The Lip-lip equality is stable under blow-up

The Lip-lip condition on metric measure spaces

On complete metric spaces that support doubling measures, we show that the validity of a Rademacher theorem for Lipschitz functions can be characterised by Keith's "Lip-lip" condition. Roughly speaking, this means that at almost every point, the infinitesmal behavior of every Lipschitz function is essentially independent of the scales used in the blow-up at that point. Moreover, the doubling property can be further weakened to a local hypothesis on the measure; we also present results in this direction. Our techniques of proof are new and may be of independent interest. They include an explici…

Tangent lines and Lipschitz differentiability spaces

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space…

Lipschitz可微空间的可嵌入性与可修正性

Embeddability and rectifiability of Lipschitz differentiability spaces

本文通过可分解束和爆破分析，证明了双Lipschitz嵌入RNP空间的Lipschitz可微空间可数可修正，并给出新的刻画及解决Le Donne问题。