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Quantitative rectifiability and Lipschitz mappings. (English) Zbl 0792.49029
The authors give some new sufficient conditions for rectifiability (“qualitative rectifiability”) of sets \(E\) in \(\mathbb{R}^ n\) and obtain some interesting results about the connections of these with previously introduced sufficient conditions, called “quantitative rectifiability” by the authors. These sufficient conditions are so-called BPLG (the set \(E\) has ‘big pieces of Lipschitz graphs’) [see the first author, Ann. Sci. Ecole Norm. Super., IV. Ser. 17, 157-189 (1984; Zbl 0537.42016); ibid. 21, No. 2, 225-258 (1988; Zbl 0655.42013)], implying condition B [see the first author, Rev. Mat. Iberoam. 4, No. 1, 73-114 (1988; Zbl 0696.42011); the first author and D. Jerison, Indiana Univ. Math. J. 39, No. 3, 831-845 (1990; Zbl 0758.42008)].
The authors establish the following important results. If a set \(E\subset\mathbb{R}^ n\) is regular, has big projections, and satisfies the Weak Geometric Lemma, then \(E\) has BPLG. This theorem 1.14 allows the authors to give a simpler proof of David’s theorem about the fact that condition B follows from BPLG. If a set \(E\subset\mathbb{R}^ n\) satisfies Condition B, then \(H_ \varepsilon\) is a Carleson set for all \(\varepsilon> 0\) and \(E\) satisfies the Local Symmetry Condition; conversely, if \(E\) is regular, \(d= n- 1\) (Hausdorff dimension of \(E\)) and \(H_ \varepsilon'\) is a Carleson set for all \(\varepsilon>0\), then \(E\) satisfies Condition E. Remember that \(H_ \varepsilon\) (respectively, \(H_ \varepsilon'\)) is the set of all \((x,t)\in E\times R_ +\) for which there are \(y_ 1,y_ 2\in B(x,t)=\bigl\{u\in \mathbb{R}^ m: \| u- x\|\leq t\bigr\}\) such that \(\text{dist}(y_ i,E)\geq\varepsilon t\) \((i=1,2)\), \(y_ 1\) to \(y_ 2\) lie in the same component of \(\mathbb{R}^{d+1}\backslash E\), and the line segment joining \(y_ 1\) and \(y_ 2\) intersects \(E\) (respectively, \({1\over 2} (y_ 1+y_ 2)\in E\)). As is well-known, the above-mentioned quantitative conditions of rectifiability of \(E\) arise when studying estimates for singular integral operators on \(E\), namely \[ T_ * f(x)=\sup_{\varepsilon>0}\Bigl| \int_{E\backslash B(x,\varepsilon)} k(x- y)f(y) d\mu(x)\Bigr|. \] It should be noticed that all the results on these topics are presented in a monograph of the authors with the title ‘Analysis of and on uniformly rectifiable sets’ [Mathematical Surveys and Monographs 38, Providence, Amer. Math. Soc. (1993)].

MSC:
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
28A75 Length, area, volume, other geometric measure theory
Keywords:
rectifiability
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