Differentiability of Lipschitz functions with respect to singular measures

Giovanni Alberti
Università di Pisa

Rademacher theorem states that every Lipschitz function is differentiable almost everywhere with respect to the Lebesgue measure. In this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by a singular measure, in particular I will show that the differentiability properties of Lipschitz functions with respect to such a measure are exactly described by its decompositions in terms of one-dimensional rectifiable measures. This result is directly related to recent work by many authors, including myself, David Bate, Marianna Csornyei, Peter Jones, Andrea Marchese, and David Preiss.

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