Illinois Journal of Mathematics

Rigidity of derivations in the plane and in metric measure spaces

Jasun Gong

Full-text: Open access

Abstract

Following the work of Weaver, we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for $k=1,2$ we show that measures on $\mathbb{R}^{k}$ that induce rank-$k$ modules of derivations must be absolutely continuous to Lebesgue measure. An analogous result holds true for measures concentrated on $k$-rectifiable sets with respect to $k$-dimensional Hausdorff measure.

These rigidity results also apply to the metric space setting and specifically, to spaces that support a doubling measure and a $p$-Poincaré inequality. Using our results for the Euclidean plane, we prove the $2$-dimensional case of a conjecture of Cheeger, which concerns the non-degeneracy of Lipschitz images of such spaces.

Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1109-1147.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395825

Digital Object Identifier
doi:10.1215/ijm/1399395825

Mathematical Reviews number (MathSciNet)
MR3231476

Zentralblatt MATH identifier
1295.46030

Subjects
Primary: 46G05: Derivatives [See also 46T20, 58C20, 58C25]
Secondary: 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56]

Citation

Gong, Jasun. Rigidity of derivations in the plane and in metric measure spaces. Illinois J. Math. 56 (2012), no. 4, 1109--1147. doi:10.1215/ijm/1399395825. https://projecteuclid.org/euclid.ijm/1399395825


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