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