Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 20, 2019

Linear extension operators between spaces of Lipschitz maps and optimal transport

  • Luigi Ambrosio EMAIL logo and Daniele Puglisi

Abstract

Motivated by the notion of K-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of K-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call K-random projection. We show that K-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak* continuous operators. Finally, we use this notion to characterize the metric spaces (X,d) such that the free space (X) has the bounded approximation propriety.

Funding statement: The second author was supported by “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INDAM).

Acknowledgements

The paper was written while the second author was visiting the Scuola Normale Superiore. He is grateful to the first author for his kind hospitality. The authors thank A. Naor and D. Zaev for useful bibliographical information.

References

[1] A. Brudnyi and Y. Brudnyi, Linear and nonlinear extensions of Lipschitz functions from subsets of metric spaces, Algebra i Analiz 19 (2007), no. 3, 106–118. 10.1090/S1061-0022-08-01003-0Search in Google Scholar

[2] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. 10.1007/BF02392270Search in Google Scholar

[3] G. Godefroy, Extensions of Lipschitz functions and Grothendieck’s bounded approximation property, North-West. Eur. J. Math. 1 (2015), 1–6. Search in Google Scholar

[4] G. Godefroy and N. Ozawa, Free Banach spaces and the approximation properties, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1681–1687. 10.1090/S0002-9939-2014-11933-2Search in Google Scholar

[5] 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. 10.36045/bbms/1457560854Search in Google Scholar

[6] P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), no. 2, 629–646. 10.1016/j.jmaa.2014.02.060Search in Google Scholar

[7] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Math. 1547, Springer, Berlin 1993. 10.1007/BFb0084355Search in Google Scholar

[8] S. C. Hille and D. T. H. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations Operator Theory 63 (2009), no. 3, 351–371. 10.1007/s00020-008-1652-zSearch in Google Scholar

[9] W. B. Johnson, J. Lindenstrauss and G. Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), no. 2, 129–138. 10.1007/BF02764938Search in Google Scholar

[10] N. J. Kalton, Spaces of Lipschitz and Holder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217. Search in Google Scholar

[11] G. Lancien and E. Pernecká, Approximation properties and Schauder decompositions in Lipschitz-free spaces, J. Funct. Anal. 264 (2013), no. 10, 2323–2334. 10.1016/j.jfa.2013.02.012Search in Google Scholar

[12] J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95. 10.1007/s00222-004-0400-5Search in Google Scholar

[13] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263–287. 10.1307/mmj/1028999141Search in Google Scholar

[14] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. 10.1090/S0002-9904-1934-05978-0Search in Google Scholar

[15] R. E. Megginson, An introduction to Banach space theory, Grad. Texts in Math. 183, Springer, New York 1998. 10.1007/978-1-4612-0603-3Search in Google Scholar

[16] A. Naor and Y. Rabani, On Lipschitz extension from finite subsets, Israel J. Math. 219 (2017), no. 1, 115–161. 10.1007/s11856-017-1475-1Search in Google Scholar

[17] S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425. 10.1007/s11117-008-2202-2Search in Google Scholar

[18] C. Villani, Optimal transport, Grundlehren Math. Wiss. 338, Springer, Berlin 2009. 10.1007/978-3-540-71050-9Search in Google Scholar

[19] N. Weaver, Lipschitz algebras, World Scientific, River Edge 1999. 10.1142/4100Search in Google Scholar

Received: 2016-09-20
Revised: 2018-11-29
Published Online: 2019-01-20
Published in Print: 2020-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/crelle-2018-0037/html
Scroll to top button