### Abstract

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i. e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon-Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.

Original language | English (US) |
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Pages (from-to) | 1017-1028 |

Number of pages | 12 |

Journal | Geometric and Functional Analysis |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2009 |

### Keywords

- Banach space
- Differentiability
- Doubling measure
- Lipschitz function
- Metric measure space
- Minimal upper gradient
- Poincaré inequality
- Radon-Nikodym property

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology