Abstract
The main result shows that the Rademacher theorem proved by J. Lindenstrauss and D. Preiss [13] (which says that, for some pairs X, Y of Banach spaces, each Lipschitz f : X → Y is \({\Gamma}\)-a.e. Fréchet differentiable) generalizes the corresponding stepanov theorem (whichsays that, forsuch X and Y, an arbitrary f : X → Y is Fréchet differentiable at \({\Gamma}\) -almost all points at which f is Lipschitz). We also present an abstract approach which shows an easy way how (in some cases) a theorem of stepanov type (for vector functions) can be inferred from the corresponding theorem of Radamacher type. Finally we present Stepanov type differentiability theorems with the assumption of pointwise directional Lipschitzness.
Similar content being viewed by others
References
Ambrosio L. Kirchheim B. (2000) Rectifiablesets in metric and Banachspaces. Math. Ann. 318, 527–555
Ball K. (1992) Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal. 2, 137–172
Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloquium Publications, 48, American Mathematicalsociety (Providence, 2000).
D. Bongiorno, Stepanoff’s theorem inseparable Banachspaces, Comment. Math. Univ. Carolin., 39 (1998), 323–335.
D. Bongiorno, Radon–Nikodým property of the range of Lipschitz extensions, Atti Sem. Mat. Fis. Univ. Modena, 48 (2000), 517–525.
J. Cheeger, Differentiability of Lipschitz functions on metric measurespaces, Geom. Funct. Anal., 9 (1999), 428–517.
J. Duda, On Gâteaux differentiability of pointwise Lipschitz mappings, Canad. Math. Bull., 51 (2008), 205–216.
J. Duda, Metric and w*-differentiability of pointwise Lipschitz mappings, Z. Anal. Anwend., 26 (2007), 341–362.
R. Engelking, General Topology, 2nd ed.,sigmaseries in Pure Mathematics, 6, Heldermann Verlag (Berlin, 1989).
J. Heinonen, Lectures on Lipschitz Analysis, Report, University of Jyväskylä Department of Mathematics andstatistics, vol. 100, University of Jyväskylä (Jyväskylä, 2005).
W. B. Johnson, J. Lindenstrauss and G.schechtman, Extensions of Lipschitz maps into Banachspaces, Israel J. Math., 54 (1986), 129–138.
Kirchheim B. (1994) Rectifiable metricspaces: localstructure and regularity of the Hausdorff measure. Proc. Amer. Math.soc. 121, 113–123
J. Lindenstrauss and D. Preiss, On Fréchet differentiability of Lipschitz maps between Banachspaces, Annals Math., 157 (2003), 257–288.
J. Lindenstrauss, D. Preiss and J. Tišer, Fréchet Differentiability of Lipschitz Maps and Poroussets in Banachspaces, Princeton University Press (Princeton, 2012).
J. Malý, Asimple proof of thestepanov theorem on differentiability almost everywhere, Exposition. Math., 17 (1999), 59–61.
A. Naor, Y. Peres, O.schramm ands.sheffield, Markov chains insmooth Banach spaces and Gromov-hyperbolic metricspaces, Duke Math. J., 134 (2006), 165–197.
D. Preiss and L. Zajíček, Directional derivatives of Lipschitz functions, Israel J. Math., 125 (2001), 1–27.
s.saks, Théorie de l’intégrale (Warsaw, 1933).
W. Wildrick and T. Zürcher,sharp differentiability results for lip, preprint (2012), http://arxiv.org/abs/1208.2133.
L. Zajíček, On \({\sigma}\) -poroussets in abstractspaces, Abstr. Appl. Anal., 2005 (2005), 509–534.
L. Zajíček, Onsets of non-differentiability of Lipschitz and convex functions, Math. Bohem., 132 (2007), 75–85.
L. Zajíček, Hadamard differentiability via Gâteaux differentiability, to appear in Proc. Amer. Math.soc., preprint (2012), http://arxiv.org/abs/1210.4715.
L. Zajíček, Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions, Comment. Math. Univ. Carolin. 55 (2014), 203–213.
L. Zajíček, Properties of Hadamard directional derivatives: Denjoy–Young–Saks theorem for functions on Banachspaces, to appear in J. Convex Analysis (2015), preprint (2013), http://arxiv.org/abs/1308.2415.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was partiallysupported by the grant GAČR P201/12/0436.
Rights and permissions
About this article
Cite this article
Malý, J., Zajíček, L. On stepanov Type differentiability Theorems. Acta Math. Hungar. 145, 174–190 (2015). https://doi.org/10.1007/s10474-014-0465-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-014-0465-6
Key words and phrases
- Rademacher’s theorem
- Stepanov’s theorem
- Fréchet differentiability
- pointwise Lipschitz function
- \({\Gamma}\) -nullset
- Gâteaux differentiability
- metric differentiability
- w*differentiability