Differentiability of Lipschitz functions in Lebesgue null sets

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• 作者：David Preiss (1)
  Gareth Speight (2)
  
  1. Mathematics Institute ; University of Warwick ; Coventry ; CV4 7AL ; UK
  2. Faculty of Sciences ; Scuola Normale Superiore ; Piazza dei Cavalieri ; 7 ; 56126聽 ; Pisa ; Italy
  
• 刊名：Inventiones Mathematicae
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：199
• 期：2
• 页码：517-559
• 全文大小：355 KB
• 参考文献：1. Alberti, G., Cs枚rnyei, M., Preiss, D.: Differentiability of Lipschitz functions, structure of null sets, and other problems. Proc. Int. Congress Math. III, 1379鈥?394 (2010)
  2. David Bate. Structure of measures in Lipschitz differentiability spaces. J. Amer. Math. Soc. arXiv:1305.3154 (to appear)
  3. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, vol. 48 (2000). ISBN 0-8218-0835-4
  4. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428鈥?17 (1999) CrossRef
  5. Cs枚rnyei, M., Jones, P.: Product formulas for measures and applications to analysis and geometry. http://www.math.sunysb.edu/Videos/dfest/PDFs/38-Jones.pdf
  6. De Pauw, T., Huovinen, P.: Points of \(\varepsilon \) -differentiability of Lipschitz functions from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n-1}\) . Bull. Lond. Math. Soc. 34(5), 539鈥?50 (2002) CrossRef
  7. Dor茅, M., Maleva, O.: A universal differentiability set in Banach spaces with separable dual. J. Funct. Anal. 261(6), 1674鈥?710 (2011)
  8. Dugundji, J.: Topology. Allyn and Bacon Inc, Boston (1966). ISBN 978-0205002719
  9. Dymond, M., Maleva, O.: Differentiability inside sets with upper Minkowski dimension one. arXiv:1305.3154
  10. Fowler, T., Preiss, D.: A simple proof of Zahorski鈥檚 description of non-differentiability sets of Lipschitz functions. Real Anal. Exchange 34(1), 127鈥?38 (2009)
  11. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext, Springer, New York (2001) CrossRef
  12. Kechris, A.S.: Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156. Springer, New York (1995). ISBN 0-387-94374-9
  13. Keith, S.: A differentiable structure for metric measure spaces. Adv. Math. 183(2), 271鈥?15 (2004) CrossRef
  14. Lindenstrauss, J., Preiss, D.: Almost Fr茅chet differentiability of finitely many Lipschitz functions. Mathematika 43(2), 393鈥?12 (1996) CrossRef
  15. Lindenstrauss, J., Preiss, D.: On Fr茅chet differentiability of Lipschitz maps between Banach spaces. Ann. Math. (2) 157(1), 257鈥?88 (2003)
  16. Lindenstrauss, J., Preiss, D., Ti拧er, J.: Fr茅chet differentiability of Lipschitz functions and porous sets in Banach spaces, Annals of Mathematics Studies, vol. 179. Princeton University Press, Princeton (2012). ISBN 978-0-691-15356-8
  17. Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal. 91(2), 312鈥?45 (1990) CrossRef
  18. Speight, G.: Surfaces meeting porous sets in positive measure. Isr. J. Math. (2013). doi:10.1007/s11856-012-0171-4
  19. Tukia, P.: Hausdorff dimension and quasisymmetric mappings. Math. Scand. 65(1), 152鈥?60 (1989)
  20. Wu, J.: Hausdorff dimension and doubling measures on metric spaces. Proc. Am. Math. Soc. 126(5), 1453鈥?459 (1998) CrossRef
  21. Zahorski, Z.: Sur l鈥檈nsemble des points de non-derivabilit茅 d鈥檜ne fonction continue. Bull. Soc. Math. France 74, 147鈥?78 (1946)
  22. Zaj铆膷ek, L.: Porosity and \(\sigma \) -porosity. Real Anal. Exchange 13(2), 314鈥?50 (1987/1988)
  23. Zaj铆膷ek, L.: On \(\sigma \) -porous sets in abstract spaces. Abstr. Appl. Anal. 2005(5), 509鈥?34 (2005) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1297

文摘

We show that if \(n>1\) then there exists a Lebesgue null set in \({\mathbb {R}}^{n}\) containing a point of differentiability of each Lipschitz function \(f:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n-1}\) ; in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of \(\sigma \) -porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.