentioned in subsection 12.2, t... |

82 |
Hedberg: Function spaces and potential theory
- Adams, I
- 1996
(Show Context)
Citation Context ...as for W 1,p (R n ) using the space L ∞ (R n ) in place of L p (R n ). 6.1. Notes. For more information about Sobolev-Poincaré inequalities, potential estimates, and related issues, see, for example, =-=[3]-=-, [172], [173], [70], [62], [135], [192]. 7. Capacity and Modulus To reiterate, functions in W 1,p (R n ) are integrable beyond the initial requirement, and they have representatives that are absolute... |

79 |
Plongements lipschitziens dans
- Assouad
- 1983
(Show Context)
Citation Context ...of a doubling metric space and Assouad dimension (under different name) can be found already in Larman’s paper [125] from 1967, but they have received more attention after Assouad’s embedding theorem =-=[14]-=- became a well known and effective tool in metric geometry. See [56], [160], [81], [161] and the references there.s38 JUHA HEINONEN Rademacher’s theorem was proved in 1919 [146]. A proof can be found,... |

76 |
Pinching constants for hyperbolic manifolds
- Gromov, Thurston
- 1987
(Show Context)
Citation Context ...ice geometric structure or not. For example, it seems to be unknown what kind of geometries can be carried on the boundary of the fundamental group of the negatively curved 4-manifolds constructed in =-=[75]-=-. Spaces with interesting features arise also in the case of compact hyperbolic manifolds with totally geodesic boundaries, or when closed hyperbolic manifolds are glued along totally geodesic submani... |

75 |
Newtonian spaces: an extension of Sobolev spaces to metric measure spaces
- Shanmugalingam
- 1999
(Show Context)
Citation Context ...n the modulus of a curve family. Independently of Cheeger, as50 JUHA HEINONEN systematic study of Sobolev spaces based on the concept of modulus and weak upper gradients was made by Shanmugalingam in =-=[163]-=-, [164]. Theorems 10.5, 10.8, and the equality C 1,p (X) = N 1,p (X), for 1 < p < ∞, are due to her; see also [98]. (The assertion for the equality C 1,p (X) = N 1,p (X) appears in [164, Theorem 4.10]... |

74 |
The Poincaré inequality for vector fields satisfying Hörmander condition
- Jerison
- 1986
(Show Context)
Citation Context ...exandrov also support (locally) a 1-Poincaré inequality. This follows from the Bonk-Lang parametrization theorem cited earlier. Carnot-Carathéodory spaces typically support a Poincaré inequality. See =-=[96]-=-, [184], [79, Section 11]. The validity of a Poincaré inequality carries over to Gromov-Hausdorff limits of metric measure spaces, where a convergence of measures has to be incorporated in the definit... |

72 |
Currents in metric spaces
- Ambrosio, Kirchheim
(Show Context)
Citation Context ...also cited earlier, see [101]. Recent monographs and surveys on analysis on metric spaces include [12], [161], [81], [13]. For the relevant literature on related but omitted topics, see [106], [180], =-=[10]-=-, [9], and the numerous references in these papers. More references will be given in the course of the article. 1.2. Acknowledgements. I would like to thank Stephen Semmes both for being a tireless ad... |

62 |
A note on the isoperimetric constant
- Buser
(Show Context)
Citation Context ...é inequality, in light of the discussion in subsection 8.8 and the fact that every complete Riemannian manifoldsNONSMOOTH CALCULUS 53 with nonnegative Ricci curvature supports a 1-Poincaré inequality =-=[41]-=-. See [47] for applications. Some singular spaces that arise as boundaries of Gromov hyperbolic groups support a Poincaré inequality; in many cases, it is not known whether or not a Poincaré inequalit... |

56 | Topology of homology manifolds
- Bryant, Ferry, et al.
- 1996
(Show Context)
Citation Context ...istantly related to Question 11.5. Finally, Semmes has also asked what kind of metric structures can exist on the exotic homology n-manifolds, n ≥ 5, constructed by Bryant, Ferry, Mio, and Weinberger =-=[39]-=-. Can they be metrized so that the hypotheses of Theorem 11.2 are valid, or just so as to have locally finite Hausdorff n-measure? 11.6. Poincaré inequality and Gromov-Hausdorff convergence. We first ... |

54 |
On Carnot-Caratheodory metrics
- Mitchell
- 1985
(Show Context)
Citation Context ...ocally compact and doubling. If X is a Riemannian manifold, then these spaces are always isometric to some Euclidean space. For a Carnot-Carathéodory space (at least under some additional assumptions =-=[136]-=-) each tangent space is a Carnot group, and for a Carnot group each tangent space is the same Carnot group. More generally, a weak tangent space (or cone) of a proper metric space (X, d) is a Gromov-H... |

53 |
Gradient Flows
- Ambrosio, Gigli, et al.
- 2005
(Show Context)
Citation Context ...ited earlier, see [101]. Recent monographs and surveys on analysis on metric spaces include [12], [161], [81], [13]. For the relevant literature on related but omitted topics, see [106], [180], [10], =-=[9]-=-, and the numerous references in these papers. More references will be given in the course of the article. 1.2. Acknowledgements. I would like to thank Stephen Semmes both for being a tireless advocat... |

51 | The Local Regularity of Solutions to Degenerate Elliptic Equations - Fabes, Kenig, et al. - 1982 |

50 |
Harmonic maps between Riemannian polyhedra, volume 142 of Cambridge Tracts in Mathematics
- Eells, Fuglede
- 2001
(Show Context)
Citation Context ...Riemannian domains, Sobolev mappings were studied in [113], [148]. For the general theory, see [89]. These studies have applications to harmonic mappings with singular targets; see for example [113], =-=[61]-=-, [120], [142], [141]. 15. Potential theory on singular spaces Cheeger’s theory, as explained in Section 12, gives the possibility to develop a theory of elliptic partial differential equations on sin... |