Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Multiscale Analysis of 1-rectifiable Measures II: Characterizations References[1] L. Ambrosio and B. Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., 318(3):527-555, 2000.[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.[3] J. Azzam and M. Mourgoglou. A characterization of 1-rectifiable doubling measures with connected supports. Anal. PDE, 9(1):99-109, 2016.[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, 2015.10.1007/s00039-015-0334-7[5] M. Badger and R. Schul. Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann., 361(3-4):1055-1072, 2015.[6] M. Badger and R. Schul. Two suficient conditions for rectifiable measures. Proc. Amer.Math. Soc., 144(6):2445-2454, 2016.[7] D. Bate. Structure of measures in Lipschitz difierentiability spaces. J. Amer. Math. Soc., 28(2):421-482, 2015.[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.[9] G. Beer. Topologies on closed and closed convex sets, volume 268 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.[10] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann., 98(1):422-464, 1928.[11] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1):296-329, 1938.[12] V. Chousionis and J. T. Tyson. Marstrand’s density theorem in the Heisenberg group. Bull. Lond. Math. Soc., 47(5):771-788, 2015.[13] G. David and S. Semmes. Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs. Astérisque, (193):152, 1991.[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.[15] G. David and T. Toro. Reifenberg parameterizations for sets with holes. Mem. Amer. Math. Soc., 215(1012):vi+102, 2012.[16] G. C. David and R. Schul. The Analyst’s traveling salesman theorem in graph inverse limits. preprint, 2016.[17] C. De Lellis. Recti_able sets, densities and tangent measures. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.[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.[19] K. J. Falconer. The geometry of fractal sets, volume 85 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.[20] H. Federer. The (', k) rectifiable subsets of n-space. Trans. Amer. Soc., 62:114-192, 1947.[21] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.[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/502[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.[24] I. Hahlomaa. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6):1894-1915, 2008.[25] P. Hajłasz and S. Malekzadeh. On conditions for unrectifiability of a metric space. Anal. Geom. Metr. Spaces, 3:1-14, 2015.[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.[27] P. W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1-15, 1990.10.1007/BF01233418[28] P. W. Jones, G. Lerman, and R. Schul. The Analyst’s traveling bandit problem in Hilbert space. in preparation.[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/626[30] B. Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113-123, 1994.[31] O. Kowalski and D. Preiss. Besicovitch-type properties of measures and submanifolds. J. Reine Angew. Math., 379:115-151, 1987.[32] J. C. Léger. Menger curvature and rectifiability. Ann. of Math. (2), 149(3):831-869, 1999.[33] G. Lerman. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl.Math., 56(9):1294- 1365, 2003.[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, 2011.10.4171/RMI/645[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/889[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.[37] A. Lorent. Rectifiability of measures with locally uniform cube density. Proc. London Math. Soc. (3), 86(1):153-249, 2003.[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.[39] J. M. Marstrand. Hausdorff two-dimensional measure in 3-space. Proc. London Math. Soc. (3), 11:91-108, 1961.[40] J. M. Marstrand. The (', s) regular subsets of n-space. Trans. Amer. Math. Soc., 113:369-392, 1964.[41] H.Martikainen and T. Orponen. Boundedness of the density normalised Jones’ square function does not imply 1-rectifiability. preprint, arXiv:1605.04091, 2016.[42] P. Mattila. Hausdorff m regular and rectifiable sets in n-space. Trans. Amer. Math. Soc., 205:263-274, 1975.[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.[44] E. F. Moore. Density ratios and (Φ, 1) recti_ability in n-space. Trans. Amer. Math. Soc., 69:324-334, 1950.[45] A. P. Morse and J. F. Randolph. The Φ rectifiable subsets of the plane. Trans. Amer. Math. Soc., 55:236-305, 1944.[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.[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.[48] A. D. Nimer. Conical 3-uniform measures: characterizations & new examples. preprint, arXiv:1608.02604, 2016.[49] K. Okikiolu. Characterization of subsets of rectifiable curves in Rn. J. London Math. Soc. (2), 46(2):336-348, 1992.[50] H. Pajot. Conditions quantitatives de rectifiabilité. Bull. Soc. Math. France, 125(1):15-53, 1997.[51] H. Pajot. Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, volume 1799 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.[52] D. Preiss. Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2), 125(3):537-643, 1987.[53] D. Preiss and J. Tišer. On Besicovitch’s 1 2 -problem. J. London Math. Soc. (2), 45(2):279-287, 1992.[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.[55] R. Schul. Subsets of rectifiable curves in Hilbert space-the analyst’s TSP. J. Anal. Math., 103:331-375, 2007.[56] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.[57] X. Tolsa. Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, volume307 of Progress in Mathematics. Birkhäuser/Springer, Cham, 2014.[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.[59] X. Tolsa. Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2), 92(1):1-18, 2015. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

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References[1] L. Ambrosio and B. Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., 318(3):527-555, 2000.[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.[3] J. Azzam and M. Mourgoglou. A characterization of 1-rectifiable doubling measures with connected supports. Anal. PDE, 9(1):99-109, 2016.[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, 2015.10.1007/s00039-015-0334-7[5] M. Badger and R. Schul. Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann., 361(3-4):1055-1072, 2015.[6] M. Badger and R. Schul. Two suficient conditions for rectifiable measures. Proc. Amer.Math. Soc., 144(6):2445-2454, 2016.[7] D. Bate. Structure of measures in Lipschitz difierentiability spaces. J. Amer. Math. Soc., 28(2):421-482, 2015.[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.[9] G. Beer. Topologies on closed and closed convex sets, volume 268 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.[10] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann., 98(1):422-464, 1928.[11] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1):296-329, 1938.[12] V. Chousionis and J. T. Tyson. Marstrand’s density theorem in the Heisenberg group. Bull. Lond. Math. Soc., 47(5):771-788, 2015.[13] G. David and S. Semmes. Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs. Astérisque, (193):152, 1991.[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.[15] G. David and T. Toro. Reifenberg parameterizations for sets with holes. Mem. Amer. Math. Soc., 215(1012):vi+102, 2012.[16] G. C. David and R. Schul. The Analyst’s traveling salesman theorem in graph inverse limits. preprint, 2016.[17] C. De Lellis. Recti_able sets, densities and tangent measures. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.[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.[19] K. J. Falconer. The geometry of fractal sets, volume 85 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.[20] H. Federer. The (', k) rectifiable subsets of n-space. Trans. Amer. Soc., 62:114-192, 1947.[21] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.[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/502[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.[24] I. Hahlomaa. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6):1894-1915, 2008.[25] P. Hajłasz and S. Malekzadeh. On conditions for unrectifiability of a metric space. Anal. Geom. Metr. Spaces, 3:1-14, 2015.[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.[27] P. W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1-15, 1990.10.1007/BF01233418[28] P. W. Jones, G. Lerman, and R. Schul. The Analyst’s traveling bandit problem in Hilbert space. in preparation.[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/626[30] B. Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113-123, 1994.[31] O. Kowalski and D. Preiss. Besicovitch-type properties of measures and submanifolds. J. Reine Angew. Math., 379:115-151, 1987.[32] J. C. Léger. Menger curvature and rectifiability. Ann. of Math. (2), 149(3):831-869, 1999.[33] G. Lerman. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl.Math., 56(9):1294- 1365, 2003.[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, 2011.10.4171/RMI/645[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/889[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.[37] A. Lorent. Rectifiability of measures with locally uniform cube density. Proc. London Math. Soc. (3), 86(1):153-249, 2003.[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.[39] J. M. Marstrand. Hausdorff two-dimensional measure in 3-space. Proc. London Math. Soc. (3), 11:91-108, 1961.[40] J. M. Marstrand. The (', s) regular subsets of n-space. Trans. Amer. Math. Soc., 113:369-392, 1964.[41] H.Martikainen and T. Orponen. Boundedness of the density normalised Jones’ square function does not imply 1-rectifiability. preprint, arXiv:1605.04091, 2016.[42] P. Mattila. Hausdorff m regular and rectifiable sets in n-space. Trans. Amer. Math. Soc., 205:263-274, 1975.[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.[44] E. F. Moore. Density ratios and (Φ, 1) recti_ability in n-space. Trans. Amer. Math. Soc., 69:324-334, 1950.[45] A. P. Morse and J. F. Randolph. The Φ rectifiable subsets of the plane. Trans. Amer. Math. Soc., 55:236-305, 1944.[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.[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.[48] A. D. Nimer. Conical 3-uniform measures: characterizations & new examples. preprint, arXiv:1608.02604, 2016.[49] K. Okikiolu. Characterization of subsets of rectifiable curves in Rn. J. London Math. Soc. (2), 46(2):336-348, 1992.[50] H. Pajot. Conditions quantitatives de rectifiabilité. Bull. Soc. Math. France, 125(1):15-53, 1997.[51] H. Pajot. Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, volume 1799 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.[52] D. Preiss. Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2), 125(3):537-643, 1987.[53] D. Preiss and J. Tišer. On Besicovitch’s 1 2 -problem. J. London Math. Soc. (2), 45(2):279-287, 1992.[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.[55] R. Schul. Subsets of rectifiable curves in Hilbert space-the analyst’s TSP. J. Anal. Math., 103:331-375, 2007.[56] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.[57] X. Tolsa. Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, volume307 of Progress in Mathematics. Birkhäuser/Springer, Cham, 2014.[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.[59] X. Tolsa. Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2), 92(1):1-18, 2015.

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