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Quantitative conditions of rectifiability. (Conditions quantitatives de rectifiabilité.) (French) Zbl 0890.28004
A set $$E\subset {\mathbf R}^n$$ is said to be $$d$$-rectifiable if $$E\subset E_{0}\cup \bigcup \Gamma_{i}$$, with the $$d$$-dimensional Hausdorff measure of $$E_{0}$$ equal to zero and $$\Gamma_{i}=\{x+A_{i}(x);x\in P_{i}\}$$, where $$A_i:P_i\rightarrow P_i^\perp$$ is a Lipschitz map and $$P_{i}$$ is a $$d$$-dimensional subspace of $${\mathbf R}^n$$. Define the numbers $$\beta_{\infty}$$, of P. W. Jones, as $\beta_{\infty} (x,t,E)=\inf_{P} \sup_{y\in E\cap B(x,t)} \biggl( {\text{ dist}(y,P) \over t}\biggr),$ if $$E\cap B(x,t)\not= \emptyset$$; if the intersection is empty, then $$\beta_{\infty} (x,t,E)=0.$$ For $$1\leq q < \infty$$, $$\beta_{q}$$ is defined as $\beta_{q}(x,t,E)=\inf_{P} \Biggl({1 \over t^{d}}\int_{E\cap B(x,t)} \biggl({\text{ dist}(y,P) \over t}\biggr)^{q} dH^{d}(y) \Biggr)^{1/q}.$ The infimum is taken over all $$d$$-planes in $${\mathbf R}^{n}$$. The number $$q$$ is to be chosen as follows: if $$d=1$$ then $$1\leq q \leq \infty$$, if $$d>1$$ then $$1\leq q < 2d/(d-2)$$.
The main result is the following:
Let $$E\subset {\mathbf R}^n$$ be a compact set with $$H^d(E)<\infty$$ and suppose that at $$H^{d}$$-almost all $$x\in E$$, $\Theta_{*}^{d}(x,E)=\liminf_{r\downarrow 0} {H^{d}(E\cup B(x,r) )\over (2r)^{d}}>0,$ $\int^{1}_{0}\beta_{q}(x,t,E)^{2}{\text{ d}t \over t}<\infty.$ Then the set $$E$$ is $$d$$-rectifiable.
Reviewer: O.Svensson (Lulea)

##### MSC:
 28A75 Length, area, volume, other geometric measure theory
##### Keywords:
rectifiable set; Jones numbers; Hausdorff measure; Lipschitz map
Full Text:
##### References:
 [1] BISHOP (C.J.) . - Harmonic measure supported on curves , Thèse, Université de Chicago, 1987 . [2] BISHOP (C.J.) . - Some conjectures concerning harmonic measure , dans Partial differential equations with minimal smoothness, IMA vol. in Math. and its Applications, Springer Verlag, t. 42, 1991 . Zbl 0792.30005 · Zbl 0792.30005 [3] BISHOP (C.J.) and JONES (P.W.) . - Harmonic measure, L2 estimates and the schwarzian derivatives , J. Analyse Math., t. 62, 1994 , p. 77-113. MR 95f:30034 | Zbl 0801.30024 · Zbl 0801.30024 · doi:10.1007/BF02835949 [4] DAVID (G.) . - Wavelets and singular integrals on curves and surfaces , Lecture Notes in Math., Springer Verlag, t. 1465, 1991 . MR 92k:42021 | Zbl 0764.42019 · Zbl 0764.42019 · doi:10.1007/BFb0091544 [5] DAVID (G.) and SEMMES (S.) . - Singular integrals and rectifiable sets in \Bbb Rn : au-delà des graphes lipschitziens , Astérisque, t. 193, 1991 . Zbl 0743.49018 · Zbl 0743.49018 [6] DAVID (G.) and SEMMES (S.) . - Analysis of and on uniformly rectifiable sets , Math. Surveys and Monographs, Amer. Math. Soc., t. 38, 1993 . MR 94i:28003 | Zbl 0832.42008 · Zbl 0832.42008 [7] FALCONER (K.J.) . - Geometry of fractal sets . - Cambridge University Press, 1984 . Zbl 0587.28004 · Zbl 0587.28004 [8] FEDERER (H.) . - Geometric measure theory . - Springer Verlag, 1969 . MR 41 #1976 | Zbl 0176.00801 · Zbl 0176.00801 [9] GARNETT (J.) . - Positive length but zero analytic capacity , Proc. Amer. Math. Soc., t. 24, 1970 , p. 696-699. MR 43 #2203 | Zbl 0208.09803 · Zbl 0208.09803 · doi:10.2307/2037304 [10] JONES (P.W.) . - Square functions, Cauchy integrals, analytic capacity, and harmonic measure , in Harmonic analysis and Partial differential equations, Lecture Notes in Math., Springer Verlag, t. 1384, 1989 , p. 24-68. MR 91b:42032 | Zbl 0675.30029 · Zbl 0675.30029 [11] JONES (P.W.) . - Rectifiable sets and the traveling salesman problem , Inventiones Math., t. 102, 1990 , p. 1-15. MR 91i:26016 | Zbl 0731.30018 · Zbl 0731.30018 · doi:10.1007/BF01233418 · eudml:143825 [12] MATTILA (P.) . - Geometry of sets and measures in euclidean spaces , Cambridge Studies in Advanced Math., Cambridge University Press, t. 44, 1995 . MR 96h:28006 | Zbl 0819.28004 · Zbl 0819.28004 [13] MATTILA (P.) , MELNIKOV (M.) and VERDERA (J.) . - The Cauchy integral , analytic capacity and uniform rectifiability, Annals of Math., t. 144, 1996 , p. 127-136. MR 97k:31004 | Zbl 0897.42007 · Zbl 0897.42007 · doi:10.2307/2118585 [14] OKIKIOLU (K.) . - Characterization of subsets of rectifiable curves in \Bbb Rn , J. London Math. Soc., t. 46, 1992 , p. 336-348. MR 93m:28008 | Zbl 0758.57020 · Zbl 0758.57020 · doi:10.1112/jlms/s2-46.2.336 [15] PAJOT (H.) . - Théorème de recouvrement par des ensembles Ahlfors-réguliers et capacité analytique , C. R. Acad. Sci. Paris, t. 323, série I, 1996 , p. 133-135. MR 97c:30034 | Zbl 0863.30033 · Zbl 0863.30033 [16] STEIN (E.M.) . - Singular integrals and differentiability properties of functions . - Princeton University Press, 1971 . Zbl 0207.13501 · Zbl 0207.13501 [17] STEIN (E.M.) and ZYGMUND (A.) . - On differentiability of functions , Studia Math., t. 23, 1964 , p. 247-283. Article | MR 28 #2176 | Zbl 0122.30203 · Zbl 0122.30203 · eudml:217074
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