A differentiable structure for metric measure spaces [microform]

The main result of this dissertation is the provision of conditions, weaker than those of Cheeger [Che99], under which a metric measure space admits a differentiable structure similar to that exhibited in [Che99]. Following the method in [Che99], the admitted differentiable structure is then used to construct a finite dimensional L(infinity) cotangent bundle over the given metric measure space. This leads to the definition of a Sobolev space H1,p over the given metric measure space, which is shown to be reflexive for p > 1. This extends the results of [Che99] to a wider collection of metric measure spaces. The new techniques developed are then applied to give an improvement for the differentiable structure presented in [Che99, Theorem 4.38].

This is achieved under the assumptions of [Che99] that the given metric measure space admits a Poincare inequality with a doubling measure. In the process, a representation is given in terms of approximate limits for the differential of functions contained in the Sobolev space H1,p defined in [Che99], and therefore N1,p defined by Shanmugalingam [Sha00]. Further applications of this representation include identifying the minimal generalized upper gradient [Che99, Definition 2.9] of an arbitrary Sobolev function, an alternate proof to that of Franchi, Haj asz and Koskela [FHK99] and Semmes [HK95b, Theorem 5.1] for the closability of the differential operator defined on Lipschitz functions in Lp, and the potentially vacuous extension of Rademacher's differentiability theorem to any doubling measure in Euclidean space.