Skip to main content
Log in

Mass Transport and Uniform Rectifiability

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript


In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance W 2 from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance W 2 which asserts that if μ and ν are probability measures in \({{\mathbb{R}^n}}\) , \({{\varphi}}\) is a radial bump function smooth enough so that \({{\int \varphi d \mu \gtrsim 1}}\) , and μ has a density bounded from above and from below on supp(\({{\varphi}}\)), then \({{W_2(\varphi \mu, a\varphi \nu) \leq cW_2(\mu, \nu)}}\) , where \({{a = \int \varphi d\mu/ \int \varphi d\nu}}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.

  2. Christ M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 6061(2), 601–628 (1990)

    MathSciNet  Google Scholar 

  3. G. David, Wavelets and Singular Integrals on Curves and Surfaces, Springer Lecture Notes in Math. 1465 (1991).

  4. G. David, S. Semmes, Singular Integrals and Rectifiable Sets in \({\mathbb {R}^n}\) : Beyond Lipschitz Graphs, Astérisque 193 (1991).

  5. G. David, S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38, American Mathematical Society, Providence, RI (1993).

  6. Jones P.W.: Square functions, Cauchy integrals, analytic capacity, and harmonic measure, in “Harmonic Analysis and Partial Differential Equations (El Escorial, 1987)”. Springer Lecture Notes in Math. 1384, 24–68 (1989)

    Article  Google Scholar 

  7. Jones P.W.: Rectifiable sets and the travelling salesman problem. Invent. Math. 102, 1–15 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Léger J.C.: Menger curvature and rectifiability. Ann. of Math. 149, 831–869 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lerman G.: Quantifying curvelike structures of measures by using L 2 Jones quantities. Comm. Pure Appl. Math. 56(9), 1294–1365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Mas, X. Tolsa, Variation for Riesz transforms and uniform rectifiability, preprint (2011).

  11. Mattila P., Melnikov M.S., Verdera J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2) 144, 127–136 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mattila P., Preiss D.: Rectifiable measures in Rn and existence of principal values for singular integrals. J. London Math. Soc. (2) 52(3), 482–496 (1995)

    MathSciNet  MATH  Google Scholar 

  13. Mattila P., Verdera J.: Convergence of singular integrals with general measures J. Eur. Math. Soc. (JEMS) 11(2), 257–271 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Okikiolu K.: Characterization of subsets of rectifiable curves in Rn. J. London Math. Soc. (2) 46(2), 336–348 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Peyre, Non asymptotic equivalence between W 2 distance and Ḣ–1 norm, preprint (2011).

  16. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.

  17. Tolsa X.: Bilipschitz maps, analytic capacity, and the Cauchy integral. Ann. of Math. 162(3), 1241–1302 (2005)

    Article  MathSciNet  Google Scholar 

  18. Tolsa X.: Uniform rectifiability, Calderón–Zygmund operators with odd kernel. and quasiorthogonality. Proc. London Math. Soc. 98(2), 393–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vihtilä M. (1996) The boundedness of Riesz s-transforms of measures in R n. Proc. Amer. Math. Soc. 124(2):3797–3804

  20. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI (2003).

  21. C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338, Springer-Verlag, Berlin (2009).

  22. A. Volberg, Calderón–Zygmund capacities and operators on nonhomogeneous spaces, CBMS Regional Conf. Ser. in Math. 100, Amer. Math. Soc., Providence (2003).

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Xavier Tolsa.

Additional information

Partially supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM-2010-16232 (Spain).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tolsa, X. Mass Transport and Uniform Rectifiability. Geom. Funct. Anal. 22, 478–527 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Keywords and phrases

2010 Mathematics Subject Classification