| Title:
|
A note on intermediate differentiability of Lipschitz functions (English) |
| Author:
|
Zajíček, Luděk |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
4 |
| Year:
|
1999 |
| Pages:
|
795-799 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Lipschitz function |
| Keyword:
|
intermediate derivative |
| Keyword:
|
$\sigma$-porous set |
| Keyword:
|
superreflexive Banach space |
| MSC:
|
46G05 |
| MSC:
|
58C20 |
| idZBL:
|
Zbl 1010.46042 |
| idMR:
|
MR1756555 |
| . |
| Date available:
|
2009-01-08T18:57:47Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119133 |
| . |
| Reference:
|
[1] Aronszajn N.: Differentiability of Lipschitzian mappings between Banach spaces.Studia Math. 57 (1976), 147-190. Zbl 0342.46034, MR 0425608 |
| Reference:
|
[2] Bates S.M., Johnson W.B., Lindenstrauss J., Preiss D., Schechtman G.: Affine approximation of Lipschitz functions and non linear quotiens.to appear. |
| Reference:
|
[3] Fabian M., Preiss D.: On intermediate differentiability of Lipschitz functions on certain Banach spaces.Proc. Amer. Math. Soc. 113 (1991), 733-740. Zbl 0743.46040, MR 1074753 |
| Reference:
|
[4] Giles J.R., Sciffer S.: Generalising generic differentiability properties from convex to locally Lipschitz functions.J. Math. Anal. Appl. 188 (1994), 833-854. Zbl 0897.46025, MR 1305489 |
| Reference:
|
[5] Preiss D.: Differentiability of Lipschitz functions in Banach spaces.J. Funct. Anal. 91 (1990), 312-345. MR 1058975 |
| Reference:
|
[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. MR 1691753 |
| Reference:
|
[7] Preiss D., Zajíček L.: Directional derivatives of Lipschitz functions.to appear. MR 1853802 |
| Reference:
|
[8] Zajíček L.: Porosity and $\sigma$-porosity.Real Analysis Exchange 13 (1987-88), 314-350. MR 0943561 |
| Reference:
|
[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. MR 1455499 |
| Reference:
|
[10] Zajíček L.: Small non-sigma-porous sets in topologically complete metric spaces.Colloq. Math. 77 (1998), 293-304. MR 1628994 |
| Reference:
|
[11] Zelený M.: The Banach-Mazur game and $\sigma$-porosity.Fund. Math. 150 (1996), 197-210. MR 1405042 |
| . |
