Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
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- by Ramón J. Aliaga, Chris Gartland, Colin Petitjean and Antonín Procházka PDF
- Trans. Amer. Math. Soc. 375 (2022), 3529-3567 Request permission
Abstract:
We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space $\mathcal {F}(M)$ over a compact metric space $M$ is a dual space if and only if $M$ is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric space $M$, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties of $\mathcal {F}(M)$ such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs has $\sigma$-finite Hausdorff 1-measure.References
- F. Albiac, J. Ansorena, M. Cúth and M. Doucha, Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces, Int. Math. Res. Not. (2021), DOI 10.1093/imrn/rnab193.
- Luigi Ambrosio and Daniele Puglisi, Linear extension operators between spaces of Lipschitz maps and optimal transport, J. Reine Angew. Math. 764 (2020), 1–21. MR 4116631, DOI 10.1515/crelle-2018-0037
- Ramón J. Aliaga, Camille Noûs, Colin Petitjean, and Antonín Procházka, Compact reduction in Lipschitz-free spaces, Studia Math. 260 (2021), no. 3, 341–359. MR 4296732, DOI 10.4064/sm200925-18-1
- Ramón J. Aliaga, Colin Petitjean, and Antonín Procházka, Embeddings of Lipschitz-free spaces into $\ell _1$, J. Funct. Anal. 280 (2021), no. 6, Paper No. 108916, 26. MR 4193768, DOI 10.1016/j.jfa.2020.108916
- Ramón J. Aliaga, Eva Pernecká, Colin Petitjean, and Antonín Procházka, Supports in Lipschitz-free spaces and applications to extremal structure, J. Math. Anal. Appl. 489 (2020), no. 1, 124128, 14. MR 4083124, DOI 10.1016/j.jmaa.2020.124128
- W. G. Bade, P. C. Curtis Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. MR 896225, DOI 10.1093/plms/s3-55_{2}.359
- David Bate, Structure of measures in Lipschitz differentiability spaces, J. Amer. Math. Soc. 28 (2015), no. 2, 421–482. MR 3300699, DOI 10.1090/S0894-0347-2014-00810-9
- David Bate, Purely unrectifiable metric spaces and perturbations of Lipschitz functions, Acta Math. 224 (2020), no. 1, 1–65. MR 4086714, DOI 10.4310/acta.2020.v224.n1.a1
- David Bate and Sean Li, Characterizations of rectifiable metric measure spaces, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 1, 1–37 (English, with English and French summaries). MR 3621425, DOI 10.24033/asens.2314
- Mario Bonk and Daniel Meyer, Quasiconformal and geodesic trees, Fund. Math. 250 (2020), no. 3, 253–299. MR 4107537, DOI 10.4064/fm749-7-2019
- J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of $L^{1}$, Israel J. Math. 37 (1980), no. 1-2, 54–75. MR 599302, DOI 10.1007/BF02762868
- B. Braga, G. Lancien, C. Petitjean, and A. Procházka, On Kalton’s interlaced graphs and nonlinear embeddings into dual Banach spaces, J. Topol. Anal. (2021), https://doi.org/10.1142/S1793525321500345.
- Bernardo Cascales, Rafael Chiclana, Luis C. García-Lirola, Miguel Martín, and Abraham Rueda Zoca, On strongly norm attaining Lipschitz maps, J. Funct. Anal. 277 (2019), no. 6, 1677–1717. MR 3985517, DOI 10.1016/j.jfa.2018.12.006
- Jeff Cheeger and Bruce Kleiner, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property, Geom. Funct. Anal. 19 (2009), no. 4, 1017–1028. MR 2570313, DOI 10.1007/s00039-009-0030-6
- Gustave Choquet, L’isométrie des ensembles dans ses rapports avec la théorie du contact et la théorie de la mesure, Mathematica, Timişoara 20 (1944), 29–64 (French). MR 0012317
- Z. Ciesielski, On the isomorphisms of the spaces $H_{\alpha }$ and $m$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 217–222 (English, with Russian summary). MR 132389
- Ştefan Cobzaş, Radu Miculescu, and Adriana Nicolae, Lipschitz functions, Lecture Notes in Mathematics, vol. 2241, Springer, Cham, 2019. MR 3931701, DOI 10.1007/978-3-030-16489-8
- Marianna Csörnyei, Jan Kališ, and Luděk Zajíček, Whitney arcs and 1-critical arcs, Fund. Math. 199 (2008), no. 2, 119–130. MR 2399495, DOI 10.4064/fm199-2-2
- A. Dalet, Free spaces over countable compact metric spaces, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3537–3546. MR 3348795, DOI 10.1090/S0002-9939-2015-12518-X
- A. Dalet, Free spaces over some proper metric spaces, Mediterr. J. Math. 12 (2015), no. 3, 973–986. MR 3376824, DOI 10.1007/s00009-014-0455-5
- Guy David and Stephen Semmes, Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications, vol. 7, The Clarendon Press, Oxford University Press, New York, 1997. Self-similar geometry through metric and measure. MR 1616732
- K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961/62), 55–66. MR 140927, DOI 10.4064/sm-21-1-55-66
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- O. Dovgoshey, O. Martio, V. Ryazanov, and M. Vuorinen, The Cantor function, Expo. Math. 24 (2006), no. 1, 1–37. MR 2195181, DOI 10.1016/j.exmath.2005.05.002
- G. C. David and V. Vellis, Bi-Lipschitz geometry of quasiconformal trees, Preprint, arXiv:2007.12297, 2020.
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- V. P. Fonf, J. Lindenstrauss, and R. R. Phelps, Infinite dimensional convexity, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 599–670. MR 1863703, DOI 10.1016/S1874-5849(01)80017-6
- Luis García-Lirola, Colin Petitjean, Antonín Procházka, and Abraham Rueda Zoca, Extremal structure and duality of Lipschitz free spaces, Mediterr. J. Math. 15 (2018), no. 2, Paper No. 69, 23. MR 3778926, DOI 10.1007/s00009-018-1113-0
- Luis García-Lirola, Antonín Procházka, and Abraham Rueda Zoca, A characterisation of the Daugavet property in spaces of Lipschitz functions, J. Math. Anal. Appl. 464 (2018), no. 1, 473–492. MR 3794100, DOI 10.1016/j.jmaa.2018.04.017
- Chris Gartland, Lipschitz free spaces over locally compact metric spaces, Studia Math. 258 (2021), no. 3, 317–342. MR 4228304, DOI 10.4064/sm200511-10-10
- A. Godard, Tree metrics and their Lipschitz-free spaces, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4311–4320. MR 2680057, DOI 10.1090/S0002-9939-2010-10421-5
- Gilles Godefroy, A survey on Lipschitz-free Banach spaces, Comment. Math. 55 (2015), no. 2, 89–118. MR 3518958, DOI 10.14708/cm.v55i2.1104
- James Hagler, A counterexample to several questions about Banach spaces, Studia Math. 60 (1977), no. 3, 289–308. MR 442651, DOI 10.4064/sm-60-3-289-308
- P. Hájek, G. Lancien, and E. Pernecká, Approximation and Schur properties for Lipschitz free spaces over compact metric spaces, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 1, 63–72. MR 3471979, DOI 10.36045/bbms/1457560854
- Piotr Hajłasz, Whitney’s example by way of Assouad’s embedding, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3463–3467. MR 1991757, DOI 10.1090/S0002-9939-03-06913-2
- Leonid G. Hanin, Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces, Proc. Amer. Math. Soc. 115 (1992), no. 2, 345–352. MR 1097344, DOI 10.1090/S0002-9939-1992-1097344-5
- P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
- R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749. MR 595102, DOI 10.1216/RMJ-1980-10-4-743
- Alejandro Illanes and Sam B. Nadler Jr., Hyperspaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 216, Marcel Dekker, Inc., New York, 1999. Fundamentals and recent advances. MR 1670250
- Thomas Morton Jenkins, BANACH SPACES OF LIPSCHITZ FUNCTIONS OF AN ABSTRACT METRIC SPACE, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (Ph.D.)–Yale University. MR 2617138
- J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147–169. MR 415289, DOI 10.1090/S0002-9947-1970-0415289-8
- N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217. MR 2068975
- Kyle Kinneberg, Conformal dimension and boundaries of planar domains, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6511–6536. MR 3660231, DOI 10.1090/tran/6944
- Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. MR 1189747, DOI 10.1090/S0002-9939-1994-1189747-7
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Daniel Meyer, Bounded turning circles are weak-quasicircles, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1751–1761. MR 2763763, DOI 10.1090/S0002-9939-2010-10634-2
- Assaf Naor and Gideon Schechtman, Planar earthmover is not in $L_1$, SIAM J. Comput. 37 (2007), no. 3, 804–826. MR 2341917, DOI 10.1137/05064206X
- Alec Norton, Functions not constant on fractal quasi-arcs of critical points, Proc. Amer. Math. Soc. 106 (1989), no. 2, 397–405. MR 969524, DOI 10.1090/S0002-9939-1989-0969524-8
- Mikhail Ostrovskii, Radon-Nikodým property and thick families of geodesics, J. Math. Anal. Appl. 409 (2014), no. 2, 906–910. MR 3103207, DOI 10.1016/j.jmaa.2013.07.067
- C. Petitjean, Lipschitz-free spaces and Schur properties, J. Math. Anal. Appl. 453 (2017), no. 2, 894–907. MR 3648263, DOI 10.1016/j.jmaa.2017.04.047
- C. Petitjean, Some aspects of the geometry of Lipschitz-free spaces, Ph.D. Thesis, Univ. Bourgogne Franche-Comté, 2018.
- Gilles Pisier, Martingales in Banach spaces, Cambridge Studies in Advanced Mathematics, vol. 155, Cambridge University Press, Cambridge, 2016. MR 3617459
- Antonín Procházka and Abraham Rueda Zoca, A characterisation of octahedrality in Lipschitz-free spaces, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 2, 569–588 (English, with English and French summaries). MR 3803112, DOI 10.5802/aif.3171
- Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
- Nik Weaver, Subalgebras of little Lipschitz algebras, Pacific J. Math. 173 (1996), no. 1, 283–293. MR 1387803, DOI 10.2140/pjm.1996.173.283
- Nik Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1832645, DOI 10.1142/4100
- Nik Weaver, Lipschitz algebras, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. Second edition of [ MR1832645]. MR 3792558, DOI 10.1142/9911
- Zhi-Ying Wen and Li-Feng Xi, The geometry of Whitney’s critical sets, Israel J. Math. 174 (2009), 303–348. MR 2581221, DOI 10.1007/s11856-009-0116-8
- Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
- Hassler Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), no. 4, 514–517. MR 1545896, DOI 10.1215/S0012-7094-35-00138-7
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- Ramón J. Aliaga
- Affiliation: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain
- ORCID: 0000-0002-2513-7711
- Email: raalva@upvnet.upv.es
- Chris Gartland
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 1238876
- Email: cgartland@math.tamu.edu
- Colin Petitjean
- Affiliation: LAMA, Univ. Gustave Eiffel, UPEM, Univ. Paris Est Creteil, CNRS, F-77447 Marne-la-Vallée, France
- MR Author ID: 1212067
- ORCID: 0000-0002-6584-5405
- Email: colin.petitjean@univ-eiffel.fr
- Antonín Procházka
- Affiliation: Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR-6623, 16, route de Gray, 25030 Besançon Cedex, France
- ORCID: 0000-0001-7762-9147
- Email: antonin.prochazka@univ-fcomte.fr
- Received by editor(s): August 4, 2021
- Received by editor(s) in revised form: October 29, 2021
- Published electronically: February 17, 2022
- Additional Notes: The first author was partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under Grant MTM2017-83262-C2-2-P. The third and fourth authors were partially supported by the French ANR project No. ANR-20-CE40-0006
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 3529-3567
- MSC (2020): Primary 51F30; Secondary 28A78, 30L05, 46B20, 46B22, 54E45
- DOI: https://doi.org/10.1090/tran/8591
- MathSciNet review: 4402669