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On stepanov Type differentiability Theorems

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Abstract

The main result shows that the Rademacher theorem proved by J. Lindenstrauss and D. Preiss [13] (which says that, for some pairs X, Y of Banach spaces, each Lipschitz f : X → Y is \({\Gamma}\)-a.e. Fréchet differentiable) generalizes the corresponding stepanov theorem (whichsays that, forsuch X and Y, an arbitrary f : X → Y is Fréchet differentiable at \({\Gamma}\) -almost all points at which f is Lipschitz). We also present an abstract approach which shows an easy way how (in some cases) a theorem of stepanov type (for vector functions) can be inferred from the corresponding theorem of Radamacher type. Finally we present Stepanov type differentiability theorems with the assumption of pointwise directional Lipschitzness.

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Authors and Affiliations

  1. Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75, Praha 8-Karlín, Czech Republic

    J. Malý & L. Zajíček

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  1. J. Malý
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  2. L. Zajíček
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Correspondence to L. Zajíček.

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The research was partiallysupported by the grant GAČR P201/12/0436.

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Malý, J., Zajíček, L. On stepanov Type differentiability Theorems. Acta Math. Hungar. 145, 174–190 (2015). https://doi.org/10.1007/s10474-014-0465-6

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  • DOI: https://doi.org/10.1007/s10474-014-0465-6

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