Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

Zajíček, Luděk

Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 203-213.

Résumé

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma$-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma$-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma$-almost everywhere Fréchet differentiable. In the proofs we use a general observation that each version of the Rademacher theorem for real functions on Banach spaces (i.e., a result on a.e. Fréchet or Gâteaux differentiability of Lipschitz functions) easily implies by a method of J. Malý a corresponding version of the Stepanov theorem (on a.e. differentiability of pointwise Lipschitz functions). Using the method of separable reduction, we extend some results to several non-separable spaces.

Détail

Classification : 46G05, 47H07, 49J50, 58C20
Keywords: cone-monotone function; Fréchet differentiability; Gâteaux differentiability; pointwise Lipschitz function; $\Gamma$-null set; quasiconvex function; separable reduction