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ISOMETRIC REPRESENTATION OF LIPSCHITZ-FREE SPACES OVER CONVEX DOMAINS IN FINITE-DIMENSIONAL SPACES

  • Marek Cúth (a1), Ondřej F. K. Kalenda (a2) and Petr Kaplický (a3)
Abstract

Let $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$ . We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$ .

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1. Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, 2nd edn. (Pure and Applied Mathematics 140 ), Elsevier (Oxford, 2003).
2. Cúth, M., Doucha, M. and Wojtaszczyk, P., On the structure of Lipschitz-free spaces. Proc. Amer. Math. Soc. 144(9) 2016, 38333846.
3. Dalet, A., Free spaces over countable compact metric spaces. Proc. Amer. Math. Soc. 143(8) 2015, 35373546.
4. Federer, H., Geometric Measure Theory (Grundlehren der mathematischen Wissenschaften, Band 153 ), Springer (New York, 1969).
5. Flores, G., Estudio de los espacios Lipschitz-libres y una caracterizacin para el caso finito-dimensional. Master Thesis, 2016, available at http://repositorio.uchile.cl/handle/2250/141350.
6. Godard, A., Tree metrics and their Lipschitz-free spaces. Proc. Amer. Math. Soc. 138(12) 2010, 43114320.
7. Godefroy, G. and Kalton, N. J., Lipschitz-free Banach spaces. Studia Math. 159(1) 2003, 121141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.
8. Godefroy, G., Lancien, G. and Zizler, V., The non-linear geometry of Banach spaces after Nigel Kalton. Rocky Mountain J. Math. 44(5) 2014, 15291583.
9. Godefroy, G. and Lerner, N., Some natural subspaces and quotient spaces of . Preprint, 2017, arXiv:1702.06049 [math.FA].
10. Hájek, P. and Pernecká, E., On Schauder bases in Lipschitz-free spaces. J. Math. Anal. Appl. 416(2) 2014, 629646.
11. Horváth, J., Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley (Reading, MA–London–Don Mills, ON, 1966).
12. Kaufmann, P. L., Products of Lipschitz-free spaces and applications. Studia Math. 226(3) 2015, 213227.
13. Kisljakov, S. V., Sobolev imbedding operators, and the nonisomorphism of certain Banach spaces. Funktsional. Anal. i Priložhen. 9(4) 1975, 2227.
14. Lancien, G. and Pernecká, E., Approximation properties and Schauder decompositions in Lipschitz-free spaces. J. Funct. Anal. 264(10) 2013, 23232334.
15. Lerner, N., A note on Lipschitz spaces. Preprint.
16. Lukeš, J. and Malý, J., Measure and Integral, 2nd edn., Matfyzpress (Prague, 2005).
17. Malý, J., Non-absolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. 193(4) 2014, 14571484.
18. Maz’ya, V., Sobolev Spaces with Applications to Elliptic Partial Differential Equations (Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) 342 ), augmented edn., Springer (Heidelberg, 2011).
19. Naor, A. and Schechtman, G., Planar earthmover is not in L 1 . SIAM J. Comput. 37(3) 2007, 804826 (electronic).
20. Novotný, A. and Straškraba, I., Introduction to the Mathematical Theory of Compressible Flow (Oxford Lecture Series in Mathematics and its Applications 27 ), Oxford University Press (Oxford, 2004).
21. Ostrovskii, M. I., Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces (De Gruyter Studies in Mathematics 49 ), De Gruyter (Berlin, 2013).
22. Pernecká, E. and Smith, R. J., The metric approximation property and Lipschitz-free spaces over subsets of ℝ N . J. Approx. Theory 199 2015, 2944.
23. Weaver, N., On the unique predual problem for Lipschitz spaces. Preprint, 2016, arXiv:1611.01812 [math.FA].
24. Zajíček, L., Fréchet differentiability, strict differentiability and subdifferentiability. Czechoslovak Math. J. 41(116, 3) 1991, 471489.
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Mathematika
  • ISSN: 0025-5793
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