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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quantitative rectifiability and Lipschitz mappings

Authors: Guy David and Stephen Semmes
Journal: Trans. Amer. Math. Soc. 337 (1993), 855-889
MSC: Primary 42B20; Secondary 49Q15
MathSciNet review: 1132876
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Abstract: The classical notion of rectifiability of sets in $ {{\mathbf{R}}^n}$ is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with $ {L^p}$ estimates for singular integral operators on sets in $ {{\mathbf{R}}^n}$. We give a criterion for one reasonably natural quantitative rectifiability condition to hold, and we use it to give a new proof of a theorem in [D3]. We also give some results on the geometric properties of a certain class of sets in $ {{\mathbf{R}}^n}$ which can be viewed as generalized hypersurfaces. Along the way we shall encounter some questions concerning the behavior of Lipschitz functions, with regard to approximation by affine functions in particular. We shall also discuss an amusing variation of the classical Lipschitz and bilipschitz conditions, which allow some singularities forbidden by the classical conditions while still forcing good behavior on substantial sets.

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  • [D1] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. Ecole Norm. Sup. 17 (1984), 157-189. MR 744071 (85k:42026)
  • [D2] -, Opérateurs d'intégrale singuliere sur les surfaces régulières, Ann. Sci. Ecole Norm. Sup. (4) 21 (1988), 225-258. MR 956767 (89m:42014)
  • [D3] -, Morceaux de graphes lipschitziennes et intégrales singulières sur un surface, Rev. Mat. Iberoamericana 4 (1988), 73-114. MR 1009120 (90h:42026)
  • [D4] -, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Math., vol. 1465, Springer-Verlag,, 1991. MR 1123480 (92k:42021)
  • [DJ] G. David and D. Jerison, Lipschitz approximations to hypersurfaces, harmonic measure, and singular integrals, Indiana Math. J. 39 (1990), 831-845. MR 1078740 (92b:42021)
  • [DS] G. David and S. Semmes, Singular integrals and rectifiable sets in $ {{\mathbf{R}}^n}$: au-delà des graphes lipschitziens, Astérisque 193 (1991).
  • [DS2] -, Analysis of and on uniformly rectifiable sets, manuscript, 1992.
  • [Do] J. R. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc. 95 (1985), 21-31. MR 796440 (86k:46046)
  • [F] H. Federer, Geometric measure theory, Springer-Verlag, 1969. MR 0257325 (41:1976)
  • [G] F. Gehring, The $ {L^p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 0402038 (53:5861)
  • [JK] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), 80-147. MR 676988 (84d:31005b)
  • [J1] P. W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic Analysis and Partial Differential Equations, edited by J. Garcia-Cuerva, Lecture Notes in Math., vol. 1384, Springer-Verlag, 1989. MR 1013815 (91b:42032)
  • [J2] -, Lipschitz and bi-Lipschitz functions, Rev. Mat. Iberoamericana 4 (1988), 115-122. MR 1009121 (90h:26016)
  • [J3] -, Rectifiable sets and the travelling salesman problem, Invent. Math. 102 (1990), 1-15. MR 1069238 (91i:26016)
  • [Ma] P. Mattila, Lecture notes on geometric measure theory, Departmento de Matemáticas, Universidad de Extremadura, 1986. MR 931079 (89e:49037)
  • [Mu] T. Murai, A real-variable method for the Cauchy transform, and analytic capacity, Lecture Notes in Math., vol. 1307, Springer-Verlag, 1988. MR 944308 (89k:30022)
  • [S1] S. Semmes, A criterion for the boundedness of singular integrals on hypersurfaces, Trans. Amer. Math. Soc. 311 (1989), 501-513. MR 948198 (89k:42017)
  • [S2] -, Chord-arc surfaces with small constant II: good parameterizations, Adv. in Math. 88 (1991), 170-199. MR 1120612 (93d:42019b)
  • [S3] -, Differentiable function theory on hypersurfaces in $ {{\mathbf{R}}^n}$ (without bounds on their smoothness), Indiana Math. J. 39 (1990), 985-1004. MR 1087182 (92m:30088)
  • [S4] -, Analysis vs geometry on a class of rectifiable hypersurfaces in $ {{\mathbf{R}}^n}$, Indiana Math. J. 39 (1990), 1005-1035. MR 1087183 (92m:30089)
  • [St] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. MR 0290095 (44:7280)
  • [V] J. Väisälä, Invariants for quasisymmetric, quasimöbius, and bilipschitz maps, J. Analyse Math. 50 (1988), 201-233.

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Keywords: Quantitative rectifiability, Lipschitz mappings
Article copyright: © Copyright 1993 American Mathematical Society

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