Differentiability of Lipschitz functions with respect to singular measures and related questions
By Rademacher theorem every Lipschitz function on the Euclidean space is differentiable almost everywhere with respect to the Lebesgue measure. In these lectures I will explain how this statement should be modified when the Lebesgue measure is replaced by a singular (bounded) measure: it turns out that in this case the differentiability properties of Lipschitz functions are exactly described by the decompositions of the measure in terms of one-dimensional rectifiable measures. This leads to the problem of understanding the structure of measures that admits many of these decompositions; to this question, however, there are only partial answers.The results I will present are directly related to recent work by many authors, including myself, David Bate, Marianna Csornyei, Peter Jones, Andrea Marchese, and David Preiss.