Differentiability and Poincare-type inequalities in metric measure spaces

作者：Bate David; Li Sean

来源：Advances in Mathematics, 2018, 333: 868-930.

DOI：10.1016/j.aim.2018.06.002

• Lipschitz
• Differentiation
• Radon-Nikodym property
• Poincare inequality
• Quasiconvexity

We demonstrate the necessity of a Poincare type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincare inequality. Using techniques similar to Cheeger Kleiner [12], we show that our conditions are also sufficient.
We also develop another characterization of RNP Lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in [5].

• 出版日期2018-7-31