Metric and $\omega^*$-Differentiability of Pointwise Lipschitz Mappings

Abstract

We study the metric and w^* -differentiability of pointwise Lipschitz mappings. First, we prove several theorems about metric and w^* -differentiability of pointwise Lipschitz mappings between \mathbb{R}^n and a Banach space X (which extend results due to Ambrosio, Kirchheim and others), then apply these to functions satisfying the spherical Rado–Reichelderfer condition, and to absolutely continuous functions of several variables with values in a Banach space. We also establish the area formula for pointwise Lipschitz functions, and for (n,\lambda) -absolutely continuous functions with values in Banach spaces. In the second part of this paper, we prove two theorems concerning metric and w^* -differentiability of pointwise Lipschitz mappings f:X\mapsto Y where X,Y are Banach spaces with X being separable (resp. X separable and Y=G^* with G separable).