IUMJ

Title: Measurable differentiable structures and the Poincare inequality

Authors: Stephen Keith

Issue: Volume 53 (2004), Issue 4, 1127-1150

Abstract:

The main result of this paper is an improvement for the differentiable structure presented in Cheeger \cite{CHEE:DIFF}*{Theorem 4.38} under the same assumptions of \cite{CHEE:DIFF} that the given metric measure space admits a Poincar'e inequality with a doubling measure. To be precise, it is shown in this paper that the coordinate functions of the differentiable structure can be taken to be distance functions. In the process, a representation is given in terms of approximate limits for the differential of functions contained in the Sobolev space $H_{1,p}$ defined in \cite{CHEE:DIFF}, and therefore $N^{1,p}$ defined by Shanmugalingam \cite{NAGE:THES}. Further application of these results includes identifying the minimal generalized upper gradient \cite{CHEE:DIFF}*{Definition 2.9} of an arbitrary Sobolev function, and an alternate proof to that of Franchi, Haj{\l}asz and Koskela \cite{PEKKA} and Semmes \cite{HEIN:SEMM}*{Theorem 5.1} for the closability of the differential operator defined on Lipschitz functions in $L^p$.