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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
rectifiability
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