A note on intermediate differentiability of Lipschitz functions

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 4, page 795-799
  • ISSN: 0010-2628

Abstract

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Let f be a Lipschitz function on a superreflexive Banach space X . We prove that then the set of points of X at which f has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces X ), but it is even σ -porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.

How to cite

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Zajíček, Luděk. "A note on intermediate differentiability of Lipschitz functions." Commentationes Mathematicae Universitatis Carolinae 40.4 (1999): 795-799. <http://eudml.org/doc/248400>.

@article{Zajíček1999,
abstract = {Let $f$ be a Lipschitz function on a superreflexive Banach space $X$. We prove that then the set of points of $X$ at which $f$ has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces $X$), but it is even $\sigma $-porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lipschitz function; intermediate derivative; $\sigma $-porous set; superreflexive Banach space; Lipschitz function; intermediate derivative; -porous set; superreflexive Banach space},
language = {eng},
number = {4},
pages = {795-799},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on intermediate differentiability of Lipschitz functions},
url = {http://eudml.org/doc/248400},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Zajíček, Luděk
TI - A note on intermediate differentiability of Lipschitz functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 4
SP - 795
EP - 799
AB - Let $f$ be a Lipschitz function on a superreflexive Banach space $X$. We prove that then the set of points of $X$ at which $f$ has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces $X$), but it is even $\sigma $-porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.
LA - eng
KW - Lipschitz function; intermediate derivative; $\sigma $-porous set; superreflexive Banach space; Lipschitz function; intermediate derivative; -porous set; superreflexive Banach space
UR - http://eudml.org/doc/248400
ER -

References

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  1. Aronszajn N., Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190. (1976) Zbl0342.46034MR0425608
  2. Bates S.M., Johnson W.B., Lindenstrauss J., Preiss D., Schechtman G., Affine approximation of Lipschitz functions and non linear quotiens, to appear. 
  3. Fabian M., Preiss D., On intermediate differentiability of Lipschitz functions on certain Banach spaces, Proc. Amer. Math. Soc. 113 (1991), 733-740. (1991) Zbl0743.46040MR1074753
  4. Giles J.R., Sciffer S., Generalising generic differentiability properties from convex to locally Lipschitz functions, J. Math. Anal. Appl. 188 (1994), 833-854. (1994) Zbl0897.46025MR1305489
  5. Preiss D., Differentiability of Lipschitz functions in Banach spaces, J. Funct. Anal. 91 (1990), 312-345. (1990) MR1058975
  6. Preiss D., Zajíček L., Sigma-porous sets in products of metric spaces and sigma-directionally porous sets in Banach spaces, Real Analysis Exchange 24 (1998-99), 295-313. (1998-99) MR1691753
  7. Preiss D., Zajíček L., Directional derivatives of Lipschitz functions, to appear. MR1853802
  8. Zajíček L., Porosity and σ -porosity, Real Analysis Exchange 13 (1987-88), 314-350. (1987-88) MR0943561
  9. Zajíček L., On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function, Comment. Math. Univ. Carolinae 32 (1997), 329-336. (1997) MR1455499
  10. Zajíček L., Small non-sigma-porous sets in topologically complete metric spaces, Colloq. Math. 77 (1998), 293-304. (1998) MR1628994
  11. Zelený M., The Banach-Mazur game and σ -porosity, Fund. Math. 150 (1996), 197-210. (1996) MR1405042

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