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Characterization of n-rectifiability in terms of Jones’ square function: part I

  • Xavier TolsaEmail author
Article

Abstract

In this paper it is shown that if \(\mu \) is a finite Radon measure in \({\mathbb R}^d\) which is n-rectifiable and \(1\le p\le 2\), then
$$\begin{aligned} \displaystyle \int _0^\infty \beta _{\mu ,p}^n(x,r)^2\,\frac{dr}{r}<\infty \quad \mathrm{for}\,\, \mu {\text {-}}\mathrm{a.e.}\,\, x\in {\mathbb R}^d, \end{aligned}$$
where
$$\begin{aligned} \displaystyle \beta _{\mu ,p}^n(x,r) = \inf _L \left( \frac{1}{r^n} \int _{\bar{B}(x,r)} \left( \frac{\mathrm{dist}(y,L)}{r}\right) ^p\,d\mu (y)\right) ^{1/p}, \end{aligned}$$
with the infimum taken over all the n-planes \(L\subset {\mathbb R}^d\). The \(\beta _{\mu ,p}^n\) coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform n-rectifiability. An analogous necessary condition for n-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance \(W_1\) is also proved.

Mathematics Subject Classification

28A75 28A78 42B20 

References

  1. 1.
    Azzam, J., David, G., Toro, T.: Wasserstein distance and the rectifiability of doubling measures: part I. Math. Ann (2014). arXiv:1408.6645
  2. 2.
    Azzam, J., David, G., Toro, T.: Wasserstein distance and the rectifiability of doubling measures: part II (2014). arXiv:1411.2512
  3. 3.
    Azzam, J., Tolsa, X.: Characterization of n-rectifiability in terms of the Jones’ square function: part II. Geom. Funct. Anal. (2015). arXiv:1501.01572
  4. 4.
    Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann, 361(3–4), 1055–1072 (2015)Google Scholar
  5. 5.
    Badger, M., Schul, R.: Two sufficient conditions for rectifiable measures. Proc. Amer. Math. Soc. (2014). arXiv:1412.8357
  6. 6.
    Chousionis, V., Garnett, J., Le, T., Tolsa, X.: Square functions and uniform rectifiability. Trans. Am. Math. Soc. (2014). arXiv:1401.3382
  7. 7.
    David, G.: Unrectifiable \(1\)-sets have vanishing analytic capacity. Revista Mat. Iberoamericana 14(2), 369–479 (1998)zbMATHCrossRefGoogle Scholar
  8. 8.
    David, G., Semmes, S.: Singular integrals and rectifiable sets in \(\mathbb{R}^{n}\): beyond lipschitz graphs. Astérisque 193, 152 (1991)Google Scholar
  9. 9.
    David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence, RI (1993)CrossRefGoogle Scholar
  10. 10.
    Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102(1), 1–15 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Léger, J.C.: Menger curvature and rectifiability. Ann. Math. 149, 831–869 (1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    Lerman, G.: Quantifying curvelike structures of measures by using \(L^2\) Jones quantities. Commun. Pure Appl. Math. 56(9), 1294–1365 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Stud. Adv. Math., vol. 44. Cambridge Univ. Press, Cambridge (1995)CrossRefGoogle Scholar
  14. 14.
    Mas, A., Tolsa, X.: Variation for Riesz transforms and uniform rectifiability. J. Eur. Math. Soc. 16(11), 2267–2321 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Nazarov, F., Tolsa, X., Volberg, A.: On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. Acta Math. 213(2), 237–321 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Okikiolu, K.: Characterization of subsets of rectifiable curves in \({R}^{n}\). J. London Math. Soc. 46(2), 336–348 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Pajot, H.: Conditions quantitatives de rectifiabilité. Bulletin de la Société Mathématique de France 125, 1–39 (1997)MathSciNetGoogle Scholar
  18. 18.
    Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. London Math. Soc. 98(2), 393–426 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Tolsa, X.: Principal values for Riesz transforms and rectifiability. J. Funct. Anal. 254(7), 1811–1863 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Tolsa, X.: Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderón-Zygmund Theory. Progress in Mathematics, vol. 307. Birkhäuser Verlag, Basel (2014)CrossRefGoogle Scholar
  21. 21.
    Tolsa, X., Toro, T.: Rectifiability via a square function and Preiss’ theorem. Int. Math. Res. Notices 2015(13), 4638–4662 (2015)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MatemàtiquesICREA and Universitat Autònoma de BarcelonaBarcelonaSpain

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