Luděk Zajíček
Remarks on Fr\'echet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

Comment.Math.Univ.Carolin. 55,2 (2014) 203-213.

Abstract:We present some consequences of a deep result of J.~Lindenstrauss and D.~Preiss on $\Gamma$-almost everywhere Fr\'echet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fr\'echet differentiable at $\Gamma$-almost every $x$ at which it is G\^ateaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma$-almost everywhere Fr\'echet 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.\ Fr\'echet or G\^ateaux differentiability of Lipschitz functions) easily implies by a method of J.~Mal\'y 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.

Keywords: cone-monotone function; Fr\'echet differentiability; G\^ateaux differentiability; pointwise Lipschitz function; $\Gamma$-null set; quasiconvex function; separable reduction
AMS Subject Classification: 46G05 47H07