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