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MiS Preprint
78/2014

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

Giovanni Alberti and Andrea Marchese

Abstract

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure $\mu$.

In particular we show that the differentiability properties of Lipschitz functions at $\mu$-almost every point are related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures.

In the process we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.

Received:
Aug 11, 2014
Published:
Aug 15, 2014
MSC Codes:
26B05, 49Q15, 26A27, 28A75, 46E35
Keywords:
Lipschitz functions, differentiability, Rademacher theorem, normal currents

Related publications

inJournal
2016 Repository Open Access
Giovanni Alberti and Andrea Marchese

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

In: Geometric and functional analysis, 26 (2016) 1, pp. 1-66