Metric and ${\omega }^{\ast }$-Differentiability of Pointwise Lipschitz Mappings

Abstract

We study the metric and $w^{\ast }$-differentiability of pointwise Lipschitz mappings. First, we prove several theorems about metric and $w^{\ast }$-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 $w^{\ast }$-differentiability of pointwise Lipschitz mappings $f:X\mapsto Y$ where $X,Y$ are Banach spaces with $X$ being separable (resp. $X$ separable and $Y=G^{\ast }$ with $G$ separable).