A differentiable structure for metric measure spaces

By S. Keith. 2004

153 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0b561bd30c74825f68a8b48ffbd0a24ee5ac5c93

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Local Currents in Metric Spaces (2011, background): “In this context we just mention the functions of bounded higher variation [8, 9, 21, 24, 27], the recent progress on (generalized) flat chains [1, 18, 36, 37], the scans from [10, 11, 16], the differentiation theory on metric measure spaces [4, 5, 22], and the derivations from [29, 30].”
Sobolev Mappings between Manifolds and Metric Spaces (2009, background): “Although the known examples show that spaces supporting a Poincaré inequality can be very exotic, surprisingly, one can prove that such spaces are always equipped with a weak differentiable structure [13, 53].”
Modulus and the Poincaré inequality on metric measure spaces (2003, background): “Some of this is covered in [Gro81,Gro99,Pet93,BS94,Pet98, CY98 ], and the rest is left to the reader.”
On a conjecture of Cheeger (2016, background): “This fact was later axiomatized by Keith [Kei04], leading to the notion of Lipschitz differentiability space, see Section 2 below.”
On quotients of spaces with Ricci curvature bounded below (2017, background): “A weaker condition, called the Lip-lip-condition was later discovered by Keith [40].”
On the differentiability of Lipschitz functions with respect to measures in the Euclidean space (2014, background): “In recent years much attention has been devoted to the differentiability of Lipschitz functions on infinite dimensional Banach spaces (see the monograph by J. Lindenstrauss, D. Preiss and J. Tǐser [16]) and on metric spaces (we just mention here the works by J. Cheeger [7], S. Keith [13] and D.…”
The Poincare inequality is an open ended condition (2008, background): “There are many important examples of such spaces (see [28], [26]), and many classical first-order theorems from Euclidean space remain true in this setting.”
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope (2012, background): “2), it is also worthwhile to mention a remarkable paper [23] by Keith, where (1.”
DIFFERENTIABLE STRUCTURES ON METRIC MEASURE SPACES: A PRIMER (2011, methodology): “Following [Kei04a], we approach this using a blow-up argument.”