Rigidity of Derivations in the Plane and in Metric Measure Spaces

Following 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 k-dimensional Euclidean space 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. Though formulated for Euclidean spaces, these rigidity results also apply to the metric space setting and specifically, to spaces that support a doubling measure and a p-Poincar\'e 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.