Zoltan M. Balogh (University of Bern)
Stepanov's differentiability theorem in metric measure spaces
Abstract.
We extend Cheeger's theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov's condition of boundedness of the upper-scaled oscillation. As consequence, we obtain the analogue of Calderon's differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincare inequality. We show by examples that a Stepanov type differentiability theorem does not hold under the boundedness assumption of the lower scaled-oscillation only and give additional conditions in terms of the lower scaled-oscillation for a function to belong to the Sobolev or Lipschitz class.

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