Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T02:07:06.804Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Normed linear spaces and Banach spaces; Banach lattices Real functions

Published online by Cambridge University Press:  04 April 2017

Marek Cúth ,
Ondřej F. K. Kalenda  and
Petr Kaplický
Marek Cúth
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email cuth@karlin.mff.cuni.cz
Ondřej F. K. Kalenda
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email kalenda@karlin.mff.cuni.cz
Petr Kaplický
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email kaplicky@karlin.mff.cuni.cz
Get access

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 $E$.

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 26A16: Lipschitz (Hölder) classes
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 538 - 552
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, 2nd edn. (Pure and Applied Mathematics 140 ), Elsevier (Oxford, 2003).Google Scholar
Cúth, M., Doucha, M. and Wojtaszczyk, P., On the structure of Lipschitz-free spaces. Proc. Amer. Math. Soc. 144(9) 2016, 3833–3846.Google Scholar
Dalet, A., Free spaces over countable compact metric spaces. Proc. Amer. Math. Soc. 143(8) 2015, 3537–3546.Google Scholar
Federer, H., Geometric Measure Theory (Grundlehren der mathematischen Wissenschaften, Band 153 ), Springer (New York, 1969).Google Scholar
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.Google Scholar
Godard, A., Tree metrics and their Lipschitz-free spaces. Proc. Amer. Math. Soc. 138(12) 2010, 4311–4320.CrossRefGoogle Scholar
Godefroy, G. and Kalton, N. J., Lipschitz-free Banach spaces. Studia Math. 159(1) 2003, 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.Google Scholar
Godefroy, G., Lancien, G. and Zizler, V., The non-linear geometry of Banach spaces after Nigel Kalton. Rocky Mountain J. Math. 44(5) 2014, 1529–1583.CrossRefGoogle Scholar
Godefroy, G. and Lerner, N., Some natural subspaces and quotient spaces of $L^{1}$ . Preprint, 2017, arXiv:1702.06049 [math.FA].Google Scholar
Hájek, P. and Pernecká, E., On Schauder bases in Lipschitz-free spaces. J. Math. Anal. Appl. 416(2) 2014, 629–646.Google Scholar
Horváth, J., Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley (Reading, MA–London–Don Mills, ON, 1966).Google Scholar
Kaufmann, P. L., Products of Lipschitz-free spaces and applications. Studia Math. 226(3) 2015, 213–227.Google Scholar
Kisljakov, S. V., Sobolev imbedding operators, and the nonisomorphism of certain Banach spaces. Funktsional. Anal. i Priložhen. 9(4) 1975, 22–27.Google Scholar
Lancien, G. and Pernecká, E., Approximation properties and Schauder decompositions in Lipschitz-free spaces. J. Funct. Anal. 264(10) 2013, 2323–2334.Google Scholar
Lerner, N., A note on Lipschitz spaces. Preprint.Google Scholar
Lukeš, J. and Malý, J., Measure and Integral, 2nd edn., Matfyzpress (Prague, 2005).Google Scholar
Malý, J., Non-absolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. 193(4) 2014, 1457–1484.Google Scholar
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).CrossRefGoogle Scholar
Naor, A. and Schechtman, G., Planar earthmover is not in L 1 . SIAM J. Comput. 37(3) 2007, 804–826 (electronic).Google Scholar
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).Google Scholar
Ostrovskii, M. I., Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces (De Gruyter Studies in Mathematics 49 ), De Gruyter (Berlin, 2013).CrossRefGoogle Scholar
Pernecká, E. and Smith, R. J., The metric approximation property and Lipschitz-free spaces over subsets of ℝ N . J. Approx. Theory 199 2015, 29–44.Google Scholar
Weaver, N., On the unique predual problem for Lipschitz spaces. Preprint, 2016, arXiv:1611.01812 [math.FA].Google Scholar
Zajíček, L., Fréchet differentiability, strict differentiability and subdifferentiability. Czechoslovak Math. J. 41(116, 3) 1991, 471–489.CrossRefGoogle Scholar