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Very weak notions of differentiability

Published online by Cambridge University Press:  21 May 2007

Luigi Ambrosio
Affiliation:
Scuola Normale Superiore, 56100 Pisa, Italy (l.ambrosio@sns.it)
Jan Malý
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Prague 8, Czech Republic Department of Mathematics, Faculty of Science, J. E. Purkyně University, Ceské mládeze 8, 40096 Ústí nad Labem, Czech Republic (maly@karlin.mff.cuni.cz)

Abstract

In this paper we study some very weak notions of differentiability arising in connection with the spatial regularity of flows associated with non-smooth vector fields. The main difference from other similar concepts, also studied in a Sobolev setting, is that the convergence of difference quotients has to be understood as convergence in measure. We show in particular that the classical approximate differentiability is a property strictly stronger than approximate differentiability in measure.

Type
Research Article
Copyright
2007 Royal Society of Edinburgh

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