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Winter 2012 Rigidity of derivations in the plane and in metric measure spaces
Jasun Gong
Illinois J. Math. 56(4): 1109-1147 (Winter 2012). DOI: 10.1215/ijm/1399395825


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.


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Jasun Gong. "Rigidity of derivations in the plane and in metric measure spaces." Illinois J. Math. 56 (4) 1109 - 1147, Winter 2012.


Published: Winter 2012
First available in Project Euclid: 6 May 2014

zbMATH: 1295.46030
MathSciNet: MR3231476
Digital Object Identifier: 10.1215/ijm/1399395825

Primary: 46G05
Secondary: 49J52

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign


Vol.56 • No. 4 • Winter 2012
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