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Keywords:
Lipschitz function; intermediate derivative; $\sigma$-porous set; superreflexive Banach space
Lipschitz function; intermediate derivative; $\sigma$-porous set; superreflexive Banach space
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.
References:
[1] Aronszajn N.: Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976), 147-190. MR 0425608 | Zbl 0342.46034
[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. MR 1074753 | Zbl 0743.46040
[4] Giles J.R., Sciffer S.: Generalising generic differentiability properties from convex to locally Lipschitz functions. J. Math. Anal. Appl. 188 (1994), 833-854. MR 1305489 | Zbl 0897.46025
[5] Preiss D.: Differentiability of Lipschitz functions in Banach spaces. J. Funct. Anal. 91 (1990), 312-345. MR 1058975
[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
[7] Preiss D., Zajíček L.: Directional derivatives of Lipschitz functions. to appear. MR 1853802
[8] Zajíček L.: Porosity and $\sigma$-porosity. Real Analysis Exchange 13 (1987-88), 314-350. MR 0943561
[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
[10] Zajíček L.: Small non-sigma-porous sets in topologically complete metric spaces. Colloq. Math. 77 (1998), 293-304. MR 1628994
[11] Zelený M.: The Banach-Mazur game and $\sigma$-porosity. Fund. Math. 150 (1996), 197-210. MR 1405042
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