The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger's theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov's condition. As a consequence, we obtain the analogue of Calderon's differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequalit

Local compactness and closedness for families of <math> <?Pub Eqn> <f> <sc>A</sc></f> </math>-harmonic functions.

We show that closed families of A -harmonic functions whose members all admit a common growth condition admits many topologies, all of which are generated by norms, so that for each of those topologies, tau, there is a set, Utau, that is open, dense and locally compact under tau. When the family is a vector space this implies that the family is finite dimensional.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132799/2/3058034.pd