We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximation. This class of targets includes separable dual spaces. We also observe that there is a straightforward extension of Pansu’s differentiation theory for Lipschitz maps between Carnot groups, [Pan89], to the most general possible class of Banach space targets, those with the Radon-Nikodym property.
|Original language||English (US)|
|Title of host publication||Inspired By S S Chern|
|Subtitle of host publication||A Memorial Volume In Honor Of A Great Mathematician|
|Publisher||World Scientific Publishing Co.|
|Number of pages||24|
|State||Published - Jan 1 2006|
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