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Zeitschrift für Analysis und ihre Anwendungen


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Volume 26, Issue 3, 2007, pp. 341–362
DOI: 10.4171/ZAA/1328

Published online: 2007-09-30

Metric and ω*-Differentiability of Pointwise Lipschitz Mappings

Jakub Duda[1]

(1) Weizmann Institute of Science, Rehovot, Israel

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 $\Rn$ 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).

Keywords: Lipschitz mappings, pointwise Lipschitz mappings, metric differentiability, ω*-differentiability, absolutely continuous functions of several variables, area formula, Radon–Nikodým property, Aronszajn null sets

Duda Jakub: Metric and ω*-Differentiability of Pointwise Lipschitz Mappings. Z. Anal. Anwend. 26 (2007), 341-362. doi: 10.4171/ZAA/1328