Abstract
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.
References
[1] L. Ambrosio and B. Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., 318(3):527-555, 2000.10.1007/s002080000122Search in Google Scholar
[2] J. Azzam, G. David, and T. Toro. Wasserstein distance and the rectifiability of doubling measures: part I. Math. Ann., 364(1- 2):151-224, 2016.Search in Google Scholar
[3] J. Azzam and M. Mourgoglou. A characterization of 1-rectifiable doubling measures with connected supports. Anal. PDE, 9(1):99-109, 2016.10.2140/apde.2016.9.99Search in Google Scholar
[4] J. Azzam and X. Tolsa. Characterization of n-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal., 25(5):1371-1412, 2015.Search in Google Scholar
[5] M. Badger and R. Schul. Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann., 361(3-4):1055-1072, 2015.Search in Google Scholar
[6] M. Badger and R. Schul. Two suficient conditions for rectifiable measures. Proc. Amer.Math. Soc., 144(6):2445-2454, 2016.10.1090/proc/12881Search in Google Scholar
[7] D. Bate. Structure of measures in Lipschitz difierentiability spaces. J. Amer. Math. Soc., 28(2):421-482, 2015.10.1090/S0894-0347-2014-00810-9Search in Google Scholar
[8] D. Bate and S. Li. Characterizations of rectifiable metric measure spaces. preprint, arXiv:1409.4242, to appear in Ann. Sci. Éc. Norm. Supèr., 2014.Search in Google Scholar
[9] G. Beer. Topologies on closed and closed convex sets, volume 268 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.10.1007/978-94-015-8149-3Search in Google Scholar
[10] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann., 98(1):422-464, 1928.10.1007/BF01451603Search in Google Scholar
[11] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1):296-329, 1938.10.1007/BF01448943Search in Google Scholar
[12] V. Chousionis and J. T. Tyson. Marstrand’s density theorem in the Heisenberg group. Bull. Lond. Math. Soc., 47(5):771-788, 2015.10.1112/blms/bdv056Search in Google Scholar
[13] G. David and S. Semmes. Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs. Astérisque, (193):152, 1991.Search in Google Scholar
[14] G. David and S. Semmes. Analysis of and on uniformly rectifiable sets, volume 38 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1993.10.1090/surv/038Search in Google Scholar
[15] G. David and T. Toro. Reifenberg parameterizations for sets with holes. Mem. Amer. Math. Soc., 215(1012):vi+102, 2012.10.1090/S0065-9266-2011-00629-5Search in Google Scholar
[16] G. C. David and R. Schul. The Analyst’s traveling salesman theorem in graph inverse limits. preprint, 2016.10.5186/aasfm.2017.4260Search in Google Scholar
[17] C. De Lellis. Recti_able sets, densities and tangent measures. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.10.4171/044Search in Google Scholar
[18] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015.10.1201/b18333Search in Google Scholar
[19] K. J. Falconer. The geometry of fractal sets, volume 85 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.Search in Google Scholar
[20] H. Federer. The (', k) rectifiable subsets of n-space. Trans. Amer. Soc., 62:114-192, 1947.10.1090/S0002-9947-1947-0022594-3Search in Google Scholar
[21] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.Search in Google Scholar
[22] F. Ferrari, B. Franchi, and H. Pajot. The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 23(2):437-480, 2007.10.4171/RMI/502Search in Google Scholar
[23] J. Garnett, R. Killip, and R. Schul. A doubling measure on Rd can charge a rectifiable curve. Proc. Amer. Math. Soc., 138(5):1673-1679, 2010.10.1090/S0002-9939-10-10234-2Search in Google Scholar
[24] I. Hahlomaa. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6):1894-1915, 2008.10.1016/j.aim.2008.07.013Search in Google Scholar
[25] P. Hajłasz and S. Malekzadeh. On conditions for unrectifiability of a metric space. Anal. Geom. Metr. Spaces, 3:1-14, 2015.10.1515/agms-2015-0001Search in Google Scholar
[26] P.W. Jones. Square functions, Cauchy integrals, analytic capacity, and harmonic measure. In Harmonic analysis and partial differential equations (El Escorial, 1987), volume 1384 of Lecture Notes in Math., pages 24-68. Springer, Berlin, 1989.10.1007/BFb0086793Search in Google Scholar
[27] P. W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1-15, 1990.10.1007/BF01233418Search in Google Scholar
[28] P. W. Jones, G. Lerman, and R. Schul. The Analyst’s traveling bandit problem in Hilbert space. in preparation.Search in Google Scholar
[29] N. Juillet. A counterexample for the geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 26(3):1035-1056, 2010.10.4171/RMI/626Search in Google Scholar
[30] B. Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113-123, 1994.Search in Google Scholar
[31] O. Kowalski and D. Preiss. Besicovitch-type properties of measures and submanifolds. J. Reine Angew. Math., 379:115-151, 1987.10.1515/crll.1987.379.115Search in Google Scholar
[32] J. C. Léger. Menger curvature and rectifiability. Ann. of Math. (2), 149(3):831-869, 1999.10.2307/121074Search in Google Scholar
[33] G. Lerman. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl.Math., 56(9):1294- 1365, 2003.10.1002/cpa.10096Search in Google Scholar
[34] G. Lerman and J. T. Whitehouse. High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities. Rev. Mat. Iberoam., 27(2):493-555, 2011.Search in Google Scholar
[35] S. Li and R. Schul. An upper bound for the length of a traveling salesman path in the Heisenberg group. Rev. Mat. Iberoam. 32(2):391-417, 2016.10.4171/RMI/889Search in Google Scholar
[36] S. Li and R. Schul. The traveling salesman problem in the Heisenberg group: Upper bounding curvature. Trans. Amer.Math. Soc., 368(7):4585-4620, 2016.Search in Google Scholar
[37] A. Lorent. Rectifiability of measures with locally uniform cube density. Proc. London Math. Soc. (3), 86(1):153-249, 2003.10.1112/S0024611502013710Search in Google Scholar
[38] A. Lorent. A Marstrand type theorem for measures with cube density in general dimension. Math. Proc. Cambridge Philos. Soc., 137(3):657-696, 2004.10.1017/S0305004104007972Search in Google Scholar
[39] J. M. Marstrand. Hausdorff two-dimensional measure in 3-space. Proc. London Math. Soc. (3), 11:91-108, 1961.10.1112/plms/s3-11.1.91Search in Google Scholar
[40] J. M. Marstrand. The (', s) regular subsets of n-space. Trans. Amer. Math. Soc., 113:369-392, 1964.10.1090/S0002-9947-1964-0166336-XSearch in Google Scholar
[41] H.Martikainen and T. Orponen. Boundedness of the density normalised Jones’ square function does not imply 1-rectifiability. preprint, arXiv:1605.04091, 2016.Search in Google Scholar
[42] P. Mattila. Hausdorff m regular and rectifiable sets in n-space. Trans. Amer. Math. Soc., 205:263-274, 1975.10.2307/1997203Search in Google Scholar
[43] P. Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.10.1017/CBO9780511623813Search in Google Scholar
[44] E. F. Moore. Density ratios and (Φ, 1) recti_ability in n-space. Trans. Amer. Math. Soc., 69:324-334, 1950.10.1090/S0002-9947-1950-0037894-0Search in Google Scholar
[45] A. P. Morse and J. F. Randolph. The Φ rectifiable subsets of the plane. Trans. Amer. Math. Soc., 55:236-305, 1944.10.1090/S0002-9947-1944-0009975-6Search in Google Scholar
[46] A. Naber and D. Valtorta. Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Of Math. (2), 185(1):131-227, 2017.10.4007/annals.2017.185.1.3Search in Google Scholar
[47] A. D. Nimer. A sharp bound on the Hausdorff dimension of the singular set of an n-uniform measure. preprint, arXiv:1510.03732, 2015.Search in Google Scholar
[48] A. D. Nimer. Conical 3-uniform measures: characterizations & new examples. preprint, arXiv:1608.02604, 2016.Search in Google Scholar
[49] K. Okikiolu. Characterization of subsets of rectifiable curves in Rn. J. London Math. Soc. (2), 46(2):336-348, 1992.10.1112/jlms/s2-46.2.336Search in Google Scholar
[50] H. Pajot. Conditions quantitatives de rectifiabilité. Bull. Soc. Math. France, 125(1):15-53, 1997.10.24033/bsmf.2298Search in Google Scholar
[51] H. Pajot. Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, volume 1799 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.10.1007/b84244Search in Google Scholar
[52] D. Preiss. Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2), 125(3):537-643, 1987.Search in Google Scholar
[53] D. Preiss and J. Tišer. On Besicovitch’s 1 2 -problem. J. London Math. Soc. (2), 45(2):279-287, 1992.10.1112/jlms/s2-45.2.279Search in Google Scholar
[54] C. A. Rogers. Hausdorff measures. CambridgeMathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1970 original, With a foreword by K. J. Falconer.Search in Google Scholar
[55] R. Schul. Subsets of rectifiable curves in Hilbert space-the analyst’s TSP. J. Anal. Math., 103:331-375, 2007.10.1007/s11854-008-0011-ySearch in Google Scholar
[56] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.Search in Google Scholar
[57] X. Tolsa. Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, volume307 of Progress in Mathematics. Birkhäuser/Springer, Cham, 2014.10.1007/978-3-319-00596-6Search in Google Scholar
[58] X. Tolsa. Characterization of n-rectifiability in terms of Jones’ square function: part I. Calc. Var. Partial Differential Equations, 54(4):3643-3665, 2015.Search in Google Scholar
[59] X. Tolsa. Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2), 92(1):1-18, 2015. 10.1112/jlms/jdv013Search in Google Scholar
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