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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